ORCID Profile
0000-0002-5246-9391
Current Organisation
UNSW
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Research Topics
In Research Link Australia (RLA), "Research Topics" refer to ANZSRC FOR and SEO codes. These topics are either sourced from ANZSRC FOR and SEO codes listed in researchers' related grants or generated by a large language model (LLM) based on their publications.
Top 5 Research Topics
ANZSRC Field of Research (FoR)
Pure Mathematics | Analysis of Algorithms and Complexity | Number Theory And Field Theory | Algebra and Number Theory | Data Security | Analysis Of Algorithms And Complexity | Computation Theory and Mathematics | Combinatorics and Discrete Mathematics (excl. Physical Combinatorics) | Algebra and number theory | Pure mathematics | Other Artificial Intelligence | Applied Discrete Mathematics | Information Systems | Cryptography | Data Encryption | Numerical analysis | Experimental mathematics | Global Information Systems | Discrete Mathematics
ANZSRC Socio-Economic Objective (SEO)
Computer software and services not elsewhere classified | Mathematical sciences | Expanding Knowledge in the Information and Computing Sciences | Expanding Knowledge in the Mathematical Sciences | Information processing services | Communication services not elsewhere classified | Application Software Packages (excl. Computer Games) | Electronic Information Storage and Retrieval Services |
Related Links
Publications
Geometric progressions in vector sumsets over finite fields
Publisher: Elsevier BV
Date: 12-2020
On the additive energy of the distance set in finite fields
Publisher: Elsevier BV
Date: 11-2016
On the largest prime factor of n! + 2n − 1
Publisher: Cellule MathDoc/CEDRAM
Date: 2005
DOI: 10.5802/JTNB.524
Polynomial Interpolation and Identity Testing from High Powers Over Finite Fields
Publisher: Springer Science and Business Media LLC
Date: 05-01-2017
Finding the group structure of elliptic curves over finite fields
Publisher: Cambridge University Press (CUP)
Date: 10-2005
DOI: 10.1017/S0004972700035048
Abstract: We show that an algorithm of V. Miller to compute the group structure of an elliptic curve over a prime finite field runs in probabilistic polynomial time for almost all curves over the field. Important to our proof are estimates for some isor sums.
On the number of distances between the coordinates of points on modular hyperbolas
Publisher: Elsevier BV
Date: 05-2008
ON GAPS BETWEEN PRIMITIVE ROOTS IN THE HAMMING METRIC
Publisher: Oxford University Press (OUP)
Date: 16-08-2012
DOI: 10.1093/QMATH/HAS022
Character sums over shifted primes
Publisher: Pleiades Publishing Ltd
Date: 10-2010
Points on curves in small boxes and applications
Publisher: Michigan Mathematical Journal
Date: 09-2014
On digit patterns in expansions of rational numbers with prime denominator
Publisher: Oxford University Press (OUP)
Date: 11-10-2013
DOI: 10.1093/QMATH/HAS027
Linear complexity of the discrete logarithm
Publisher: Springer Science and Business Media LLC
Date: 2003
Distribution of matrices with restricted entries over finite fields
Publisher: Elsevier BV
Date: 09-2007
Role of Graphitic Bowls in Temperature Dependent Fullerene Formation
Publisher: American Chemical Society (ACS)
Date: 30-11-2022
Abstract: Fullerenes are used extensively in organic electronics as electron acceptors among other uses however, there are still several key mysteries regarding their formation such as the importance of graphitic intermediates and the thermokinetics of initial cage formation. To this end, we have conducted density functional tight binding molecular dynamics (DFTB-MD) calculations on disintegrated
Close values of shifted modular inversions and the decisional modular inversion hidden number problem
Publisher: American Institute of Mathematical Sciences (AIMS)
Date: 2015
Dynamical Systems Generated by Rational Functions
Publisher: Springer Berlin Heidelberg
Date: 2003
On the discrepancy and linear complexity of some counter-dependent recurrence sequences
Publisher: Springer Berlin Heidelberg
Date: 2006
DOI: 10.1007/11863854_25
On the convex hull of the points on multivariate modular hyperbolas
Publisher: Elsevier BV
Date: 02-2017
Disjointness of the Möbius Transformation and Möbius Function
Publisher: Springer Science and Business Media LLC
Date: 07-02-2019
Piatetski-Shapiro sequences
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2013
DOI: 10.4064/AA157-1-3
On the Linear Complexity of the Power Generator
Publisher: Springer Science and Business Media LLC
Date: 2001
On a question of Erdös and Graham
Publisher: Springer Science and Business Media LLC
Date: 06-2002
Smooth shifted monomial products
Publisher: University of Debrecen/ Debreceni Egyetem
Date: 12-2011
Sums with the Möbius function twisted by characters with powerful moduli
Publisher: American Mathematical Society (AMS)
Date: 23-09-2020
DOI: 10.1090/TRAN/7914
Abstract: In a recent work, the authors (2016) have combined classical ideas of A. G. Postnikov (1956) and N. M. Korobov (1974) to derive improved bounds on short character sums for certain nonprincipal characters with powerful moduli. In the present paper, these results are used to bound sums with the Möbius function twisted by characters of the same type, which complements and improves some earlier work of B. Green (2012). To achieve this, we obtain a series of results about the size and zero-free region of L L -functions with the same class of moduli.
CREW PRAM complexity of modular inversion
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2000
Energy bounds, bilinear forms and their applications in function fields
Publisher: Elsevier BV
Date: 09-2022
Large Weyl sums and Hausdorff dimension
Publisher: Elsevier BV
Date: 06-2022
On the distribution of values and zeros of polynomial systems over arbitrary sets
Publisher: Elsevier BV
Date: 09-2013
On the construction of solutions of systems of linear ordinary differential equations in the neighbourhood of a regular singularity
Publisher: Elsevier BV
Date: 03-1992
Elements of large order on varieties over prime finite fields
Publisher: Cellule MathDoc/CEDRAM
Date: 2014
DOI: 10.5802/JTNB.880
Incomplete character sums and a special class of permutations
Publisher: Cellule MathDoc/CEDRAM
Date: 2001
DOI: 10.5802/JTNB.303
Codes correcting restricted errors
Publisher: Springer Science and Business Media LLC
Date: 24-11-2018
Periodic Sequences with Maximal Linear Complexity and Almost Maximal k-Error Linear Complexity
Publisher: Springer Berlin Heidelberg
Date: 2003
On one property of a multiplicative transducer of pseudo-random numbers
Publisher: Elsevier BV
Date: 1983
Random walks and bisections in random circulant graphs
Publisher: Springer Berlin Heidelberg
Date: 2012
Pseudoprime reductions of elliptic curves
Publisher: Cambridge University Press (CUP)
Date: 14-07-2008
Bounds of Gauss sums in finite fields
Publisher: American Mathematical Society (AMS)
Date: 02-06-2004
DOI: 10.1090/S0002-9939-04-07133-3
Abstract: We consider Gauss sums of the form \[ G n ( a ) = ∑ x ∈ F p m χ ( x n ) G_n(a) = \sum _{x \in \mathbb {F}_{p^m}} \chi (x^n) \] with a nontrivial additive character χ ≠ χ 0 \chi \ne \chi _0 of a finite field F p m \mathbb {F}_{p^m} of p m p^m elements of characteristic p p . The classical bound | G n ( a ) | ≤ ( n − 1 ) p m / 2 |G_n(a)| \le (n-1) p^{m/2} becomes trivial for n ≥ p m / 2 + 1 n \ge p^{m/2} + 1 . We show that, combining some recent bounds of Heath-Brown and Konyagin with several bounds due to Deligne, Katz, and Li, one can obtain the bound on | G n ( a ) | |G_n(a)| which is nontrivial for the values of n n of order up to p m / 2 + 1 / 6 p^{m/2 + 1/6} . We also show that for almost all primes one can obtain a bound which is nontrivial for the values of n n of order up to p m / 2 + 1 / 2 p^{m/2 + 1/2} .
Pseudorandom numbers and hash functions from iterations of multivariate polynomials
Publisher: Springer Science and Business Media LLC
Date: 09-01-2010
An estimate of incomplete multiplicative character sum of polynomials
Publisher: Walter de Gruyter GmbH
Date: 1992
Predicting the Inversive Generator
Publisher: Springer Berlin Heidelberg
Date: 2003
Power series approximations to Fekete polynomials
Publisher: Elsevier BV
Date: 10-2017
On the divisibility of Fermat quotients
Publisher: Michigan Mathematical Journal
Date: 08-2010
Prime divisors of sequences associated to elliptic curves
Publisher: Cambridge University Press (CUP)
Date: 02-2005
Small exponent point groups on elliptic curves
Publisher: Cellule MathDoc/CEDRAM
Date: 2006
DOI: 10.5802/JTNB.554
Squarefree parts of discriminants of trinomials
Publisher: Springer Science and Business Media LLC
Date: 06-2014
On the average value of divisor sums in arithmetic progressions
Publisher: Oxford University Press (OUP)
Date: 2005
DOI: 10.1155/IMRN.2005.1
On the Security o Lenstra’ s Variant o DSA without Long Inversions
Publisher: Springer Berlin Heidelberg
Date: 2001
On the linear complexity of the naor—reingold pseudo-random function
Publisher: Springer Berlin Heidelberg
Date: 1999
On Constructing Primitive Roots in Finite Fields With Advice
Publisher: Institute of Electrical and Electronics Engineers (IEEE)
Date: 11-2018
On the number of zeros of exponential polynomials and related questions
Publisher: Cambridge University Press (CUP)
Date: 12-1992
DOI: 10.1017/S0004972700012065
Abstract: We apply Straßmann's theorem to p –adic power series satisfying linear differential equations with polynomial coefficients and note that our approach leads to our estimating the number of integer zeros of polynomials on a given interval and thence to an investigation of the number of p –adic small values of a function on such an interval, that is, of the number of solutions of a congruence modulo p r .
Distribution of consecutive modular roots of an integer
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2008
DOI: 10.4064/AA134-1-6
On the singularity of the demjanenko matrix of quotients of fermat curves
Publisher: American Mathematical Society (AMS)
Date: 07-2016
DOI: 10.1090/PROC12717
On bivariate polynomial factorization over finite fields
Publisher: American Mathematical Society (AMS)
Date: 1993
DOI: 10.1090/S0025-5718-1993-1176716-3
Abstract: This paper shows that a recently proposed approach of D. Q. Wan to bivariate factorization over finite fields, the univariate factoring algorithm of V. Shoup, and the new bound of this paper for the average number of irreducible isors of polynomials of a given degree over a finite field can be used to design a bivariate factoring algorithm that is polynomial for "almost all" bivariate polynomials.
Smooth Orders and Cryptographic Applications
Publisher: Springer Berlin Heidelberg
Date: 2002
Finding points on curves over finite fields
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2003
Distribution of elliptic twin primes in isogeny and isomorphism classes
Publisher: Elsevier BV
Date: 04-2014
Chinese Remaindering for Algebraic Numbers in a Hidden Field
Publisher: Springer Berlin Heidelberg
Date: 2002
On the maximal difference between an element and its inverse in residue rings
Publisher: American Mathematical Society (AMS)
Date: 08-06-2005
DOI: 10.1090/S0002-9939-05-07962-1
Abstract: We investigate the distribution of n − M ( n ) n - M(n) where \[ M ( n ) = max { | a − b | : 1 ≤ a , b ≤ n − 1 \ and\ a b ≡ 1 ( mod n ) } . M(n)=\max \left \{ \, \left | a-b\right |\ :\ 1 \leq a,b\leq n-1 \textrm {\ and\ } ab \equiv 1\pmod n\right \}. \] Exponential sums provide a natural tool for obtaining upper bounds on this quantity. Here we use results about the distribution of integers with a isor in a given interval to obtain lower bounds on n − M ( n ) n - M(n) . We also present some heuristic arguments showing that these lower bounds are probably tight, and thus our technique can be a more appropriate tool to study n − M ( n ) n - M(n) than a more traditional way using exponential sums.
Hidden number problem with hidden multipliers, timed-release crypto, and noisy exponentiation
Publisher: American Mathematical Society (AMS)
Date: 03-02-2003
Character sums and nonlinear recurrence sequences
Publisher: Elsevier BV
Date: 06-2006
COUNTING INTEGERS with A SMOOTH TOTIENT
Publisher: Oxford University Press (OUP)
Date: 10-09-2019
Abstract: In an earlier paper we considered the distribution of integers $n$ for which Euler’s totient function at $n$ has all small prime factors. Here we obtain an improvement that is likely to be best possible.
Bounds of multiplicative character sums with Fermat quotients of primes
Publisher: Cambridge University Press (CUP)
Date: 07-02-2011
DOI: 10.1017/S000497271000198X
Abstract: Given a prime p , the Fermat quotient q p ( u ) of u with gcd ( u , p )=1 is defined by the conditions We derive a new bound on multiplicative character sums with Fermat quotients q p ( ℓ ) at prime arguments ℓ .
Elliptic Curves over Finite Fields: Number Theoretic and Cryptographic Aspects
Publisher: Springer US
Date: 2013
On the elliptic curve analogue of the sum-product problem
Publisher: Elsevier BV
Date: 07-2008
A Variant of NTRU with Non-invertible Polynomials
Publisher: Springer Berlin Heidelberg
Date: 2002
The uniform distribution of fractional parts of recurrent sequences
Publisher: Steklov Mathematical Institute
Date: 30-06-1979
On some approximation problems concerning sparse polynomials over finite fields
Publisher: Elsevier BV
Date: 05-1996
Mitochondrial DNA mutations associated with aminoglycoside ototoxicity
Publisher: Informa UK Limited
Date: 2006
The distribution of primitive roots in finite fields
Publisher: Steklov Mathematical Institute
Date: 28-02-1990
On the energy of some circulant graphs
Publisher: Elsevier BV
Date: 04-2006
Algebraic entropy, automorphisms and sparsity of algebraic dynamical systems and pseudorandom number generators
Publisher: American Mathematical Society (AMS)
Date: 30-09-2013
DOI: 10.1090/S0025-5718-2013-02780-9
Abstract: We present several general results that show how algebraic dynamical systems with a slow degree growth and also rational automorphisms can be used to construct stronger pseudorandom number generators. We then give several concrete constructions that illustrate the applicability of these general results.
Statistics of different reduction types of fermat curves
Publisher: Informa UK Limited
Date: 03-07-2013
New estimates for exponential sums over multiplicative subgroups and intervals in prime fields
Publisher: Elsevier BV
Date: 10-2020
On the restricted divisor function in arithmetic progressions
Publisher: European Mathematical Society - EMS - Publishing House GmbH
Date: 2012
DOI: 10.4171/RMI/675
Collisions in fast generation of ideal classes and points on hyperelliptic and elliptic curves
Publisher: Springer Science and Business Media LLC
Date: 12-11-2005
On values taken by the largest prime factor of shifted primes
Publisher: Cambridge University Press (CUP)
Date: 02-2007
DOI: 10.1017/S1446788700017511
Abstract: Let P denote the set of prime numbers, and let P ( n ) denote the largest prime factor of an integer n 1. We show that, for every real number , there exists a constant c (η) 1 such that for every integer a ≠ 0, the set has relative asymptotic density one in the set of all prime numbers. Moreover, in the range , one can take c (η) = 1+ε for any fixed ε 0. In particular, our results imply that for every real number 0.486 ≤ b.thetav ≤ 0.531, the relation P ( q − a ) ≍ q θ holds for infinitely many primes q . We use this result to derive a lower bound on the number of distinct prime isor of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map q ↦ P ( q - a ) for a 0, and show that for infinitely many primes q , this map can be iterated at least (log log q ) 1+o(1) times before it terminates.
On the Distribution of Pseudopowers
Publisher: Canadian Mathematical Society
Date: 06-2010
Abstract: An x -pseudopower to base g is a positive integer that is not a power of g , yet is so modulo p for all primes p ≤ x . We improve an upper bound for the least such number, due to E. Bach, R. Lukes, J. Shallit, and H. C. Williams. The method is based on a combination of some bounds of exponential sums with new results about the average behaviour of the multiplicative order of g modulo prime numbers.
Arithmetic properties of Apéry numbers
Publisher: Wiley
Date: 08-07-2008
DOI: 10.1112/JLMS/JDN031
A SPARSITY RESULT FOR THE DYNAMICAL MORDELL–LANG CONJECTURE IN POSITIVE CHARACTERISTIC
Publisher: Cambridge University Press (CUP)
Date: 23-02-2021
DOI: 10.1017/S0004972721000083
Abstract: We prove a quantitative partial result in support of the dynamical Mordell–Lang conjecture (also known as the DML conjecture ) in positive characteristic. More precisely, we show the following: given a field K of characteristic p , a semiabelian variety X defined over a finite subfield of K and endowed with a regular self-map $\\Phi :X{\\longrightarrow } X$ defined over K , a point $\\alpha \\in X(K)$ and a subvariety $V\\subseteq X$ , then the set of all nonnegative integers n such that $\\Phi ^n(\\alpha )\\in V(K)$ is a union of finitely many arithmetic progressions along with a subset S with the property that there exists a positive real number A (depending only on X , $\\Phi $ , $\\alpha $ and V ) such that for each positive integer M , $$\\begin{align*}\\scriptsize\\#\\{n\\in S\\colon n\\le M\\}\\le A\\cdot (1+\\log M)^{\\dim V}.\\end{align*}$$
Irrationality of Power Series for Various Number Theoretic Functions
Publisher: Springer Science and Business Media LLC
Date: 23-05-2005
On routing in circulant graphs
Publisher: Springer Berlin Heidelberg
Date: 1999
Average order in cyclic groups
Publisher: Cellule MathDoc/CEDRAM
Date: 2004
DOI: 10.5802/JTNB.436
On the number of Eisenstein polynomials of bounded height
Publisher: Springer Science and Business Media LLC
Date: 17-04-2013
Prime divisors of shifted factorials
Publisher: Wiley
Date: 12-2005
On the nonlinearity of the sequence of signs of Kloosterman sums
Publisher: Cambridge University Press (CUP)
Date: 06-2005
DOI: 10.1017/S0004972700038405
Abstract: It is known that Kloosterman sums with prime denominator p take real values, so one can define a sequence of signs of such sums. Several pseudorandom properties of this sequence have recently been studied by Fouvry, Michel, Rivat and Sárközy. Here we use one of their results to estimate a certain important characteristic of this sequence which is also of cryptographic interest.
On the concentration of points on modular hyperbolas and exponential curves
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2010
DOI: 10.4064/AA142-1-5
Bounds on the Fourier coefficients of the weighted sum function
Publisher: Elsevier BV
Date: 07-2007
Density of non-residues in Burgess-type intervals and applications
Publisher: Wiley
Date: 02-2008
DOI: 10.1112/BLMS/BDM111
On the average sensitivity of testing square-free numbers
Publisher: Springer Berlin Heidelberg
Date: 1999
Quantum period reconstruction of approximate sequences
Publisher: Elsevier BV
Date: 09-2007
On the convex hull of solutions to polynomial congruences
Publisher: Elsevier BV
Date: 2012
A hidden number problem in small subgroups
Publisher: American Mathematical Society (AMS)
Date: 05-04-2005
Counting solvable $S$-unit equations
Publisher: American Mathematical Society (AMS)
Date: 24-09-2021
DOI: 10.1090/PROC/15674
Abstract: We obtain upper bounds on the number of finite sets S \\mathcal {S} of primes below a given bound for which various 2 2 variable S \\mathcal {S} -unit equations have a solution.
On the distribution of irreducible trinomials
Publisher: Canadian Mathematical Society
Date: 12-2011
Abstract: We obtain new results about the number of trinomials t n + at + b with integer coefficients in a box ( a , b ) ∈ [ C , C + A ] × [ D , D + B ] that are irreducible modulo a prime p . As a by-product we show that for any p there are irreducible polynomials of height at most p 1/2+ o (1) , improving on the previous estimate of p 2/3+ o (1) obtained by the author in 1989.
Number of prime divisors of recurrence sequences
Publisher: Springer Science and Business Media LLC
Date: 07-1985
DOI: 10.1007/BF01137461
Rank Statistics for a Family of Elliptic Curves over a Function Field
Publisher: International Press of Boston
Date: 2010
On the Convex Closure of the Graph of Modular Inversions
Publisher: Informa UK Limited
Date: 2008
On some exponential sums with exponential and rational functions
Publisher: Rocky Mountain Mathematics Consortium
Date: 02-2013
Ratios of Small Integers in Multiplicative Subgroups of Residue Rings
Publisher: Informa UK Limited
Date: 29-02-2016
Chowla and Sarnak conjectures for Kloosterman sums
Publisher: Wiley
Date: 16-08-2023
Abstract: We formulate several analogs of the Chowla and Sarnak conjectures, which are widely known in the setting of the Möbius function, in the setting of Kloosterman sums. We then show that for Kloosterman sums, in some cases, these conjectures can be established unconditionally.
On character sums with distances on the upper half plane over a finite field
Publisher: Elsevier BV
Date: 12-2009
On the statistical properties of Diffie-Hellman distributions
Publisher: Springer Science and Business Media LLC
Date: 12-2000
On Group Structures Realized by Elliptic Curves over Arbitrary Finite Fields
Publisher: Informa UK Limited
Date: 03-2012
On the construction of a primitive normal basis in a finite field
Publisher: IOP Publishing
Date: 28-02-1990
Bilinear character sums and sum-product problems on elliptic curves
Publisher: Cambridge University Press (CUP)
Date: 12-01-2010
DOI: 10.1017/S0013091508000771
Abstract: Let E be an ordinary elliptic curve over a finite field q of q elements. We improve a bound on bilinear additive character sums over points on E , and obtain its analogue for bilinear multiplicative character sums. We apply these bounds to some variants of the sum-product problem on E .
Some divisibilities amongst the terms of linear recurrences
Publisher: Springer Science and Business Media LLC
Date: 12-2006
DOI: 10.1007/BF02960862
On the exponential large sieve inequality for sparse sequences modulo primes
Publisher: Elsevier BV
Date: 03-2018
On pseudorandom numbers from multivariate polynomial systems
Publisher: Elsevier BV
Date: 09-2010
Catalan and Apéry numbers in residue classes
Publisher: Elsevier BV
Date: 07-2006
Values of arithmetical functions equal to a sum of two squares
Publisher: Oxford University Press (OUP)
Date: 06-2005
DOI: 10.1093/QMATH/HAH039
Number of different prime divisors of recurrence sequences
Publisher: Springer Science and Business Media LLC
Date: 10-1987
DOI: 10.1007/BF01138309
Movement through slits: Cellular migration via the Slit family
Publisher: Wiley
Date: 20-12-2002
DOI: 10.1002/BIES.10199
Abstract: First isolated in the fly and now characterised in vertebrates, the Slit proteins have emerged as pivotal components controlling the guidance of axonal growth cones and the directional migration of neuronal precursors. As well as extensive expression during development of the central nervous system (CNS), the Slit proteins exhibit a striking array of expression sites in non-neuronal tissues, including the urogenital system, limb primordia and developing eye. Zebrafish Slit has been shown to mediate mesodermal migration during gastrulation, while Drosophila slit guides the migration of mesodermal cells during myogenesis. This suggests that the actions of these secreted molecules are not simply confined to the sphere of CNS development, but rather act in a more general fashion during development and throughout the lifetime of an organism. This review focuses on the non-neuronal activities of Slit proteins, highlighting a common role for the Slit family in cellular migration.
Least totient in a residue class
Publisher: Wiley
Date: 04-05-2007
DOI: 10.1112/BLMS/BDM027
Elliptic curves in isogeny classes
Publisher: Elsevier BV
Date: 10-2018
Some counting questions for matrices with restricted entries
Publisher: Elsevier BV
Date: 2010
On the values of Kloosterman sums
Publisher: Institute of Electrical and Electronics Engineers (IEEE)
Date: 06-2009
A Nonuniform Algorithm for the Hidden Number Problem in Subgroups
Publisher: Springer Berlin Heidelberg
Date: 2004
Classical and quantum function reconstruction via character evaluation
Publisher: No publisher found
Date: 2004
New improved algorithm for the iterative solution of a system of linear algebraic equations
Publisher: Elsevier BV
Date: 07-1982
On Gaussian sums for finite fields and elliptic curves
Publisher: Springer-Verlag
Date: 1992
DOI: 10.1007/BFB0034335
Squares in Piatetski-Shapiro sequences
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2017
Distribution of values of polynomial Fermat quotients
Publisher: Elsevier BV
Date: 2013
Double exponential sums with exponential functions
Publisher: World Scientific Pub Co Pte Lt
Date: 16-10-2017
DOI: 10.1142/S179304211750141X
Abstract: We obtain several estimates for double rational exponential sums modulo a prime [Formula: see text] with products [Formula: see text] where both [Formula: see text] and [Formula: see text] run through short intervals and [Formula: see text] is fixed integer. We also obtain some new estimates for the number of points on exponential modular curves [Formula: see text] and similar.
On some generalisations of the Erdos distance problem over finite fields
Publisher: Cambridge University Press (CUP)
Date: 04-2006
DOI: 10.1017/S0004972700038867
Abstract: We use exponential sums to obtain new lower bounds on the number of distinct distances defined by all pairs of points ( a, b ) ∈ A × B for two given sets where is a finite field of q elements and n ≥ 1 is an integer.
Estimates for Wieferich numbers
Publisher: Springer Science and Business Media LLC
Date: 18-09-2007
On the uniformity of distribution of the RSA pairs
Publisher: American Mathematical Society (AMS)
Date: 12-06-2001
The Insecurity of the Elliptic Curve Digital Signature Algorithm with Partially Known Nonces
Publisher: Springer Science and Business Media LLC
Date: 2003
Counting curves and their projections
Publisher: ACM Press
Date: 1993
Generating safe primes
Publisher: Walter de Gruyter GmbH
Date: 2013
Congruences with intervals and subgroups modulo a prime
Publisher: Michigan Mathematical Journal
Date: 09-2015
Fixed points of the subset sum pseudorandom number generators
Publisher: Springer Science and Business Media LLC
Date: 23-03-2023
DOI: 10.1007/S10623-023-01209-5
Abstract: We give upper bounds on the power moments of the number of fixed points of a family of subset sum pseudorandom number generators, introduced by Rueppel (Analysis and design of stream ciphers, Springer-Verlag, Berlin, 1986).
Noisy Chinese remaindering in the Lee norm
Publisher: Elsevier BV
Date: 04-2004
Distribution of elements of cosets of small subgroups and applications
Publisher: Oxford University Press (OUP)
Date: 27-05-2012
DOI: 10.1093/IMRN/RNR097
Cryptographic Applications of Sparse Polynomials over Finite Rings
Publisher: Springer Berlin Heidelberg
Date: 2001
Partitions into two Lehmer numbers
Publisher: Springer Science and Business Media LLC
Date: 09-06-2009
Multiplicative character sums with the sum of g-ary Digits Function
Publisher: Springer Science and Business Media LLC
Date: 04-2006
Smooth values of shifted primes in arithmetic progressions
Publisher: Michigan Mathematical Journal
Date: 12-2004
On the length of critical orbits of stable quadratic polynomials
Publisher: American Mathematical Society (AMS)
Date: 08-2010
On the distribution of arguments of Gauss sums
Publisher: Tokyo Institute of Technology, Department of Mathematics
Date: 03-2009
Circuit complexity of testing square-free numbers
Publisher: Springer Berlin Heidelberg
Date: 1999
Arithmetic functions with linear recurrence sequences
Publisher: Elsevier BV
Date: 08-2007
Non-residues and primitive roots in beatty sequences
Publisher: Cambridge University Press (CUP)
Date: 06-2006
DOI: 10.1017/S0004972700035449
Abstract: We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence where α,β ∈ ℝ, and α is irrational. In particular, our bounds imply that for every fixed ε 0, if p is sufficiently large and p ½+ε ≤ N ≤ p , then among the first N elements of ℬ α,β , there are N /2+ o ( N ) quadratic non-residues modulo p . When more information is available about the Diophantine properties of α, then the error term o ( N ) admits a sharper estimate.
On linear recurrence sequences with polynomial coefficients
Publisher: Cambridge University Press (CUP)
Date: 05-1996
DOI: 10.1017/S0017089500031372
Abstract: We consider sequences (A h ) defined over the field ℚ of rational numbers and satisfying a linear homogeneous recurrence relation with polynomial coefficients s j . We shall assume without loss of generality, as we may, that the s j , are defined over ℤ and the initial values A 0 A ] …, A n − 1 are integer numbers. Also, without loss of generality we may assume that S 0 and S n have no non-negative integer zero. Indeed, any other case can be reduced to this one by making a shift h → h – l – 1 where l is an upper bound for zeros of the corresponding polynomials (and which can be effectively estimated in terms of their heights)
Components and projections of curves over finite fields
Publisher: Springer Berlin Heidelberg
Date: 1994
Bounds of incomplete multiple Kloosterman sums
Publisher: Elsevier BV
Date: 09-2007
On the lower bound of the linear complexity over F/sub p/ of Sidelnikov sequences
Publisher: Institute of Electrical and Electronics Engineers (IEEE)
Date: 07-2006
On the distribution of rational functions along a curve over
Publisher:
Elsevier BV
Date:
06-2005
Publisher: Elsevier BV
Date: 06-2005
Polynomial Gauss sums
Publisher: American Mathematical Society (AMS)
Date: 17-03-2005
Arithmetic Properties of Integers in Chains and Reflections of g-ary Expansions
Publisher: Informa UK Limited
Date: 31-10-2018
On the Győry-Sárközy-Stewart conjecture in function fields
Publisher: Institute of Mathematics, Czech Academy of Sciences
Date: 20-04-2018
On a completely uniform distribution
Publisher: Elsevier BV
Date: 1979
ON POINT SETS IN VECTOR SPACES OVER FINITE FIELDS THAT DETERMINE ONLY ACUTE ANGLE TRIANGLES
Publisher: Cambridge University Press (CUP)
Date: 21-10-2009
DOI: 10.1017/S0004972709000719
Abstract: For three points $\\vec {u}$ , $\\vec {v}$ and $\\vec {w}$ in the n -dimensional space 𝔽 n q over the finite field 𝔽 q of q elements we give a natural interpretation of an acute angle triangle defined by these points. We obtain an upper bound on the size of a set 𝒵 such that all triples of distinct points $\\vec {u}, \\vec {v}, \\vec {w} \\in \\cZ $ define acute angle triangles. A similar question in the real space ℛ n dates back to P. Erdős and has been studied by several authors.
Quadratic non-residues in short intervals
Publisher: American Mathematical Society (AMS)
Date: 31-03-2015
Approximate Polynomial gcd: Small Degree and Small Height Perturbations
Publisher: Springer Berlin Heidelberg
Date: 2008
Pseudoprime values of the Fibonacci sequence, polynomials and the Euler function
Publisher: Elsevier BV
Date: 12-2006
Theory of sigma bond resonance in flat boron materials
Publisher: Springer Science and Business Media LLC
Date: 31-03-2023
DOI: 10.1038/S41467-023-37442-8
Abstract: In chemistry, theory of aromaticity or π bond resonance plays a central role in intuitively understanding the stability and properties of organic molecules. Here we present an analogue theory for σ bond resonance in flat boron materials, which allows us to determine the distribution of two-center two-electron and three-center two-electron bonds without quantum calculations. Based on this theory, three rules are proposed to draw the Kekulé-like bonding configurations for flat boron materials and to explore their properties intuitively. As an application of the theory, a simple explanation of why neutral borophene with ~1/9 hole has the highest stability and the effect of charge doping on borophene’s optimal hole concentration is provided with the assumption of σ and π orbital occupation balance. Like the aromaticity theory for carbon materials, this theory greatly deepens our understanding on boron materials and paves the way for the rational design of various boron-based materials.
Short cycles in repeated exponentiation modulo a prime
Publisher: Springer Science and Business Media LLC
Date: 31-10-2009
Products in residue classes
Publisher: International Press of Boston
Date: 2008
EXPONENTIAL AND CHARACTER SUMS WITH MERSENNE NUMBERS
Publisher: Cambridge University Press (CUP)
Date: 02-2012
DOI: 10.1017/S1446788712000109
Abstract: We give new bounds on sums of the form ∑ n ≤ N Λ( n )exp (2 πiag n / m ) and ∑ n ≤ N Λ( n ) χ ( g n + a ), where Λ is the von Mangoldt function, m is a natural number, a and g are integers coprime to m , and χ is a multiplicative character modulo m . In particular, our results yield bounds on the sums ∑ p ≤ N exp (2 πiaM p / m ) and ∑ p ≤ N χ ( M p ) with Mersenne numbers M p =2 p −1, where p is prime.
Erratum: Period of the power generator and small values of Carmichael's function (Mathematics of Computation)
Publisher: American Mathematical Society (AMS)
Date: 28-05-2002
On RSA moduli with almost half of the bits prescribed
Publisher: Elsevier BV
Date: 09-2008
POLYNOMIAL VALUES IN SUBFIELDS AND AFFINE SUBSPACES OF FINITE FIELDS
Publisher: Oxford University Press (OUP)
Date: 08-12-2014
DOI: 10.1093/QMATH/HAU032
On the distribution of inversive congruential pseudorandom numbers in parts of the period
Publisher: American Mathematical Society (AMS)
Date: 12-06-2001
DOI: 10.1090/S0025-5718-00-01273-4
Abstract: The inversive congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present the first nontrivial bounds on the discrepancy of in idual sequences of inversive congruential pseudorandom numbers in parts of the period. The proof is based on a new bound for certain incomplete exponential sums.
On the fixed points of the map x→xx modulo a prime
Publisher: International Press of Boston
Date: 2015
On the size of the Gelfond exponent
Publisher: Elsevier BV
Date: 04-2010
The Insecurity of the Digital Signature Algorithm with Partially Known Nonces
Publisher: Springer Science and Business Media LLC
Date: 06-2002
Graphs with integral spectrum
Publisher: Elsevier BV
Date: 2009
Exponential sums and prime divisors of sparse integers
Publisher: Springer Science and Business Media LLC
Date: 09-2008
On the Uniformity of Distribution of the ElGamal Signature
Publisher: Springer Science and Business Media LLC
Date: 04-2002
On the asymptotic effectiveness of Weil descent attacks
Publisher: Walter de Gruyter GmbH
Date: 2010
DOI: 10.1515/JMC.2010.007
Pseudorandom graphs from elliptic curves
Publisher: Springer Berlin Heidelberg
Date: 2008
Identity testing and interpolation from high powers of polynomials of large degree over finite fields
Publisher: Elsevier BV
Date: 12-2018
Circuit and Decision Tree Complexity of Some Number Theoretic Problems
Publisher: Elsevier BV
Date: 08-2001
Sums of algebraic trace functions twisted by arithmetic functions
Publisher: Mathematical Sciences Publishers
Date: 12-02-2020
Communications of the moscow mathematical society: On the note of convergence of Newton's interpolation process and the power of certain codes
Publisher: Steklov Mathematical Institute
Date: 30-04-1984
Distribution of modular inverses and multiples of small integers and the Sato-Tate conjecture on average
Publisher: Michigan Mathematical Journal
Date: 06-2008
Multiplicative character sums and products of sparse integers in residue classes
Publisher: Springer Science and Business Media LLC
Date: 24-05-2012
On the distribution of kloosterman sums
Publisher: American Mathematical Society (AMS)
Date: 02-11-2008
Character sums and deterministic polynomial root finding in finite fields
Publisher: American Mathematical Society (AMS)
Date: 10-04-2015
DOI: 10.1090/MCOM/2946
Almost primes of the form ⌊ p
Publisher:
Elsevier BV
Date:
03-2016
Publisher: Elsevier BV
Date: 03-2016
Cancellations amongst Kloosterman sums
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2016
Multiplicative congruences with variables from short intervals
Publisher: Springer Science and Business Media LLC
Date: 10-2014
Bilinear forms with exponential sums with binomials
Publisher: Elsevier BV
Date: 07-2018
On a ternary quadratic form over primes
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2011
DOI: 10.4064/AA150-3-5
Short Kloosterman Sums for Polynomials over Finite Fields
Publisher: Canadian Mathematical Society
Date: 04-2003
Abstract: We extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring [ x ]/ M ( x ) for collections of polynomials either of the form f −1 g −1 or of the form f −1 g −1 + afg , where f and g are polynomials coprime to M and of very small degree relative to M , and a is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields.
On Some Properties of the Shrinking Generator
Publisher: Springer Science and Business Media LLC
Date: 2001
Prime divisors of some shifted products
Publisher: Hindawi Limited
Date: 2005
Abstract: We study prime isors of various sequences of positive integers A ( n ) + 1 , n = 1 , … , N , such that the ratios a ( n ) = A ( n ) / A ( n − 1 ) have some number-theoretic or combinatorial meaning. In the case a ( n ) = n , we obviously have A ( n ) = n ! , for which several new results about prime isors of n ! + 1 have recently been obtained.
ON THE SOLVABILITY OF BILINEAR EQUATIONS IN FINITE FIELDS
Publisher: Cambridge University Press (CUP)
Date: 09-2008
DOI: 10.1017/S0017089508004382
Abstract: We consider the equation over a finite field q of q elements, with variables from arbitrary sets $\\cA,\\cB, \\cC, \\cD \\subseteq \\F_q$ . The question of solvability of such and more general equations has recently been considered by Hart and Iosevich, who, in particular, prove that if for some absolute constant C 0, then above equation has a solution for any λ ∈ q *. Here we show that using bounds of multiplicative character sums allows us to extend the class of sets which satisfy this property.
MULTIPLE EXPONENTIAL AND CHARACTER SUMS WITH MONOMIALS
Publisher: Wiley
Date: 27-05-2014
On the Lang-Trotter and Sato-Tate conjectures on average for polynomial families of elliptic curves
Publisher: Michigan Mathematical Journal
Date: 09-2013
Groups generated by iterations of polynomials over finite fields
Publisher: Cambridge University Press (CUP)
Date: 05-06-2016
DOI: 10.1017/S0013091515000097
Abstract: Given a finite field of q elements, we consider a trajectory of the map associated with a polynomial ]. Using bounds of character sums, under some mild condition on f , we show that for an appropriate constant C 0 no N ⩾ Cq ½ distinct consecutive elements of such a trajectory are contained in a small subgroup of , improving the trivial lower bound . Using a different technique, we also obtain a similar result for very small values of N . These results are multiplicative analogues of several recently obtained bounds on the length of intervals containing N distinct consecutive elements of such a trajectory.
On the multidimensional distribution of the subset sum generator of pseudorandom numbers
Publisher: American Mathematical Society (AMS)
Date: 02-09-2003
Erratum: Least totient in a residue class (Bulletin of the London Mathematical Society (2007) 39 (425-432))
Publisher: Wiley
Date: 30-04-2008
DOI: 10.1112/BLMS/BDN037
On the Distribution of Diffie--Hellman Triples with Sparse Exponents
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2001
Coincidences in the values of the Euler and Carmichael functions
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2006
DOI: 10.4064/AA122-3-1
Communication complexity of some number theoretic functions
Publisher: Elsevier BV
Date: 08-2007
Finding elliptic curves with a subgroup of prescribed size
Publisher: World Scientific Pub Co Pte Ltd
Date: 16-11-2016
DOI: 10.1142/S1793042117500099
Abstract: Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime [Formula: see text] and positive integer [Formula: see text], outputs an elliptic curve [Formula: see text] over the finite field [Formula: see text] for which the cardinality of [Formula: see text] is isible by [Formula: see text]. The running time of the algorithm is [Formula: see text], and this leads to more efficient constructions of rational functions over [Formula: see text] whose image is small relative to [Formula: see text]. We also give an unconditional version of the algorithm that works for almost all primes [Formula: see text], and give a probabilistic algorithm with subexponential time complexity.
Circulant graphs and GCD and LCM of subsets
Publisher: Elsevier BV
Date: 02-2015
On the consecutive powers of a primitive root: Gaps and exponential sums
Publisher: Wiley
Date: 08-11-2012
On the Uniformity of Distribution of the Naor–Reingold Pseudo-Random Function
Publisher: Elsevier BV
Date: 04-2001
On the Multiplicative Orders of γ and γ+γ−1 over finite fields
Publisher: Elsevier BV
Date: 04-2001
Functional graphs of polynomials over finite fields
Publisher: Elsevier BV
Date: 2016
Smooth numbers with few nonzero binary digits
Publisher: Canadian Mathematical Society
Date: 20-06-2023
DOI: 10.4153/S0008439523000504
Abstract: We use bounds of character sums and some combinatorial arguments to show the abundance of very smooth numbers which also have very few nonzero binary digits.
On the degree growth in some polynomial dynamical systems and nonlinear pseudorandom number generators
Publisher: American Mathematical Society (AMS)
Date: 2010
Bilinear sums of Kloosterman sums, multiplicative congruences and average values of the divisor function over families of arithmetic progressions
Publisher: Springer Science and Business Media LLC
Date: 20-02-2020
Correcting noisy exponentiation black-boxes modulo a prime
Publisher: Elsevier BV
Date: 06-2013
On the power generator and its multivariate analogue
Publisher: Elsevier BV
Date: 04-2012
On the linear complexity of bounded integer sequences over different moduli
Publisher: Elsevier BV
Date: 12-2005
Elliptic Twin Prime Conjecture
Publisher: Springer Berlin Heidelberg
Date: 2009
The rate of convergence of some iterational processes
Publisher: Elsevier BV
Date: 1984
On the distribution of solutions to linear equations
Publisher: University of Zagreb, Faculty of Science, Department of Mathematics
Date: 21-05-2009
DOI: 10.3336/GM.44.1.02
On the Uniformity of Distribution of the Elliptic Curve ElGamal Signature
Publisher: Elsevier BV
Date: 10-2002
An explicit polynomial analogue of Romanoff's theorem
Publisher: Elsevier BV
Date: 03-2017
On the Waring problem with Dickson polynomials in finite fields
Publisher: American Mathematical Society (AMS)
Date: 11-03-2011
DOI: 10.1090/S0002-9939-2011-10843-8
Abstract: We improve recent results of D. Gomez and A. Winterhof on the Waring problem with Dickson polynomials in finite fields. Our approach is based on recent advances in arithmetic combinatorics in arbitrary finite fields due to A. Glibichuk and M. Rudnev.
Distribution of exponential functions with k-full exponent modulo a prime
Publisher: Elsevier BV
Date: 2004
ON SOME MULTIPLE CHARACTER SUMS
Publisher: Wiley
Date: 2017
On the uniformity of distribution of the decryption exponent in fixed encryption exponent RSA
Publisher: Elsevier BV
Date: 11-2004
Exponential sums and rational points on complete intersections
Publisher: Wiley
Date: 12-1990
The Sato-Tate distribution in thin families of elliptic curves over high degree extensions of finite fields
Publisher: World Scientific Pub Co Pte Lt
Date: 21-03-2019
DOI: 10.1142/S1793042119500246
Abstract: Over the last two decades, there has been a wave of activity establishing the Sato-Tate kind of distribution in various families of elliptic curves over prime fields. Typically the goal here is to prove this for families which are as thin as possible. We consider a function field analogue of this question, that is, for high degree extensions of a finite field where new effects allow us to study families, which are much thinner that those typically investigated over prime fields.
Bounds of double multiplicative character sums and gaps between residues of exponential functions
Publisher: Elsevier BV
Date: 10-2016
On the linear complexity and multidimensional distribution of congruential generators over elliptic curves
Publisher: Springer Science and Business Media LLC
Date: 04-2005
Bilinear character sums over elliptic curves
Publisher: Elsevier BV
Date: 2008
On the density of integer points on generalised Markoff–Hurwitz and Dwork hypersurfaces
Publisher: Springer Science and Business Media LLC
Date: 07-11-2016
Florianptic curves
Publisher: Duke University Press
Date: 07-2004
Testing set proportionality and the Ádám isomorphism of circulant graphs
Publisher: Elsevier BV
Date: 06-2006
On the distribution of the power generator
Publisher: American Mathematical Society (AMS)
Date: 17-10-2001
DOI: 10.1090/S0025-5718-00-01283-7
Abstract: We present a new method to study the power generator of pseudorandom numbers modulo a Blum integer m m . This includes as special cases the RSA generator and the Blum–Blum–Shub generator. We prove the uniform distribution of these, provided that the period t ≥ m 3 / 4 + δ t\\ge m^{3/4 + \\delta } with fixed δ 0 \\delta 0 and, under the same condition, the uniform distribution of a positive proportion of the leftmost and rightmost bits. This sharpens and generalizes previous results which dealt with the RSA generator, provided the period t ≥ m 23 / 24 + δ t\\ge m^{23/24 + \\delta } . We apply our results to deduce that the period of the binary sequence of the rightmost bit has exponential length.
MULTIPLICATIVE CHARACTER SUMS OF A CLASS OF NONLINEAR RECURRENCE VECTOR SEQUENCES
Publisher: World Scientific Pub Co Pte Lt
Date: 09-2011
DOI: 10.1142/S1793042111004484
Abstract: We estimate multiplicative character sums along the orbits of a class of nonlinear recurrence vector sequences. In the one-dimensional case, only much weaker estimates are known and our results have no one-dimensional analogs.
On the Glasner property for matrices with polynomial entries
Publisher: Elsevier BV
Date: 2023
Average multiplicative orders of elements modulo n
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2003
DOI: 10.4064/AA109-4-7
Enumeration of certain varieties over a finite field
Publisher: American Mathematical Society (AMS)
Date: 21-04-2014
DOI: 10.1090/S0002-9939-2014-11999-X
Abstract: Let F q \\mathbb {F}_q be a finite field of q q elements. E. Howe has shown that there is a natural correspondence between the isogeny classes of two-dimensional ordinary abelian varieties over F q \\mathbb {F}_q which do not contain a principally polarized variety and pairs of positive integers ( a , b ) (a,b) satisfying q = a 2 + b q = a^2 + b , where gcd ( q , b ) = 1 \\gcd (q,b)=1 and all prime isors ℓ \\ell of b b are in the arithmetic progression ℓ ≡ 1 ( mod 3 ) \\ell \\equiv 1 \\pmod 3 . This arithmetic criterion allows us to give good upper bounds, and for many finite fields good lower bounds, for the frequency of occurrence of isogeny classes of varieties having this property.
Unlikely intersections over finite fields: Polynomial orbits in small subgroups
Publisher: American Institute of Mathematical Sciences (AIMS)
Date: 2020
DOI: 10.3934/DCDS.2020070
Communication complexity and fourier coefficients of the Diffie-Hellman key
Publisher: Springer Berlin Heidelberg
Date: 2000
DOI: 10.1007/10719839_27
MULTIPLICATIVE ORDER OF GAUSS PERIODS
Publisher: World Scientific Pub Co Pte Lt
Date: 06-2010
DOI: 10.1142/S1793042110003290
Abstract: We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of von zur Gathen and Shparlinski.
On the Counting Function of Elliptic Carmichael Numbers
Publisher: Canadian Mathematical Society
Date: 14-03-2014
Abstract: We give an upper bound for the number of elliptic Carmichael numbers n ≤ x that were recently introduced by J. H. Silverman in the case of an elliptic curve without complex multiplication (non CM). We also discuss several possible further improvements.
On the exponential sum—product problem
Publisher: Elsevier BV
Date: 06-2008
On the Average Distribution of Inversive Pseudorandom Numbers
Publisher: Elsevier BV
Date: 10-2002
Sums with convolutions of Dirichlet characters
Publisher: Springer Science and Business Media LLC
Date: 18-06-2010
Covering Sets for Limited-Magnitude Errors
Publisher: Institute of Electrical and Electronics Engineers (IEEE)
Date: 09-2014
Bilinear forms in Weyl sums for modular square roots and applications
Publisher: Elsevier BV
Date: 12-2020
Exponential function analogue of kloosterman sums
Publisher: Rocky Mountain Mathematics Consortium
Date: 12-2004
On quadratic fields generated by discriminants of irreducible trinomials
Publisher: American Mathematical Society (AMS)
Date: 2010
On vanishing Fermat quotients and a bound of the Ihara sum
Publisher: Tokyo Institute of Technology, Department of Mathematics
Date: 03-2013
On the average energy of circulant graphs
Publisher: Elsevier BV
Date: 04-2008
An Average Bound for Character Sums with Some Counter-Dependent Recurrence Sequences
Publisher: Rocky Mountain Mathematics Consortium
Date: 10-2009
DENOMINATORS of BERNOULLI POLYNOMIALS
Publisher: Wiley
Date: 2018
Orders of Gauss Periods in Finite Fields
Publisher: Springer Science and Business Media LLC
Date: 30-04-1998
Prime numbers with Beatty sequences
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2009
DOI: 10.4064/CM115-2-1
Lower bounds for periods of ducci sequences
Publisher: Cambridge University Press (CUP)
Date: 22-11-2020
DOI: 10.1017/S0004972719001187
Abstract: A Ducci sequence is a sequence of integer $n$ -tuples obtained by iterating the map $$\begin{eqnarray}D:(a_{1},a_{2},\ldots ,a_{n})\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\ldots ,|a_{n}-a_{1}|).\end{eqnarray}$$ Such a sequence is eventually periodic and we denote by $P(n)$ the maximal period of such sequences for given odd $n$ . We prove a lower bound for $P(n)$ by counting certain partitions. We then estimate the size of these partitions via the multiplicative order of two modulo $n$ .
Exponential sums and the distribution of inversive congruential pseudorandom numbers with prime-power modulus
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2000
Arithmetic properties of the Ramanujan function
Publisher: Springer Science and Business Media LLC
Date: 02-2006
DOI: 10.1007/BF02829735
Distribution of polynomial discriminants modulo a prime
Publisher: Springer Science and Business Media LLC
Date: 21-08-2015
Bounds of trilinear sums with Kloosterman fractions
Publisher: Springer Science and Business Media LLC
Date: 06-2021
On the Distribution of Values of Recurring Sequences and the Bell Numbers in Finite Fields
Publisher: Elsevier BV
Date: 1991
The distribution of the fractional parts of recurrent sequences
Publisher: Elsevier BV
Date: 1981
On the distribution of power residues and primitive elements in some nonlinear recurring sequences
Publisher: Wiley
Date: 09-06-2003
Residue classes modulo a prime number in a field of algebraic numbers
Publisher: Springer Science and Business Media LLC
Date: 04-1988
DOI: 10.1007/BF01139128
On two functions arising in the study of the euler and carmichael quotients
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2017
An authentication scheme based on roots of sparse polynomials
Publisher: IEEE
Date: 2003
Security of the most significant bits of the Shamir message passing scheme
Publisher: American Mathematical Society (AMS)
Date: 14-06-2002
Certain exponential sums and random walks on elliptic curves
Publisher: Canadian Mathematical Society
Date: 04-2005
Abstract: For a given elliptic curve E, we obtain an upper bound on the discrepancy of sets of multiples z s G where zs runs through a sequence Z = ( z 1 , … , z T ) such that kz 1 , … , kz T is a permutation of z 1 , … , z T , both sequences taken modulo t , for sufficiently many distinct values of k modulo t . We apply this result to studying an analogue of the power generator over an elliptic curve. These results are elliptic curve analogues of those obtained for multiplicative groups of finite fields and residue rings.
Counting the values taken by algebraic exponential polynomials
Publisher: American Mathematical Society (AMS)
Date: 1999
DOI: 10.1090/S0002-9939-99-04728-0
Abstract: We prove an effective mean-value theorem for the values of a non-degenerate, algebraic exponential polynomial in several variables. These objects generalise simultaneously the fundamental ex les of linear recurrence sequences and sums of S S -units. The proof is based on an effective, uniform estimate for the deviation of the exponential polynomial from its expected value. This estimate is also used to obtain a non-effective asymptotic formula counting the norms of these values below a fixed bound.
On the Sato-Tate conjecture on average for some families of elliptic curves
Publisher: Walter de Gruyter GmbH
Date: 30-06-2013
Lang-trotter and sato-tate distributions in single and double parametric families of elliptic curves
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2015
DOI: 10.4064/AA170-4-1
Prescribing the binary digits of squarefree numbers and quadratic residues
Publisher: American Mathematical Society (AMS)
Date: 05-05-2017
DOI: 10.1090/TRAN/6903
Short character sums with Beatty sequences
Publisher: International Press of Boston
Date: 2006
Computing Jacobi Symbols modulo Sparse Integers and Polynomials and Some Applications
Publisher: Elsevier BV
Date: 08-2000
Upper and Lower Bounds for Higher Moments of Theta Functions
Publisher: Oxford University Press (OUP)
Date: 25-01-2014
DOI: 10.1093/QMATH/HAV039
Period of the power generator and small values of Carmichael's function
Publisher: American Mathematical Society (AMS)
Date: 13-07-2001
DOI: 10.1090/S0025-5718-00-01282-5
Abstract: Consider the pseudorandom number generator u n ≡ u n − 1 e ( mod m ) , 0 ≤ u n ≤ m − 1 , n = 1 , 2 , … , \\begin{equation*} u_n\\equiv u_{n-1}^e\\pmod {m},\\quad 0\\le u_n\\le m-1,\\quad n=1,2,\\ldots , \\end{equation*} where we are given the modulus m m , the initial value u 0 = ϑ u_0=\\vartheta and the exponent e e . One case of particular interest is when the modulus m m is of the form p l pl , where p , l p,l are different primes of the same magnitude. It is known from work of the first and third authors that for moduli m = p l m=pl , if the period of the sequence ( u n ) (u_n) exceeds m 3 / 4 + ε m^{3/4+\\varepsilon } , then the sequence is uniformly distributed. We show rigorously that for almost all choices of p , l p,l it is the case that for almost all choices of ϑ , e \\vartheta ,e , the period of the power generator exceeds ( p l ) 1 − ε (pl)^{1-\\varepsilon } . And so, in this case, the power generator is uniformly distributed. We also give some other cryptographic applications, namely, to ruling-out the cycling attack on the RSA cryptosystem and to so-called time-release crypto. The principal tool is an estimate related to the Carmichael function λ ( m ) \\lambda (m) , the size of the largest cyclic subgroup of the multiplicative group of residues modulo m m . In particular, we show that for any Δ ≥ ( log log N ) 3 \\Delta \\ge (\\log \\log N)^3 , we have λ ( m ) ≥ N exp ( − Δ ) \\lambda (m)\\ge N\\exp (-\\Delta ) for all integers m m with 1 ≤ m ≤ N 1\\le m\\le N , apart from at most N exp ( − 0.69 ( Δ log Δ ) 1 / 3 ) N\\exp \\left (-0.69\\left (\\Delta \\log \\Delta \\right )^{1/3}\\right ) exceptions.
Products of Small Integers in Residue Classes and Additive Properties of Fermat Quotients
Publisher: Oxford University Press (OUP)
Date: 18-06-2016
DOI: 10.1093/IMRN/RNV182
On the product of small Elkies primes
Publisher: American Mathematical Society (AMS)
Date: 12-2015
On structure complexity of normal basis of finite field
Publisher: Springer Berlin Heidelberg
Date: 1987
Distribution of nonresidues and primitive roots in recurrent sequences
Publisher: Springer Science and Business Media LLC
Date: 11-1978
DOI: 10.1007/BF01141537
On the Number of Sparse RSA Exponents
Publisher: Elsevier BV
Date: 08-2002
Incomplete exponential sums and Diffie–Hellman triples
Publisher: Cambridge University Press (CUP)
Date: 22-02-2006
On bilinear exponential and character sums with reciprocals of polynomials
Publisher: Wiley
Date: 2016
On congruences with products of variables from short intervals and applications
Publisher: Pleiades Publishing Ltd
Date: 03-2018
Average distribution of prime ideals in families of number fields
Publisher: Springer Science and Business Media LLC
Date: 09-2008
Sets with integral distances in finite fields
Publisher: American Mathematical Society (AMS)
Date: 17-11-2009
DOI: 10.1090/S0002-9947-09-05004-1
Abstract: Given a positive integer n n , a finite field F q \\mathbb F_q of q q elements ( q q odd), and a non-degenerate quadratic form Q Q on F q n \\mathbb {F}_q^n , in this paper we study the largest possible cardinality of subsets E ⊆ F q n \\mathcal {E} \\subseteq \\mathbb {F}_q^n with pairwise integral Q Q -distances that is, for any two vectors {x} = ( x 1 , … , x n ) , {y} = ( y 1 , … , y n ) ∈ E \\textbf {{x}}=(x_1, \\ldots ,x_n), \\textbf {{y}}=(y_1,\\ldots ,y_n) \\in \\mathcal {E} , one has \\[ Q ( {x} − {y} ) = u 2 Q(\\textbf {{x}}-\\textbf {{y}})=u^2 \\] for some u ∈ F q u \\in \\mathbb F_q .
On finding primitive roots in finite fields
Publisher: Elsevier BV
Date: 05-1996
Cayley graphs generated by small degree polynomials over finite fields
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2015
DOI: 10.1137/14095813X
Erratum to “Character sums and nonlinear recurrence sequences”
Publisher: Elsevier BV
Date: 05-2007
Multiplicative character sums with the Euler function
Publisher: Akademiai Kiado Zrt.
Date: 06-2009
DOI: 10.1556/SSCMATH.46.2009.2.1087
Abstract: We give upper bounds for sums of multiplicative characters modulo an integer q ≧ 2 with the Euler function ϕ ( n ) and with the shifted largest prime isor P ( n ) + a of integers n ≦ x .
On small gaps between the elements of multiplicative subgroups of finite fields
Publisher: Springer Science and Business Media LLC
Date: 22-03-2016
The large sieve inequality with exponential functions and the distribution of mersenne numbers modulo primes
Publisher: Oxford University Press (OUP)
Date: 2005
On the linear complexity profile of the power generator
Publisher: Institute of Electrical and Electronics Engineers (IEEE)
Date: 2000
DOI: 10.1109/18.868485
Non-linear Complexity of the Naor–Reingold Pseudo-random Function
Publisher: Springer Berlin Heidelberg
Date: 2000
DOI: 10.1007/10719994_5
Level curves of rational functions and unimodular points on rational curves
Publisher: American Mathematical Society (AMS)
Date: 21-01-2020
DOI: 10.1090/PROC/14928
Abstract: We obtain an improvement and broad generalisation of a result of N. Ailon and Z. Rudnick (2004) on common zeros of shifted powers of polynomials. Our approach is based on reducing this question to a more general question of counting intersections of level curves of complex functions. We treat this question via classical tools of complex analysis and algebraic geometry.
On a New Exponential Sum
Publisher: Canadian Mathematical Society
Date: 03-2001
Abstract: Let p be prime and let be of multiplicative order t modulo p . We consider exponential sums of the form and prove that for any ε 0
Distribution of some sequences of points on elliptic curves
Publisher: Walter de Gruyter GmbH
Date: 2007
DOI: 10.1515/JMC.2007.001
On the Distribution of Atkin and Elkies Primes
Publisher: Springer Science and Business Media LLC
Date: 19-02-2014
On the g-Ary Expansions of Apéry, Motzkin, Schröder and Other Combinatorial Numbers
Publisher: Springer Science and Business Media LLC
Date: 12-2010
ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS
Publisher: Cambridge University Press (CUP)
Date: 08-2008
DOI: 10.1017/S1446788708000323
Abstract: We show that, for every x exceeding some explicit bound depending only on k and N , there are at least C ( k , N ) x /log 17 x positive and negative coefficients a ( n ) with n ≤ x in the Fourier expansion of any non-zero cuspidal Hecke eigenform of even integral weight k ≥2 and squarefree level N that is a newform, where C ( k , N ) depends only on k and N . From this we deduce the existence of a sign change in a short interval.
Erratum: Sum-product estimates and multiplicative orders ofΓ and Γ + Γ-1 in finite fields (Bulletin of the Australian Mathematical Society)
Publisher: Cambridge University Press (CUP)
Date: 21-02-2013
On the exponent of the group of points on elliptic curves in extension fields
Publisher: Oxford University Press (OUP)
Date: 2005
Quantum noisy rational function reconstruction
Publisher: Springer Berlin Heidelberg
Date: 2005
DOI: 10.1007/11533719_43
PARAMETERS OF INTEGRAL CIRCULANT GRAPHS AND PERIODIC QUANTUM DYNAMICS
Publisher: World Scientific Pub Co Pte Lt
Date: 06-2007
DOI: 10.1142/S0219749907002918
Abstract: The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system whose hamiltonian is identical to the adjacency matrix of a circulant graph is periodic if and only if all eigenvalues of the graph are integers (that is, the graph is integral). Motivated by this observation, we focus on relevant properties of integral circulant graphs. Specifically, we bound the number of vertices of integral circulant graphs in terms of their degree, characterize bipartiteness and give exact bounds for their diameter. Additionally, we prove that circulant graphs with odd order do not allow perfect state transfer.
Counting additive decompositions of quadratic residues in finite fields
Publisher: Adam Mickiewicz University (Euclid)
Date: 03-2015
Kloosterman paths of prime powers moduli, II
Publisher: Societe Mathematique de France
Date: 2020
DOI: 10.24033/BSMF.2802
VSH and multiplicative modular relations between small primes with polynomial exponents
Publisher: Springer Science and Business Media LLC
Date: 18-03-2014
On Stern’s Attack Against Secret Truncated Linear Congruential Generators
Publisher: Springer Berlin Heidelberg
Date: 2005
DOI: 10.1007/11506157_5
On arithmetical properties of solutions of norm equations
Publisher: Steklov Mathematical Institute
Date: 30-06-1989
Visible points on multidimensional modular hyperbolas
Publisher: Elsevier BV
Date: 09-2008
Some additive combinatorics problems in matrix rings
Publisher: Springer Science and Business Media LLC
Date: 10-02-2010
On Coincidences Among Quadratic Fields Generated by the Shanks Sequence
Publisher: Oxford University Press (OUP)
Date: 19-12-2016
DOI: 10.1093/QMATH/HAW054
Trilinear forms with double Kloosterman sums
Publisher: World Scientific Pub Co Pte Lt
Date: 22-08-2018
DOI: 10.1142/S1793042118501312
Abstract: We obtain several estimates for trilinear form with double Kloosterman sums. In particular, these bounds show the existence of nontrivial cancellations between such sums.
On the g-Ary expansions of middle binomial coefficients and catalan numbers
Publisher: Rocky Mountain Mathematics Consortium
Date: 08-2011
Complexity of some arithmetic problems for binary polynomials
Publisher: Springer Science and Business Media LLC
Date: 06-2003
Character sums and congruences with n!
Publisher: American Mathematical Society (AMS)
Date: 29-06-2004
DOI: 10.1090/S0002-9947-04-03612-8
Abstract: We estimate character sums with n ! n! , on average, and in idually. These bounds are used to derive new results about various congruences modulo a prime p p and obtain new information about the spacings between quadratic nonresidues modulo p p . In particular, we show that there exists a positive integer n ≪ p 1 / 2 + ε n\ll p^{1/2+\varepsilon } such that n ! n! is a primitive root modulo p p . We also show that every nonzero congruence class a ≢ 0 ( mod p ) a \not \equiv 0 \pmod p can be represented as a product of 7 factorials, a ≡ n 1 ! … n 7 ! ( mod p ) a \equiv n_1! \ldots n_7! \pmod p , where max { n i | i = 1 , … , 7 } = O ( p 11 / 12 + ε ) \max \{n_i \ |\ i=1,\ldots , 7\}=O(p^{11/12+\varepsilon }) , and we find the asymptotic formula for the number of such representations. Finally, we show that products of 4 factorials n 1 ! n 2 ! n 3 ! n 4 ! , n_1!n_2!n_3!n_4!, with max { n 1 , n 2 , n 3 , n 4 } = O ( p 6 / 7 + ε ) \max \{n_1, n_2, n_3, n_4\}=O(p^{6/7+\varepsilon }) represent “almost all” residue classes modulo p, and that products of 3 factorials n 1 ! n 2 ! n 3 ! n_1!n_2!n_3! with max { n 1 , n 2 , n 3 } = O ( p 5 / 6 + ε ) \max \{n_1, n_2, n_3\}=O(p^{5/6+\varepsilon }) are uniformly distributed modulo p p .
On the values of the divisor function
Publisher: Springer Science and Business Media LLC
Date: 05-11-2007
On some dynamical systems in finite fields and residue rings
Publisher: American Institute of Mathematical Sciences (AIMS)
Date: 2007
On the singularity of generalised Vandermonde matrices over finite fields
Publisher: Elsevier BV
Date: 04-2005
Congruences with intervals and arbitrary sets
Publisher: Springer Science and Business Media LLC
Date: 14-12-2019
On multiplicatively dependent vectors of algebraic numbers
Publisher: American Mathematical Society (AMS)
Date: 02-2018
DOI: 10.1090/TRAN/7115
Abstract: In this paper, we give several asymptotic formulas for the number of multiplicatively dependent vectors of algebraic numbers of fixed degree, or within a fixed number field, and bounded height.
On the number of isogeny classes of pairing-friendly elliptic curves and statistics of MNT curves
Publisher: American Mathematical Society (AMS)
Date: 29-09-2011
Sparse polynomial approximation in finite fields
Publisher: ACM
Date: 06-07-2001
On the greatest common divisor of shifted sets
Publisher: Elsevier BV
Date: 09-2015
Selective Forgery of RSA Signatures with Fixed-Pattern Padding
Publisher: Springer Berlin Heidelberg
Date: 2002
On the spectral Ádám property for circulant graphs
Publisher: Elsevier BV
Date: 06-2002
Distribution of roots of polynomial congruences
Publisher: Hindawi Limited
Date: 2007
DOI: 10.1155/2007/37853
Abstract: For a prime p , we obtain an upper bound on the discrepancy of fractions r / p , where r runs through all of roots modulo p of all monic univariate polynomials of degree d whose vector of coefficients belongs to a d -dimensional box ℬ . The bound is nontrivial starting with boxes ℬ of size | ℬ | ≥ p d / 2 + ɛ for any fixed ɛ 0 and sufficiently large p .
On the value set of fermat quotients
Publisher: American Mathematical Society (AMS)
Date: 04-2012
On the Distribution of the Diffie–Hellman Pairs
Publisher: Elsevier BV
Date: 04-2002
Multiples of squares in short intervals
Publisher: Adam Mickiewicz University (Euclid)
Date: 03-2016
On the distribution of solutions to polynomial congruences
Publisher: Springer Science and Business Media LLC
Date: 10-2012
Number Theoretic Designs for Directed Regular Graphs of Small Diameter
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2004
Distribution of exponential functions with squarefull exponent in residue rings
Publisher: Elsevier BV
Date: 06-2004
ARITHMETIC AND GEOMETRIC PROGRESSIONS IN PRODUCT SETS OVER FINITE FIELDS
Publisher: Cambridge University Press (CUP)
Date: 12-2008
DOI: 10.1017/S0004972708000695
Abstract: Given two sets ${\\mathcal A}, {\\mathcal B} \\subseteq \\mathbb {F}_q$ of elements of the finite field 𝔽 q of q elements, we show that the product set contains an arithmetic progression of length k ≥3 provided that k p , where p is the characteristic of 𝔽 q , and # 𝒜 # ℬ≥3 q 2 d −2/ k . We also consider geometric progressions in a shifted product set 𝒜ℬ+ h , for f ∈𝔽 q , and obtain a similar result.
On the hidden shifted power problem
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2012
DOI: 10.1137/110850414
On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states
Publisher: AIP Publishing
Date: 08-2005
DOI: 10.1063/1.1998831
Abstract: We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension n consisting of n2 operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and, despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in Cn which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory.
Exponential sums with Catalan numbers and middle binomial coefficients
Publisher: Elsevier BV
Date: 2007
On the distribution of the number of points on algebraic curves in extensions of finite fields
Publisher: International Press of Boston
Date: 2010
Modular hyperbolas
Publisher: Springer Science and Business Media LLC
Date: 11-2012
A Refinement of the Burgess Bound for Character Sums
Publisher: Michigan Mathematical Journal
Date: 05-2020
Interpolation and approximation of polynomials in finite fields over a short interval from noisy values
Publisher: Informa UK Limited
Date: 03-07-2014
On the set of distances between two sets over finite fields
Publisher: Hindawi Limited
Date: 2006
Abstract: We use bounds of exponential sums to derive new lower bounds on the number of distinct distances between all pairs of points ( x , y ) ∈ × ℬ for two given sets , ℬ ∈ F q n , where F q is a finite field of q elements and n ≥ 1 is an integer.
On exponential sums and group generators for elliptic curves over finite fields
Publisher: Springer Berlin Heidelberg
Date: 2000
DOI: 10.1007/10722028_24
Chinese remaindering with multiplicative noise
Publisher: Springer Science and Business Media LLC
Date: 21-06-2007
On certain exponential sums and the distribution of Diffie-Hellman triples
Publisher: Wiley
Date: 06-1999
Corrections to “Value sets of sparse polynomials”
Publisher: Canadian Mathematical Society
Date: 02-11-2022
DOI: 10.4153/S0008439521000928
Abstract: We give a corrected version of our previous lower bound on the value set of binomials (Canad. Math. Bull., v.63, 2020, 187–196). The other results are not affected.
Bilinear sums with exponential functions
Publisher: American Mathematical Society (AMS)
Date: 04-03-2009
On the spline-based method for experimental data deconvolution
Publisher: Elsevier BV
Date: 05-1983
An Extremely Small and Efficient Identification Scheme
Publisher: Springer Berlin Heidelberg
Date: 2000
DOI: 10.1007/10718964_31
Elliptic curves with low embedding degree
Publisher: Springer Science and Business Media LLC
Date: 20-09-2006
On the distribution of nonlinear recursive congruential pseudorandom numbers of higher orders
Publisher: Springer Berlin Heidelberg
Date: 1999
On the Insecurity of a Server-Aided RSA Protocol
Publisher: Springer Berlin Heidelberg
Date: 2001
On the Bit Security of NTRUEncrypt
Publisher: Springer Berlin Heidelberg
Date: 18-12-2002
On the distribution of pseudorandom numbers and vectors generated by inversive methods
Publisher: Springer Science and Business Media LLC
Date: 03-2000
Subgroups generated by rational functions in finite fields
Publisher: Springer Science and Business Media LLC
Date: 29-10-2014
Constructing Dominating Sets in Circulant Graphs
Publisher: Springer Science and Business Media LLC
Date: 05-02-2018
On the distribution of modular inverses from short intervals
Publisher: Wiley
Date: 27-09-2023
DOI: 10.1112/MTK.12224
Exponential sums over Mersenne numbers
Publisher: Wiley
Date: 2004
Counting dihedral and quaternionic extensions
Publisher: American Mathematical Society (AMS)
Date: 11-01-2011
DOI: 10.1090/S0002-9947-2011-05233-5
Abstract: We give asymptotic formulas for the number of biquadratic extensions of Q \\mathbb {Q} that admit a quadratic extension which is a Galois extension of Q \\mathbb {Q} with a prescribed Galois group, for ex le, with a Galois group isomorphic to the quaternionic group. Our approach is based on a combination of the theory of quadratic equations with some analytic tools such as the Siegel–Walfisz theorem and the double oscillations theorem.
Sums of inverses in thin sets of finite fields
Publisher: American Mathematical Society (AMS)
Date: 28-12-2018
DOI: 10.1090/PROC/13915
Linear congruences with ratios
Publisher: American Mathematical Society (AMS)
Date: 20-01-2016
DOI: 10.1090/PROC/12949
Abstract: We use new bounds of double exponential sums with ratios of integers from prescribed intervals to get an asymptotic formula for the number of solutions to congruences \\[ ∑ j = 1 n a j x j y j ≡ a 0 ( mod p ) , \\sum _{j=1}^n a_j \\frac {x_j}{y_j} \\equiv a_0 \\pmod p, \\] with variables from rather general sets.
Distribution of Nonlinear Congruential Pseudorandom Numbers Modulo Almost Squarefree Integers
Publisher: Springer Science and Business Media LLC
Date: 27-02-2006
Exponential sums over points of elliptic curves
Publisher: Elsevier BV
Date: 07-2014
SOME DIVISIBILITY PROPERTIES OF THE EULER FUNCTION
Publisher: Cambridge University Press (CUP)
Date: 29-11-2005
Character sums with division polynomials
Publisher: Canadian Mathematical Society
Date: 12-2012
Abstract: We obtain nontrivial estimates of quadratic character sums of ision polynomials Ψ n ( P ), n = 1, 2, … , evaluated at a given point P on an elliptic curve over a finite field of q elements. Our bounds are nontrivial if the order of P is at least q 1/2+∈ for some fixed ∈ 0. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences that was recently brought up by K. Lauter and the second author.
Bounds of Trilinear and Quadrilinear Exponential Sums
Publisher: Springer Science and Business Media LLC
Date: 07-2019
On the size of the Jacobians of curves over finite fields
Publisher: Springer Science and Business Media LLC
Date: 12-2008
ON CURVES OVER FINITE FIELDS WITH JACOBIANS OF SMALL EXPONENT
Publisher: World Scientific Pub Co Pte Ltd
Date: 10-2008
DOI: 10.1142/S1793042108001687
Abstract: We show that finite fields over which there is a curve of a given genus g ≥ 1 with its Jacobian having a small exponent, are very rare. This extends a recent result of Duke in the case of g = 1. We also show that when g = 1 or g = 2, our lower bounds on the exponent, valid for almost all finite fields 𝔽 q and all curves over 𝔽 q , are best possible.
On Artin′s Conjecture over Function Fields
Publisher: Elsevier BV
Date: 10-1995
On the Multidimensional Distribution of the Naor–Reingold Pseudo-Random Function
Publisher: American Mathematical Society (AMS)
Date: 17-01-2014
Arithmetic properties of φ(n)/λ(n) and the structure of the multiplicative group modulo n
Publisher: European Mathematical Society - EMS - Publishing House GmbH
Date: 2006
DOI: 10.4171/CMH/40
The Sato–Tate distribution in families of elliptic curves with a rational parameter of bounded height
Publisher: Elsevier BV
Date: 04-2017
On the Linear Complexity of the Naor-Reingold Pseudo-random Function from Elliptic Curves
Publisher: Springer Science and Business Media LLC
Date: 2001
Noisy interpolation of sparse polynomials in finite fields
Publisher: Springer Science and Business Media LLC
Date: 07-2005
Concentration of points on curves in finite fields
Publisher: Springer Science and Business Media LLC
Date: 27-04-2013
On an iterative process for the numerical solution of systems of linear algebraic equations
Publisher: Elsevier BV
Date: 1982
Cyclotomic coefficients: Gaps and jumps
Publisher: Elsevier BV
Date: 06-2016
Random walks, bisections and gossiping in circulant graphs
Publisher: Springer Science and Business Media LLC
Date: 11-07-2014
Arithmetic functions on Beatty sequences
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2009
DOI: 10.4064/AA136-1-6
Cancellations between kloosterman sums modulo a prime power with prime arguments
Publisher: Wiley
Date: 2019
Products with variables from low-dimensional affine spaces and shifted power identity testing in finite fields
Publisher: Elsevier BV
Date: 08-2014
Sum-product estimates and multiplicative orders of γ and γ+γ -1 in finite fields
Publisher: Cambridge University Press (CUP)
Date: 30-11-2012
DOI: 10.1017/S0004972711002887
Abstract: Using a recent result on the sum–product problem, we estimate the number of elements γ in a prime finite field such that both γ and γ + γ −1 are of small order.
Geometric properties of points on modular hyperbolas
Publisher: American Mathematical Society (AMS)
Date: 09-07-2010
DOI: 10.1090/S0002-9939-2010-10561-0
Abstract: Given an integer n ≥ 2 n\\ge 2 , let H n \\mathcal {H}_n be the set \\[ H n = { ( a , b ) : a b ≡ 1 ( mod n ) , 1 ≤ a , b ≤ n − 1 } \\mathcal {H}_n= \\{(a,b) \\ : \\ ab \\equiv 1 \\pmod n,\\ 1\\le a,b \\le n-1\\} \\] and let M ( n ) M(n) be the maximal difference of b − a b-a for ( a , b ) ∈ H n (a,b) \\in \\mathcal {H}_n . We prove that for almost all n n , n − M ( n ) = O ( n 1 / 2 + o ( 1 ) ) . n-M(n)=O\\left (n^{1/2+o(1)}\\right ). We also improve some previously known upper and lower bounds on the number of vertices of the convex closure of H n \\mathcal {H}_n .
Bisecting and gossiping in circulant graphs
Publisher: Springer Berlin Heidelberg
Date: 2004
On Polynomial Representations of Boolean Functions Related to Some Number Theoretic Problems
Publisher: Springer Berlin Heidelberg
Date: 2001
Constructing elements of large order in finite fields
Publisher: Springer Berlin Heidelberg
Date: 1999
Approximate polynomial gcd: Small degree and small height perturbations
Publisher: Elsevier BV
Date: 08-2010
Orbits of polynomial dynamical systems modulo primes
Publisher: American Mathematical Society (AMS)
Date: 26-12-2017
DOI: 10.1090/PROC/13904
Abstract: We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over C \\mathbb {C} . Applying recent results of Baker and DeMarco (2011) and of Ghioca, Krieger, Nguyen and Ye (2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang (2015). As a by-product, we also slightly improve a result of Silverman (2008) and recover a result of Akbary and Ghioca (2009) as special extreme cases of our estimates.
GCD of random linear combinations
Publisher: Springer Science and Business Media LLC
Date: 03-07-2006
Some doubly exponential sums over Zm
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2002
DOI: 10.4064/AA105-4-4
Orbits of Algebraic Dynamical Systems in Subgroups and Subfields
Publisher: Springer International Publishing
Date: 2017
On smooth square-free numbers in arithmetic progressions
Publisher: Wiley
Date: 23-01-2020
DOI: 10.1112/JLMS.12297
Erratum: Pseudoprime reductions of elliptic curves (Mathematical Proceedings of the Cambridge Philosophical Society (2009) 146 (513-522))
Publisher: Cambridge University Press (CUP)
Date: 03-02-2012
DOI: 10.1017/S0305004111000399
Abstract: Unfortunately, there are two inaccuracies in the argument of [ CLS ]. First, the statements of Lemmas 3, 4, 6, and 7 of [ CLS ] hold only under the additional condition gcd( m , M E ) = 1 for some integer M E ≥ 1 depending only on E . Second, the isibility condition (3·6) in [ CLS ] implies that t b (ℓ) | n E ( p )−1 (rather than t b (ℓ) | n E ( p ), as it was erroneously claimed on p. 519 in [ CLS ]). In particular, instead of the isibility ℓ t b (ℓ) | n E ( p ) (see the last displayed formula on p. 519 in [ CLS ]), we conclude that for every prime ℓ | L there is an integer a ℓ such that (0.1) However, the final result is correct and can easily be recovered. To do so, we remark that under the condition gcd( m , M E ) =1, we have full analogues of Lemmas 6, 7, 9, and 10 of [ CLS ] for the function Π( x m , a ) defined as the number of primes p ≤ x with n E ( p ) ≡ a (mod m ) (rather than just for Π( x m ) = Π( x m ,0) as in [ CLS ]). Define ρ*( n ) as the largest square-free isor of n which is relatively prime to M E . We then derive from (0.1) above that Therefore (0.2) Since we see that (0.2) above implies the bound (3·7) from [ CLS ], and the result now follows without any further changes.
Additive decompositions of subgroups of finite fields
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2013
DOI: 10.1137/130924470
On the average number of square-free values of polynomials
Publisher: Canadian Mathematical Society
Date: 12-2013
Abstract: We obtain an asymptotic formula for the number of square-free integers in N consecutive values of polynomials on average over integral polynomials of degree at most k and of height at most H , where H ≥ N k-1+ε for some fixed ε 0. In idual results of this kind for polynomials of degree k 3, due to A. Granville (1998), are only known under the ABC-conjecture.
Character sums over shifted smooth numbers
Publisher: American Mathematical Society (AMS)
Date: 02-05-2007
DOI: 10.1090/S0002-9939-07-08785-0
Abstract: We give nontrivial bounds in various ranges for character sums of the form \\[ ∑ n ∈ S ( x , y ) χ ( n + a ) , gcd ( a , p ) = 1 , \\sum _{n\\in \\mathcal S(x,y)}\\chi (n +a), \\qquad \\gcd (a,p) = 1, \\] where χ \\chi is a nontrivial multiplicative character modulo a prime p p and S ( x , y ) \\mathcal S(x,y) is the set of positive integers n ≤ x n\\le x that are isible only by primes q ≤ y q \\le y .
Effective results on linear dependence for elliptic curves
Publisher: Mathematical Sciences Publishers
Date: 13-03-2018
Uniform Distribution of Fractional Parts Related to Pseudoprimes
Publisher: Canadian Mathematical Society
Date: 06-2009
Abstract: We estimate exponential sums with the Fermat-like quotients where g and n are positive integers, n is composite, and P ( n ) is the largest prime factor of n . Clearly, both f g ( n ) and h g ( n ) are integers if n is a Fermat pseudoprime to base g , and if n is a Carmichael number, this is true for all g coprime to n . Nevertheless, our bounds imply that the fractional parts ﹛ f g ( n )﹜ and ﹛ h g ( n )﹜ are uniformly distributed, on average over g for f g ( n ), and in idually for h g ( n ). We also obtain similar results with the functions and .
A deterministic test for permutation polynomials
Publisher: Springer Science and Business Media LLC
Date: 06-1992
DOI: 10.1007/BF01202000
Average distribution of k‐free numbers in arithmetic progressions
Publisher: Wiley
Date: 25-05-2020
INFINITE HILBERT CLASS FIELD TOWERS OVER CYCLOTOMIC FIELDS
Publisher: Cambridge University Press (CUP)
Date: 2008
DOI: 10.1017/S0017089507003977
Abstract: We use a result of Y. Furuta to show that for almost all positive integers m , the cyclotomic field $\\Q(\\exp(2 \\pi i/m))$ has an infinite Hilbert p -class field tower with high rank Galois groups at each step, simultaneously for all primes p of size up to about (log log m ) 1 + o (1) . We also use a recent result of B. Schmidt to show that for infinitely many m there is an infinite Hilbert p -class field tower over $\\Q(\\exp(2 \\pi i/m))$ for some p ≥ m 0.3385 + o (1) . These results have immediate applications to the isibility properties of the class number of $\\Q(\\exp(2 \\pi i/m))$ .
ON THE SQUARE-FREE PARTS OF ⌊en!⌋
Publisher: Cambridge University Press (CUP)
Date: 05-2007
DOI: 10.1017/S0017089507003734
Abstract: In this note, we show that if we write ⌊ en! ⌋ = s ( n ) u ( n ) 2 , where s ( n ) is square-free then has at least C log log N distinct prime factors for some absolute constant C 0 and sufficiently large N . A similar result is obtained for the total number of distinct primes iding the m th power-free part of s ( n ) as n ranges from 1 to N , where m ≥ 3 is a positive integer. As an application of such results, we give an upper bound on the number of n ≤ N such that ⌊ en! ⌋ is a square.
Pseudoprime Cullen and Woodall numbers
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2007
DOI: 10.4064/CM107-1-5
On the computational hardness of testing square-freeness of sparse polynomials
Publisher: Springer Berlin Heidelberg
Date: 1999
MULTIPLICATIVE CHARACTER SUMS WITH TWICE-DIFFERENTIABLE FUNCTIONS
Publisher: Oxford University Press (OUP)
Date: 02-08-2008
DOI: 10.1093/QMATH/HAN023
Orders of points in families of elliptic curves
Publisher: American Mathematical Society (AMS)
Date: 28-01-2020
DOI: 10.1090/PROC/14901
Discriminants of complex multiplication fields of elliptic curves over finite fields
Publisher: Canadian Mathematical Society
Date: 09-2007
Abstract: We show that, for most of the elliptic curves E over a prime finite field p of p elements, the discriminant D ( E ) of the quadratic number field containing the endomorphism ring of E over p is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over p .
Character sums with subsequence sums
Publisher: Springer Science and Business Media LLC
Date: 11-2007
Optimal quantum algorithm for polynomial interpolation
Publisher: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH, Wadern/Saarbruecken, Germany
Date: 2016
On a sum involving the Euler function
Publisher: Elsevier BV
Date: 09-2019
On solutions to some polynomial congruences in small boxes
Publisher: Cambridge University Press (CUP)
Date: 07-08-2014
DOI: 10.1017/S0004972713000671
Abstract: We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$ . In particular, we obtain nontrivial results about the number of solutions in boxes with the side length below ${p}^{1/ 2} $ , which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.
Secure Bilinear Diffie-Hellman Bits
Publisher: Springer Berlin Heidelberg
Date: 2004
Small discriminants of complex multiplication fields of elliptic curves over finite fields
Publisher: Institute of Mathematics, Czech Academy of Sciences
Date: 06-2015
On the distribution of angles of the Salié sums
Publisher: Cambridge University Press (CUP)
Date: 04-2007
DOI: 10.1017/S0004972700039150
Abstract: For a prime p and integers a and b , we consider Salié sums where χ2( x ) is a quadratic character and x¯ is the modular inversion of x , that is, x x¯≡ 1 (mod p ). One can naturally associate with S p ( a, b ) a certain angle θ p ( a, b ) ∈ [0, π]. We show that, for any fixed ε 0, these angles are uniformly distributed in [0, π] when a and b run over arbitrary sets , ℬ ⊆ {0, 1, …, p − 1} such that there are at least p 1+ε quadratic residues modulo p among the products ab , where ( a, b ) ∈ × ℬ.
Evasive properties of sparse graphs and some linear equations in primes
Publisher: Elsevier BV
Date: 08-2014
On gaps between quadratic non-residues in the Euclidean and Hamming metrics
Publisher: Elsevier BV
Date: 11-2013
Linear complexity of the Naor-Reingold pseudo-random function
Publisher: Elsevier BV
Date: 12-2000
Estimates of Gaussian sums
Publisher: Springer Science and Business Media LLC
Date: 07-1991
DOI: 10.1007/BF01156612
Computing components and projections of curves over finite fields
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 1999
Finding irreducible and primitive polynomials
Publisher: Springer Science and Business Media LLC
Date: 12-1993
DOI: 10.1007/BF01200150
Hidden Number Problem with the Trace and Bit Security of XTR and LUC
Publisher: Springer Berlin Heidelberg
Date: 2002
On Some Uniformity of Distribution Properties of ESIGN
Publisher: Elsevier BV
Date: 04-2001
Bounds on short character sums and L-functions with characters to a powerful modulus
Publisher: Springer Science and Business Media LLC
Date: 10-2019
On the hardness of approximating the permanent of structured matrices
Publisher: Springer Science and Business Media LLC
Date: 06-2002
On the value set of the Ramanujan function
Publisher: Springer Science and Business Media LLC
Date: 12-2005
On the error term of a lattice counting problem
Publisher: Elsevier BV
Date: 2018
The insecurity of nyberg–rueppel and other DSA-like signature schemes with partially known nonces
Publisher: Springer Berlin Heidelberg
Date: 2001
On abelian multiplicatively dependent points on a curve in a torus
Publisher: Oxford University Press (OUP)
Date: 26-09-2017
DOI: 10.1093/QMATH/HAX045
On products of primes and almost primes in arithmetic progressions
Publisher: Springer Science and Business Media LLC
Date: 26-03-2013
CONGRUENCES AND EXPONENTIAL SUMS WITH THE SUM OF ALIQUOT DIVISORS FUNCTION
Publisher: World Scientific Pub Co Pte Lt
Date: 12-2008
DOI: 10.1142/S179304210800178X
Abstract: We give bounds on the number of integers 1 ≤ n ≤ N such that p | s(n), where p is a prime and s(n) is the sum of aliquot isors function given by s(n) = σ(n) - n, where σ(n) is the sum of isors function. Using this result, we obtain nontrivial bounds in certain ranges for rational exponential sums of the form [Formula: see text]
On the Correlation of Binary M-sequences
Publisher: Springer Science and Business Media LLC
Date: 1999
On the nonlinearity of linear recurrence sequences
Publisher: Elsevier BV
Date: 04-2006
Fractional parts of Dedekind sums
Publisher: World Scientific Pub Co Pte Lt
Date: 10-05-2016
DOI: 10.1142/S179304211650069X
Abstract: Using a recent improvement by Bettin and Chandee to a bound of Duke, Friedlander and Iwaniec [Bilinear forms with Kloosterman fractions, Invent. Math. 128 (1997) 23–43] on double exponential sums with Kloosterman fractions, we establish a uniformity of distribution result for the fractional parts of Dedekind sums [Formula: see text] with [Formula: see text] and [Formula: see text] running over rather general sets. Our result extends earlier work of Myerson [Dedekind sums and uniform distribution, J. Number Theory 28 (1988) 233–239] and Vardi [A relation between Dedekind sums and Kloosterman sums, Duke Math. J. 55 (1987) 189–197]. Using different techniques, we also study the least denominator of the collection of Dedekind sums [Formula: see text] on average for [Formula: see text].
Metric theory of Weyl sums
Publisher: Springer Science and Business Media LLC
Date: 05-01-2022
Arithmetic properties of numbers with restricted digits
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2004
DOI: 10.4064/AA112-4-1
Linear equations with rational fractions of bounded height and stochastic matrices
Publisher: Oxford University Press (OUP)
Date: 06-10-0012
DOI: 10.1093/QMATH/HAX049
On the complexity of exact counting of dynamically irreducible polynomials
Publisher: Elsevier BV
Date: 07-2020
Prime divisors of palindromes
Publisher: Springer Science and Business Media LLC
Date: 11-2005
On squares in polynomial products
Publisher: Springer Science and Business Media LLC
Date: 02-10-2008
Prime divisors in Beatty sequences
Publisher: Elsevier BV
Date: 04-2007
On the typical size and cancellations among the coefficients of some modular forms
Publisher: Cambridge University Press (CUP)
Date: 19-04-2019
DOI: 10.1017/S0305004117000780
Abstract: We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato–Tate density. Ex les of such sequences come from coefficients of several L -functions of elliptic curves and modular forms. In particular, we show that |τ( n )| ⩽ n 11/2 (log n ) −1/2+ o (1) for a set of n of asymptotic density 1, where τ( n ) is the Ramanujan τ function while the standard argument yields log 2 instead of −1/2 in the power of the logarithm. Another consequence of our result is that in the number of representations of n by a binary quadratic form one has slightly more than square-root cancellations for almost all integers n . In addition, we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato–Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally and might be within reach unconditionally using the currently established potential automorphy.
ON QUADRATIC FIELDS GENERATED BY POLYNOMIALS
Publisher: Cambridge University Press (CUP)
Date: 29-06-2023
DOI: 10.1017/S0004972723000606
Abstract: Let $f(X) \\in {\\mathbb Z}[X]$ be a polynomial of degree $d \\ge 2$ without multiple roots and let ${\\mathcal F}(N)$ be the set of Farey fractions of order N . We use bounds for some new character sums and the square-sieve to obtain upper bounds, pointwise and on average, on the number of fields ${\\mathbb Q}(\\sqrt {f(r)})$ for $r\\in {\\mathcal F}(N)$ , with a given discriminant.
On some applications of finitely generated semi-groups
Publisher: Springer Berlin Heidelberg
Date: 1994
Analogues of the Balog–Wooley Decomposition for Subsets of Finite Fields and Character Sums with Convolutions
Publisher: Springer Science and Business Media LLC
Date: 02-02-2019
Coefficients of primitive polynomials
Publisher: Springer Science and Business Media LLC
Date: 12-1985
DOI: 10.1007/BF01157011
Zero testing of p-adic and modular polynomials
Publisher: Elsevier BV
Date: 02-2000
Double sums of Kloosterman sums in finite fields
Publisher: Elsevier BV
Date: 11-2019
Editor’s Note
Publisher: American Society of Civil Engineers (ASCE)
Date: 02-2006
Fermat quotients: Exponential sums, value set and primitive roots
Publisher: Wiley
Date: 18-08-2011
DOI: 10.1112/BLMS/BDR058
On sums of Kloosterman and Gauss sums
Publisher: American Mathematical Society (AMS)
Date: 28-02-2019
DOI: 10.1090/TRAN/7506
Divisor sums of generalised exponential polynomials
Publisher: Canadian Mathematical Society
Date: 03-1996
Abstract: A study is made of sums of reciprocal norms of integral and prime ideal isors of algebraic integer values of a generalised exponential polynomial. This includes the important special cases of linear recurrence sequences and general sums of S-units. In the case of an integral binary recurrence sequence, similar (but stronger) results were obtained by P. Erdős, P. Kiss and C. Pomerance.
Character sums with smooth numbers
Publisher: Springer Science and Business Media LLC
Date: 05-03-2018
On parameters of some graphs from finite fields
Publisher: Elsevier BV
Date: 11-1993
On the density of some special primes
Publisher: Walter de Gruyter GmbH
Date: 2009
DOI: 10.1515/JMC.2009.016
Catalytic Growth of Ultralong Graphene Nanoribbons on Insulating Substrates
Publisher: Wiley
Date: 07-06-2022
Abstract: Graphene nanoribbons (GNRs) with widths of a few nanometers are promising candidates for future nanoelectronic applications due to their structurally tunable bandgaps, ultrahigh carrier mobilities, and exceptional stability. However, the direct growth of micrometer‐long GNRs on insulating substrates, which is essential for the fabrication of nanoelectronic devices, remains an immense challenge. Here, the epitaxial growth of GNRs on an insulating hexagonal boron nitride (h‐BN) substrate through nanoparticle‐catalyzed chemical vapor deposition is reported. Ultranarrow GNRs with lengths of up to 10 µm are synthesized. Remarkably, the as‐grown GNRs are crystallographically aligned with the h‐BN substrate, forming 1D moiré superlattices. Scanning tunneling microscopy reveals an average width of 2 nm and a typical bandgap of ≈1 eV for similar GNRs grown on conducting graphite substrates. Fully atomistic computational simulations support the experimental results and reveal a competition between the formation of GNRs and carbon nanotubes during the nucleation stage, and van der Waals sliding of the GNRs on the h‐BN substrate throughout the growth stage. This study provides a scalable, single‐step method for growing micrometer‐long narrow GNRs on insulating substrates, thus opening a route to explore the performance of high‐quality GNR devices and the fundamental physics of 1D moiré superlattices.
On RSA moduli with prescribed bit patterns
Publisher: Springer Science and Business Media LLC
Date: 04-2006
Explicit form of Cassels' p-adic Embedding Theorem for Number Fields
Publisher: Canadian Mathematical Society
Date: 10-2015
Abstract: In this paper, we give a general explicit form of Cassels’ p -adic embedding theorem for number fields. We also give its reûned form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial f over ℤ, we give a general unconditional upper bound for the smallest prime number p such that f has a simple root modulo p .
On the modular inversion hidden number problem
Publisher: Elsevier BV
Date: 04-2012
On polynomials of prescribed height in finite fields
Publisher: IOP Publishing
Date: 28-02-1989
On the Unpredictability of Bits of the Elliptic Curve Diffie-Hellman Scheme
Publisher: Springer Berlin Heidelberg
Date: 2001
Double exponential sums over thin sets
Publisher: American Mathematical Society (AMS)
Date: 31-10-2001
Distribution of harmonic sums and Bernoulli polynomials modulo a prime
Publisher: Springer Science and Business Media LLC
Date: 28-03-2006
On the Structure of Graphs of Markoff Triples
Publisher: Oxford University Press (OUP)
Date: 04-2020
Abstract: We sharpen the bounds of J. Bourgain, A. Gamburd and P. Sarnak (2016) on the possible number of nodes outside the ‘giant component’ and on the size of in idual connected components in the suitably defined functional graph of Markoff triples modulo $p$. This is a step towards the conjecture that there are no such nodes at all.
Polynomial values in small subgroups of finite fields
Publisher: European Mathematical Society - EMS - Publishing House GmbH
Date: 2016
DOI: 10.4171/RMI/909
On the average distribution of pseudorandom numbers generated by nonlinear permutations
Publisher: American Mathematical Society (AMS)
Date: 05-2011
On a generalisation of a Lehmer problem
Publisher: Springer Science and Business Media LLC
Date: 16-10-2008
On Gauss sums and the evaluation of Stechkin’s constant
Publisher: American Mathematical Society (AMS)
Date: 20-11-2015
DOI: 10.1090/MCOM3056
Security of most significant bits of gx2
Publisher: Elsevier BV
Date: 07-2002
Exponential sums with consecutive modular roots of an integer
Publisher: Oxford University Press (OUP)
Date: 11-07-2011
DOI: 10.1093/QMATH/HAP023
On the cycle structure of repeated exponentiation modulo a prime
Publisher: Elsevier BV
Date: 08-2004
On one sequence of iteration parameters
Publisher: Elsevier BV
Date: 1985
ON THE LARGEST PRIME FACTOR OF THE MERSENNE NUMBERS
Publisher: Cambridge University Press (CUP)
Date: 17-04-2009
DOI: 10.1017/S0004972709000033
Abstract: Let P ( k ) be the largest prime factor of the positive integer k . In this paper, we prove that the series is convergent for each constant α /2, which gives a more precise form of a result of C. L. Stewart [‘On isors of Fermat, Fibonacci, Lucas and Lehmer numbers’, Proc. London Math. Soc. 35 (3) (1977), 425–447].
Counting curves and their projections
Publisher: Springer Science and Business Media LLC
Date: 03-1996
DOI: 10.1007/BF01202042
On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average
Publisher: Wiley
Date: 2015
DOI: 10.1112/S1461157015000017
Abstract: For an elliptic curve $E/\\mathbb{Q}$ without complex multiplication we study the distribution of Atkin and Elkies primes $\\ell$ , on average, over all good reductions of $E$ modulo primes $p$ . We show that, under the generalized Riemann hypothesis, for almost all primes $p$ there are enough small Elkies primes $\\ell$ to ensure that the Schoof–Elkies–Atkin point-counting algorithm runs in $(\\log p)^{4+o(1)}$ expected time.
Distribution of farey fractions in residue classes and lang-trotter conjectures on average
Publisher: No publisher found
Date: 2008
Product sets of rationals, multiplicative translates of subgroups in residue rings, and fixed points of the discrete logarithm
Publisher: Oxford University Press (OUP)
Date: 08-07-2008
DOI: 10.1093/IMRN/RNN090
Counting irreducible binomials over finite fields
Publisher: Elsevier BV
Date: 03-2016
On a multiplicative generator of pseudorandom numbers
Publisher: Elsevier BV
Date: 1984
Polynomial representations of the Diffie-Hellman mapping
Publisher: Cambridge University Press (CUP)
Date: 06-2001
DOI: 10.1017/S0004972700019547
Abstract: We obtain lower bounds on the degrees of polynomials representing the Diffie-Hellman mapping ( g x , g y ) → g xy , where g is a primitive root of a finite field q of q elements. These bounds are exponential in terms of log q . In particular, these results can be used to obtain lower bounds on the parallel arithmetic complexity of breaking the Diffie-Hellman cryptosystem. The method is based on bounds of numbers of solutions of some polynomial equations.
The Sato–Tate distribution in thin parametric families of elliptic curves
Publisher: Springer Science and Business Media LLC
Date: 08-03-2018
Average normalisations of elliptic curves
Publisher: Cambridge University Press (CUP)
Date: 12-2002
DOI: 10.1017/S0004972700040211
Abstract: Ciet, Quisquater, and Sica have recently shown that every elliptic curve E over a finite field 𝔽 p is isomorphic to a curve y 2 = x 3 + ax + b with a and b of size O ( p ¾ ). In this paper, we show that almost all elliptic curves satisfy the stronger bound O ( p ⅔ ). The problem is motivated by cryptographic considerations.
ON SOME WEIGHTED AVERAGE VALUES OF L-FUNCTIONS
Publisher: Cambridge University Press (CUP)
Date: 26-02-2009
DOI: 10.1017/S0004972708001020
Abstract: Let q ≥2 and N ≥1 be integers. W. Zhang recently proved that for any fixed ε and q ε ≤ N ≤ q 1/2−ε , where the sum is taken over all nonprincipal characters χ modulo q , L (1, χ ) denotes the L -functions corresponding to χ , and α q = q o (1) is some explicit function of q . Here we improve this result and show that the same asymptotic formula holds in the essentially full range q ε ≤ N ≤ q 1−ε .
On shifted Eisenstein polynomials
Publisher: Springer Science and Business Media LLC
Date: 24-09-2014
On the smallest pseudopower
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2009
DOI: 10.4064/AA140-1-3
On the Generalised Hidden Number Problem and Bit Security of XTR
Publisher: Springer Berlin Heidelberg
Date: 2001
Distribution of inverses in polynomial rings
Publisher: Elsevier BV
Date: 2001
Value Sets of Sparse Polynomials
Publisher: Canadian Mathematical Society
Date: 24-09-2020
DOI: 10.4153/S0008439519000316
Abstract: We obtain a new lower bound on the size of the value set $\\mathscr{V}(f)=f(\\mathbb{F}_{p})$ of a sparse polynomial $f\\in \\mathbb{F}_{p}[X]$ over a finite field of $p$ elements when $p$ is prime. This bound is uniform with respect to the degree and depends on some natural arithmetic properties of the degrees of the monomial terms of $f$ and the number of these terms. Our result is stronger than those that can be extracted from the bounds on multiplicities of in idual values in $\\mathscr{V}(f)$ .
Uniform distribution of the fractional part of the average prime divisor
Publisher: Walter de Gruyter GmbH
Date: 18-01-2005
On Multiplicative Dependence of Values of Rational Functions and a Generalization of the Northcott Theorem
Publisher: Michigan Mathematical Journal
Date: 06-2019
Sato-Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height
Publisher: Springer Science and Business Media LLC
Date: 09-2009
Number Theory and Related Fields
Publisher: Springer New York
Date: 2013
Generalization of Chebyshev polynomials
Publisher: Springer Science and Business Media LLC
Date: 1990
DOI: 10.1007/BF00971170
Isomorphism classes of elliptic curves over a finite field in some thin families
Publisher: International Press of Boston
Date: 2012
On the Ádám Conjecture on Circulant Graphs
Publisher: Springer Berlin Heidelberg
Date: 1998
Security of polynomial transformations of the Diffie–Hellman key
Publisher: Elsevier BV
Date: 2004
On the maximal difference between an element and its inverse modulo n
Publisher: Springer Science and Business Media LLC
Date: 2003
On the distribution of points on the generalized markoff-hurwitz and dwork hypersurfaces
Publisher: World Scientific Pub Co Pte Lt
Date: 22-01-2014
DOI: 10.1142/S1793042113500863
Abstract: We use bounds of mixed character sum modulo a prime p to study the distribution of points on the hypersurface [Formula: see text] for some polynomials f i ∈ ℤ[X] that are not constant modulo a prime p and integers k i with gcd (k i , p-1) = 1, i = 1, …, n. In the case of [Formula: see text] the above congruence is known as the Markoff–Hurwitz hypersurface, while for [Formula: see text] it is known as the Dwork hypersurface. In particular, we obtain non-trivial results about the number of solution in boxes with the side length below p 1/2 , which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.
Truncations of L-functions in residue classes
Publisher: Cambridge University Press (CUP)
Date: 05-2006
New Bounds of Weyl Sums
Publisher: Oxford University Press (OUP)
Date: 18-11-2019
DOI: 10.1093/IMRN/RNZ293
Abstract: We augment the method of Wooley (2016) by some new ideas and in a series of results, improve his metric bounds on the Weyl sums and the discrepancy of fractional parts of real polynomials with partially prescribed coefficients. We also extend these results and ideas to principally new and very general settings of arbitrary orthogonal projections of the vectors of the coefficients $(u_1, \\ldots , u_d)$ onto a lower-dimensional subspace. This new point of view has an additional advantage of yielding an upper bound on the Hausdorff dimension of sets of large Weyl sums. Among other technical innovations, we also introduce a “self-improving” approach, which leads to an infinite series of monotonically decreasing bounds, converging to our final result.
Subset sum pseudorandom numbers: fast generation and distribution
Publisher: Walter de Gruyter GmbH
Date: 2009
DOI: 10.1515/JMC.2009.007
Tate-shafarevich groups and frobenius fields of reductions of elliptic curves
Publisher: Oxford University Press (OUP)
Date: 02-2010
DOI: 10.1093/QMATH/HAP001
Achievements and Challenges of Graphene Chemical Vapor Deposition Growth
Publisher: Wiley
Date: 11-05-2022
Abstract: Graphene, since the first successful exfoliation of graphite, has continuously attracted attention due to its remarkable properties and applications. Recently, the research focus on graphene synthesis has been directed to the controllable synthesis of large‐area and high‐quality graphene. In the last decade, there has been great progress in the chemical vapor deposition (CVD) growth of graphene. Theoretical investigations have led to an enhanced understanding of puzzles on hydrocarbon species stability, key reaction pathways, the role of hydrogen gas, the morphology of graphene islands, and the alignment of graphene on substrates. Experimentally, high‐quality graphene is epitaxially grown on both insulating and metal substrates. Progress has also been reported on low‐temperature graphene growth and on controlling the thickness and stacking of graphene layers. In this review, the authors summarize the previous theoretical and experimental studies on graphene CVD growth and discuss the future challenges on the growth of graphene i) on insulating substrates, ii) at low temperature, iii) with controllable thickness, and iv) with selected stacking twist angles. The authors assert that the key to the continuous advancement of graphene growth is the synergy of experimental and theoretical investigations.
On the Distribution of Orbits of $\mathrm{PGL_2(q)}$ in ${\mathbbF}_{q^n}$ and the Klapper Conjecture
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2010
DOI: 10.1137/090770746
Hausdorff dimension of the large values of Weyl sums
Publisher: Elsevier BV
Date: 09-2020
On short products of primes in arithmetic progressions
Publisher: American Mathematical Society (AMS)
Date: 16-11-2019
DOI: 10.1090/PROC/14289
A Lower Bound for Primality
Publisher: Elsevier BV
Date: 03-2001
On the distribution and lattice structure of nonlinear congruential pseudorandom numbers
Publisher: Elsevier BV
Date: 07-1999
Averaging Operators Over Homogeneous Varieties Over Finite Fields
Publisher: Springer Science and Business Media LLC
Date: 24-02-2015
Why Carbon Nanotubes Grow
Publisher: American Chemical Society (ACS)
Date: 17-03-2022
DOI: 10.1021/JACS.2C00879
Abstract: Despite three decades of intense research efforts, the most fundamental question "why do carbon nanotubes grow?" remains unanswered. In fact, carbon nanotubes (CNTs) should not grow since the encapsulation of a catalyst with graphitic carbon is energetically more favorable than CNT growth in every aspect. Here, we answer this question using a theoretical model based on extensive first-principles and molecular dynamics calculations. We reveal a historically overlooked yet fundamental aspect of the CNT-catalyst interface, viz., that the interfacial energy of the CNT-catalyst edge is contact angle-dependent. The contact angle increases via graphitic cap lift-off, drastically decreasing the interfacial formation energy by up to 6-9 eV/nm, overcoming van der Waals cap-catalyst adhesion, and driving CNT growth. Mapping this remarkable and simple interplay allows us to understand, for the first time, why CNTs grow.
On quadratic fields generated by polynomials
Publisher: Springer Science and Business Media LLC
Date: 02-10-2008
Counting Co-cyclic lattices
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2016
DOI: 10.1137/15M103950X
On the number of Diophantine m-tuples in finite fields
Publisher: Elsevier BV
Date: 09-2023
On the Naor-Reingold pseudo-random function from elliptic curves
Publisher: Springer Science and Business Media LLC
Date: 08-2000
Double character sums over subgroups and intervals
Publisher: Cambridge University Press (CUP)
Date: 15-05-2014
DOI: 10.1017/S0004972714000227
Abstract: We estimate double sums $$\\begin{eqnarray}S_{{\\it\\chi}}(a,{\\mathcal{I}},{\\mathcal{G}})=\\mathop{\\sum }\\limits_{x\\in {\\mathcal{I}}}\\mathop{\\sum }\\limits_{{\\it\\lambda}\\in {\\mathcal{G}}}{\\it\\chi}(x+a{\\it\\lambda}),\\quad 1\\leq a -1,\\end{eqnarray}$$ with a multiplicative character ${\\it\\chi}$ modulo $p$ where ${\\mathcal{I}}=\\{1,\\dots ,H\\}$ and ${\\mathcal{G}}$ is a subgroup of order $T$ of the multiplicative group of the finite field of $p$ elements. A nontrivial upper bound on $S_{{\\it\\chi}}(a,{\\mathcal{I}},{\\mathcal{G}})$ can be derived from the Burgess bound if $H\\geq p^{1/4+{\\it\\varepsilon}}$ and from some standard elementary arguments if $T\\geq p^{1/2+{\\it\\varepsilon}}$ , where ${\\it\\varepsilon} $ is arbitrary. We obtain a nontrivial estimate in a wider range of parameters $H$ and $T$ . We also estimate double sums $$\\begin{eqnarray}T_{{\\it\\chi}}(a,{\\mathcal{G}})=\\mathop{\\sum }\\limits_{{\\it\\lambda},{\\it\\mu}\\in {\\mathcal{G}}}{\\it\\chi}(a+{\\it\\lambda}+{\\it\\mu}),\\quad 1\\leq a -1,\\end{eqnarray}$$ and give an application to primitive roots modulo $p$ with three nonzero binary digits.
Erratum: Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithm (International Mathematics Research Notices)
Publisher: Oxford University Press (OUP)
Date: 2009
DOI: 10.1093/IMRN/RNP041
On functional graphs of quadratic polynomials
Publisher: Informa UK Limited
Date: 29-11-2019
SYSTEMS OF CONGRUENCES WITH PRODUCTS OF VARIABLES FROM SHORT INTERVALS
Publisher: Cambridge University Press (CUP)
Date: 11-11-2016
DOI: 10.1017/S0004972715001240
Abstract: We obtain an upper bound for the number of solutions to the system of $m$ congruences of the type $$\\begin{eqnarray}\\displaystyle \\mathop{\\prod }_{i=1}^{{\\it\\nu}}(x_{i}+s_{i})\\equiv {\\it\\lambda}_{j}~(\\text{mod }p)\\quad j=1,\\ldots ,m, & & \\displaystyle \\nonumber\\end{eqnarray}$$ modulo a prime $p$ , with variables $1\\leq x_{i}\\leq h$ , $i=1,\\ldots ,{\\it\\nu}$ and arbitrary integers $s_{j},{\\it\\lambda}_{j}$ , $j=1,\\ldots ,m$ , for a parameter $h$ significantly smaller than $p$ . We also mention some applications of this bound.
On the distribution of points on multidimensional modular hyperbolas
Publisher: Project Euclid
Date: 03-2007
DOI: 10.3792/PJAA.83.5
On the bit security of the Diffie-Hellman key
Publisher: Springer Science and Business Media LLC
Date: 07-11-2006
Order of torsion for reduction of linearly independent points for a family of Drinfeld modules
Publisher: Elsevier BV
Date: 04-2022
An identification scheme based on sparse polynomials
Publisher: Springer Berlin Heidelberg
Date: 2000
Complexity of inverting the Euler function
Publisher: American Mathematical Society (AMS)
Date: 23-01-2006
DOI: 10.1090/S0025-5718-06-01826-6
Abstract: Given an integer n n , how hard is it to find the set of all integers m m such that φ ( m ) = n \\varphi (m) = n , where φ \\varphi is the Euler totient function? We present a certain basic algorithm which, given the prime number factorization of n n , in polynomial time “on average” (that is, ( log n ) O ( 1 ) (\\log n)^{O(1)} ), finds the set of all such solutions m m . In fact, in the worst case this set of solutions is exponential in log n \\log n , and so cannot be constructed by a polynomial time algorithm. In the opposite direction, we show, under a widely accepted number theoretic conjecture, that the Partition Problem , an NP -complete problem, can be reduced in polynomial (in the input size) time to the problem of deciding whether φ ( m ) = n \\varphi (m) = n has a solution, for polynomially (in the input size of the Partition Problem ) many values of n n (where the prime factorizations of these n n are given). What this means is that the problem of deciding whether there even exists a solution m m to φ ( m ) = n \\varphi (m) = n , let alone finding any or all such solutions, is very likely to be intractable. Finally, we establish close links between the problem of inverting the Euler function and the integer factorization problem.
Distribution of short subsequences of inversive congruential pseudorandom numbers modulo $2^t$
Publisher: American Mathematical Society (AMS)
Date: 09-09-2019
DOI: 10.1090/MCOM/3467
Exponential sums over points of elliptic curves with reciprocals of primes
Publisher: Wiley
Date: 13-07-2012
Reductions modulo primes of systems of polynomial equations and algebraic dynamical systems
Publisher: American Mathematical Society (AMS)
Date: 03-05-2018
DOI: 10.1090/TRAN/7437
Abstract: We give bounds for the number and the size of the primes p p such that a reduction modulo p p of a system of multivariate polynomials over the integers with a finite number T T of complex zeros does not have exactly T T zeros over the algebraic closure of the field with p p elements. We apply these bounds to study the periodic points and the intersection of orbits of algebraic dynamical systems over finite fields. In particular, we establish some links between these problems and the uniform dynamical Mordell–Lang conjecture.
On reducing a system of equations to a single equation
Publisher: ACM
Date: 04-07-2004
Some arithmetic properties of recurrence sequences
Publisher: Springer Science and Business Media LLC
Date: 06-1990
DOI: 10.1007/BF01170895
On numbers n dividing the nth term of a linear recurrence
Publisher: Cambridge University Press (CUP)
Date: 23-02-2012
DOI: 10.1017/S0013091510001355
Abstract: We give upper and lower bounds on the count of positive integers n ≤ x iding the n th term of a non-degenerate linearly recurrent sequence with simple roots.
On Exponential Sums with Sparse Polynomials and Rational Functions
Publisher: Elsevier BV
Date: 10-1996
Congruences and Rational Exponential Sums with the Euler Function
Publisher: Rocky Mountain Mathematics Consortium
Date: 10-2006
On the embedding degree of reductions of an elliptic curve
Publisher: Elsevier BV
Date: 06-2009
ON the ARITHMETIC STRUCTURE of RATIONAL NUMBERS in the CANTOR SET
Publisher: Cambridge University Press (CUP)
Date: 27-04-2020
DOI: 10.1017/S0004972720000386
Abstract: We obtain a lower bound on the largest prime factor of the denominator of rational numbers in the Cantor set. This gives a stronger version of a recent result of Schleischitz [‘On intrinsic and extrinsic rational approximation to Cantor sets’, Ergodic Theory Dyn. Syst. to appear] obtained via a different argument.
Distances from differences of roots of polynomials to the nearest integers
Publisher: Elsevier BV
Date: 09-1992
On decimations of ℓ-sequences
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2004
Multiplicative Energy of Shifted Subgroups and Bounds On Exponential Sums with Trinomials in Finite Fields
Publisher: Canadian Mathematical Society
Date: 20-11-2018
Abstract: We prove a new bound on collinear triples in subgroups of prime finite fields and use it to give some new bounds on exponential sums with trinomials.
On small solutions to quadratic congruences
Publisher: Elsevier BV
Date: 06-2011
Bounds on average values of double incomplete Kloosterman sums
Publisher: Elsevier BV
Date: 10-2019
On quadratic fields generated by the Shanks sequence
Publisher: Cambridge University Press (CUP)
Date: 23-09-2009
DOI: 10.1017/S001309150700123X
Abstract: Let u ( n )= f ( g n ), where g 1 is integer and f ( X ) ∈ ℤ[ X ] is non-constant and has no multiple roots. We use the theory of $\\mathcal{S}$ -unit equations as well as bounds for character sums to obtain a lower bound on the number of distinct fields among $\\mathbb{Q}(\\sqrt{u(n)})$ for n ∈ $\\{M+1,\\dots,M+N\\}$ . Fields of this type include the Shanks fields and their generalizations.
The average sensitivity of square-freeness
Publisher: Springer Science and Business Media LLC
Date: 11-2000
DOI: 10.1007/PL00001600
Constructions of Approximately Mutually Unbiased Bases
Publisher: Springer Berlin Heidelberg
Date: 2006
DOI: 10.1007/11682462_72
Waring problem with factorials
Publisher: Cambridge University Press (CUP)
Date: 04-2005
DOI: 10.1017/S0004972700038223
Abstract: We show that any residue class λ modulo p can be represented in the form n 1 ! +…+ n ℓ ! ≡ λ (mod p ) with ℓ = O ((log p ) 3 log log p ).
Distribution of exponential functions modulo a prime power
Publisher: Elsevier BV
Date: 10-2014
Orders of gauss periods in finite fields
Publisher: Springer Berlin Heidelberg
Date: 1995
DOI: 10.1007/BFB0015425
APPROXIMATION BY SEVERAL RATIONALS
Publisher: Cambridge University Press (CUP)
Date: 04-2008
DOI: 10.1017/S0004972708000373
Abstract: Following T. H. Chan, we consider the problem of approximation of a given rational fraction a / q by sums of several rational fractions a 1 / q 1 ,…, a n / q n with smaller denominators. We show that in the special cases of n =3 and n =4 and certain admissible ranges for the denominators q 1 ,…, q n , one can improve a result of T. H. Chan by using a different approach.
Distributional properties of the largest prime factor
Publisher: Michigan Mathematical Journal
Date: 12-2005
ON STABLE QUADRATIC POLYNOMIALS
Publisher: Cambridge University Press (CUP)
Date: 29-03-2012
DOI: 10.1017/S001708951200002X
Abstract: We recall that a polynomial f ( X ) ∈ K [ X ] over a field K is called stable if all its iterates are irreducible over K . We show that almost all monic quadratic polynomials f ( X ) ∈ ℤ[ X ] are stable over ℚ. We also show that the presence of squares in so-called critical orbits of a quadratic polynomial f ( X ) ∈ ℤ[ X ] can be detected by a finite algorithm this property is closely related to the stability of f ( X ). We also prove there are no stable quadratic polynomials over finite fields of characteristic 2 but they exist over some infinite fields of characteristic 2.
On irreducible polynomials of small height over finite fields
Publisher: Springer Science and Business Media LLC
Date: 11-1996
DOI: 10.1007/BF01293260
Character sums with fermat quotients
Publisher: Oxford University Press (OUP)
Date: 24-08-2011
DOI: 10.1093/QMATH/HAQ028
Pseudorandomness and Dynamics of Fermat Quotients
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2011
DOI: 10.1137/100798466
ON FINDING SOLUTIONS TO EXPONENTIAL CONGRUENCES
Publisher: Cambridge University Press (CUP)
Date: 27-12-2019
DOI: 10.1017/S0004972718001302
Abstract: We improve some previously known deterministic algorithms for finding integer solutions $x,y$ to the exponential equation of the form $af^{x}+bg^{y}=c$ over finite fields.
Theory of sigma bond resonance in flat boron materials
Publisher: Research Square Platform LLC
Date: 15-12-2022
DOI: 10.21203/RS.3.RS-2380764/V1
Abstract: In chemistry, theory of aromaticity and π bond resonance plays a central role in intuitively understanding the stability and properties of organic molecules. Here we present an analogue theory for σ bond resonance in flat boron materials, which allows us to determine the distribution of two-center two-electron and three-center two-electron bonds without quantum calculations. Based on this theory, three rules are proposed to draw the Kekulé-like bonding configurations for flat boron materials and to explore their properties intuitively. As an application of this theory, the mystery of why neutral borophene with ~ 1/9 hole has the highest stability and the effect of charge doping on borophene’s optimal hole concentration are understood with the assumption of σ and π orbital occupation balance. Like the aromaticity theory for carbon materials, this theory greatly deepens our understanding on boron materials and paves the way for the rational design of various boron-based materials.
On the concentration of points of polynomial maps and applications
Publisher: Springer Science and Business Media LLC
Date: 27-11-2011
On the generalized joint linear complexity profile of a class of nonlinear pseudorandom multisequences
Publisher: American Institute of Mathematical Sciences (AIMS)
Date: 2010
Character sums over integers with restricted $g$-ary digits
Publisher: Duke University Press
Date: 07-2002
Square-free values of the Carmichael function
Publisher: Elsevier BV
Date: 11-2003
Geometric progressions in sumsets over finite fields
Publisher: Springer Science and Business Media LLC
Date: 06-2007
On irreducible divisors of iterated polynomials
Publisher: European Mathematical Society - EMS - Publishing House GmbH
Date: 2014
DOI: 10.4171/RMI/809
On the number of solutions of exponential congruences
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2011
DOI: 10.4064/AA148-1-7
Divisor problem in arithmetic progressions modulo a prime power
Publisher: Elsevier BV
Date: 02-2018
On primitive elements in finite fields and on elliptic curves
Publisher: IOP Publishing
Date: 28-02-1992
Twisted exponential sums over points of elliptic curves
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2011
DOI: 10.4064/AA148-1-6
Character sums with exponential functions
Publisher: Wiley
Date: 12-2000
Values of the Euler Function in Various Sequences
Publisher: Springer Science and Business Media LLC
Date: 06-04-2005
On the fractional parts of an/n
Publisher: Wiley
Date: 30-09-2013
DOI: 10.1112/BLMS/BDS084
Multiplicative Dependence Among Iterated Values of Rational Functions Modulo Finitely Generated Groups
Publisher: Oxford University Press (OUP)
Date: 27-05-2019
DOI: 10.1093/IMRN/RNZ091
Abstract: We study multiplicative dependence between elements in orbits of algebraic dynamical systems over number fields modulo a finitely generated multiplicative subgroup of the field. We obtain a series of results, many of which may be viewed as a blend of Northcott’s theorem on boundedness of preperiodic points and Siegel’s theorem on finiteness of solutions to $S$-unit equations.
Sparsity of curves and additive and multiplicative expansion of rational maps over finite fields
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2019
Exponential sums and congruences with factorials
Publisher: Walter de Gruyter GmbH
Date: 07-2005
MULTIPLICATIVE ORDERS IN ORBITS OF POLYNOMIALS OVER FINITE FIELDS
Publisher: Cambridge University Press (CUP)
Date: 23-10-2018
DOI: 10.1017/S0017089517000222
Abstract: We show, under some natural restrictions, that orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime p . We also show that for all but finitely many initial points either the multiplicative order of this point or the length of the orbit it generates (both modulo a large prime p ) is large. The approach is based on the results of Dvornicich and Zannier ( Duke Math. J. 139 (2007), 527–554) and Ostafe (2017) on roots of unity in polynomial orbits over the algebraic closure of the field of rational numbers.
Double Character Sums with Intervals and Arbitrary Sets
Publisher: Pleiades Publishing Ltd
Date: 11-2018
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