Continued Fractions and Torsion on Hyperelliptic Curves. Scientific advance should not blindly add to our knowledge; a true advance brings insights that collapse different issues into one. Understanding more is to need to remember less. For an important class of examples, this project identifies the study of a fundamental invariant of a quadratic number field, its regulator and hence its class number, with maximum torsion on the Jacobian variety of an hyperelliptic curve. The investigator's meth ....Continued Fractions and Torsion on Hyperelliptic Curves. Scientific advance should not blindly add to our knowledge; a true advance brings insights that collapse different issues into one. Understanding more is to need to remember less. For an important class of examples, this project identifies the study of a fundamental invariant of a quadratic number field, its regulator and hence its class number, with maximum torsion on the Jacobian variety of an hyperelliptic curve. The investigator's methods will surprise some longstanding problems into submission and in particular will lead them to reveal full data on torsion on hyperelliptic curves of low genus.
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Lattices as a constructive and destructive cryptographic tool. The project is driven by the great number of potential applications of deep mathematical and algorithmic methods to different areas of modern cryptography. These areas provide a solid platform for more applied fields such as Computer and Information Security and E-commerce. It will lead to commercialisation and everyday-life improvements.
Homomorphic cryptography: computing on encrypted data. This project is driven by the groundbreaking applications of a new cryptographic technology that allows analysis of encrypted (scrambled) data without needing to decrypt (unscramble) it first. The results of this project can be used to enable secure remote data storage, electronic auctions and voting, and protecting medical records.