On the Use of the Negation Map in the Pollard Rho Method
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We explore a few algebraic and geometric structures, through certain questions posed by modern cryptography. We focus on the cases of discrete logarithms in finite fields of small characteristic, the structure of isogeny graphs of ordinary abelian varietie ...
Post-quantum cryptography is a branch of cryptography which deals with cryptographic algorithms whose hardness assumptions are not based on problems known to be solvable by a quantum computer, such as the RSA problem, factoring or discrete logarithms.This ...
Inspired by the work of Lang-Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over Q and by the subsequent generalization of Cojocaru-Davis-Silverberg-Stange to generic abelian varieties, we study the analogous que ...
We provide new explicit examples of lattice sphere packings in dimensions 54, 55, 162, 163, 486 and 487 that are the densest known so far, using Kummer families of elliptic curves over global function fields.In some cases, these families of elliptic curves ...
Abelian varieties are fascinating objects, combining the fields of geometry and arithmetic. While the interest in abelian varieties has long time been of purely theoretic nature, they saw their first real-world application in cryptography in the mid 1980's ...
Recently, several striking advances have taken place regarding the discrete logarithm problem (DLP) in finite fields of small characteristic, despite progress having remained essentially static for nearly thirty years, with the best known algorithms being ...
Current cryptographic solutions will become obsolete with the arrival of large-scale universal quantum computers. As a result, the National Institute of Standards and Technology supervises a post-quantum standardization process which involves evaluating ca ...
In 2013 and 2014 a revolution took place in the understanding of the discrete logarithm problem (DLP) in finite fields of small characteristic. Consequently, many cryptosystems based on cryptographic pairings were rendered completely insecure, which serves ...
Fix a prime number l. Graphs of isogenies of degree a power of l are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse gr ...
Isogeny-based cryptography is an instance of post-quantum cryptography whose fundamental problem consists of finding an isogeny between two (isogenous) elliptic curves E and E′. This problem is closely related to that of computing the endomorphism ring of ...