Formes quadratiques de discriminants emboîtés
It addresses security analysis for cryptographic systems using quadratic forms, but appears incremental as it builds on existing methods for known bottlenecks.
This paper tackles the problem of analyzing cryptographic systems like Nice by establishing precise links between quadratic forms with discriminants D and Df^2, and introduces the concept of semi-equivalence for binary quadratic forms to characterize these relationships.
Quadratic forms with embedded discriminants. Integral binary quadratic forms have multiple applications, for example in factorization or cryptography. The Nice family of cryptographic systems makes use of quadratic forms with different discriminants $\pm p$, and $\pm pq^2$ where $p$, $q$ are large primes. This paper shows the precise links between forms with $D$ discriminant and forms with $Df^2$ discriminant, which are crucial in the analysis of the systems Nice and theirs attacks. We also introduce the notion of semi-equivalence of binary quadratic forms, and give some characterizations of semi-equivalent forms, which are useful in the analysis of these attacks. ----- Les formes quadratiques binaires fournissent un moyen explicite pour manipuler des idéaux de corps quadratiques, et leurs applications pratiques sont multiples. De nombreux algorithmes de factorisation les utilisent. Elle sont aussi utilisées en cryptographie, en particulier pour les systèmes Nice. Les systèmes de chiffrement Nice utilisent des formes quadratiques de discriminants $\pm p$ et $\pm pq^2$ où $p$ et $q$ sont des nombres premiers. Cet article précise les liens entre les formes de discriminant $D$ et celles de discriminant $Df^2$, ce qui est essentiel pour l'analyse de Nice et de ses attaques. Il introduit aussi la notion de formes quadratiques semi-équivalentes et en explicite plusieurs caractérisations, utiles pour l'analyse de ces attaques.