Description
This talk is about computing discrete logarithms in non-prime finite fields. These fields arise in pairing-based cryptography. In this setting, the pairing-friendly curve is defined over GF(q) and the pairing takes its values in an extension GF(q^k), where k is the embedding degree.<br/> Fr example, GF(p^2) is the embedding field of supersingular elliptic curves in large characteristic; GF(p^3), GF(p^4), GF(p^6) are the embedding fields of MNT curves, and GF(p^12) is the embedding field of the well-known Barreto-Naehrig curves. In small characteristic, GF(2^(4*n)), GF(3^(6*m)) are considered. To compute discrete logarithms in these fields, one uses the Number Field Sieve algorithm (NFS) in large characteristic (e.g. GF(p^2)), the NFS-High-Degree variant (NFS-HD) in medium characteristic (e.g. GF(p^12)) and the Quasi Polynomial-time Algorithm (QPA) in small characteristic when applicable. These algorithms are made of four steps: polynomial selection, relation collection, linear algebra modulo the prime order of the target group and finally, individual logarithm computation.<br/> All these finite fields are extensions, hence have subfields. We use their structure to speed-up the individual discrete logarithm computation. We obtain an important speed-up in practice and the best case is when the embedding degree k is even. We will illustrate the improvements with the practical case of GF(p^4) with p^4 of 400 bits (120 decimal digits).
Prochains exposés
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Verification of Rust Cryptographic Implementations with Aeneas
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From secure communications to online banking, cryptography is the cornerstone of most modern secure applications. Unfortunately, cryptographic design and implementation is notoriously error-prone, with a long history of design flaws, implementation bugs, and high-profile attacks. To address this issue, several projects proposed the use of formal verification techniques to statically ensure the[…] -
On the average hardness of SIVP for module lattices of fixed rank
Orateur : Radu Toma - Sorbonne Université
In joint work with Koen de Boer, Aurel Page, and Benjamin Wesolowski, we study the hardness of the approximate Shortest Independent Vectors Problem (SIVP) for random module lattices. We use here a natural notion of randomness as defined originally by Siegel through Haar measures. By proving a reduction, we show it is essentially as hard as the problem for arbitrary instances. While this was[…] -
Endomorphisms via Splittings
Orateur : Min-Yi Shen - No Affiliation
One of the fundamental hardness assumptions underlying isogeny-based cryptography is the problem of finding a non-trivial endomorphism of a given supersingular elliptic curve. In this talk, we show that the problem is related to the problem of finding a splitting of a principally polarised superspecial abelian surface. In particular, we provide formal security reductions and a proof-of-concept[…]-
Cryptography
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