Description
The Polynomial Modular Number System (PMNS) is an integer number system that aims to speed up arithmetic operations modulo a prime number p. This system is defined by a tuple (p, n, g, r, E), where p, n, g and r are positive integers, and E is a polynomial with integer coefficients, having g as a root modulo p.
Arithmetic operations in PMNS rely heavily on Euclidean lattices. Modular reduction in this system is done using a lattice of zeros L (here, the set of polynomials in Z[X], with degrees smaller than n, having g as a root modulo p).
Many works have shown that the PMNS can be an efficient alternative to the classical representation for modular arithmetic and cryptographic size integers.
In this presentation, we first present the PMNS and its arithmetic. Next, we introduce new properties of the lattice L, regarding a Montgomery-like coefficient reduction method. Then, we study the redundancy in the PMNS and explain how to choose the parameters for the desired redundancy in the system. Finally, we show how to use some properties of Euclidean lattices for efficient modular arithmetic and equality test within the PMNS.
Reference: F. Y. Dosso, A. Berzati, N. El Mrabet, and J. Proy. PMNS revisited for consistent redundancy and equality test. Cryptology ePrint Archive, Paper 2023/1231, (\url{https://eprint.iacr.org/2023/1231})
Infos pratiques
Prochains exposés
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Cycles of pairing-friendly abelian varieties
Orateur : Maria Corte-Real Santos - ENS Lyon
A promising avenue for realising scalable proof systems relies on the existence of 2-cycles of pairing-friendly elliptic curves. More specifically, such a cycle consists of two elliptic curves E/Fp and E’/Fq that both have a low embedding degree and also satisfy q = #E(Fp) and p = #E’(Fq). These constraints turn out to be rather restrictive; in the decade that has passed since 2-cycles were first[…]-
Cryptography
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