Pascal Véron

Jean-Marc Robert, Pascal Véron

Code-based cryptography is one of the main propositions for the post-quantum cryptographic context, and several protocols of this kind have been submitted on the NIST platform. Among them, BIKE and HQC are part of the five alternate candidates selected in the third round of the NIST standardization process in the KEM category. These two schemes make use of multiplication of large polynomials over binary rings, and due to the polynomial size (from 10,000 to 60,000 bits), this operation is one of the costliest during key generation, encapsulation, or decapsulation mechanisms. In this work, we revisit the different existing constant-time algorithms for arbitrary polynomial multiplication. We explore the different Karatsuba and Toom-Cook constructions in order to determine the best combinations for each polynomial degree range, in the context of AVX2 and AVX512 instruction sets. This leads to different kernels and constructions in each case. In particular, in the context of AVX512, we use the VPCLMULQDQ instruction, which is a vectorized binary polynomial multiplication instruction. This instruction deals with up to four polynomial (of degree up to 63) multiplications, the four results being stored in one single 512-bit word. This allows to divide by roughly 3 the retired instruction number of the operation in comparison with the AVX2 instruction set implementations, while the speedup is up to 39% in terms of processor clock cycles. These results are different than the ones estimated in Drucker (Fast multiplication of binary polynomials with the forthcoming vectorized vpclmulqdq instruction, 2018). To illustrate the benefit of the new VPCLMULQDQ instruction, we used the HQC code to evaluate our approaches. When implemented in the HQC protocol, for the security levels 128, 192, and 256, our approaches provide up to 12% speedup, for key pair generation.

Yoann Marquer, Tania Richmond, Pascal Véron

The iterative conditional branchings appear in various sensitive algorithms, like the modular exponentiation in the RSA cryptosystem or the scalar multiplication in ellipticcurve cryptography. In this paper, we abstract away the desirable security properties achieved by the Montgomery ladder, and formalize systems of equations necessary to obtain what we call the semi-interleaved and fully-interleaved ladder properties. This fruitful approach allows us to design novel fault-injection attacks, able to obtain some/all bits of the secret against different ladders, including the common Montgomery ladder. We also demonstrate the generality of our approach by applying the ladder equations to the modular exponentiation and the scalar multiplication, both in the semi-and fully-interleaved cases, thus proposing novel and more secure algorithms.

Laurent-Stéphane Didier, Fanga-Yssouf Dosso, Pascal Véron

The Adapted Modular Number System (AMNS) is a sytem of representation of integers to speed up arithmetic operations modulo a prime p. Such a system can be defined by a tuple (p, n, γ, ρ, E) where E is in Z[X]. In [13] conditions are given to build AMNS with E(X) = {X^n +1}. In this paper, we generalize their results and show how to generate multiple AMNS for a given prime p with E(X)={X^n-λ} and λ in Z. Moreover, we propose a complete set of algorithms without conditional branching to perform arithmetic and conversion operations in the AMNS, using a Montgomery-like method described in [15].