An Alternative Approach for SIDH Arithmetic

Abstract

In this paper, we present new algorithms for the field arithmetic layers of supersingular isogeny Diffie-Hellman; one of the fifteen remaining candidates in the NIST post-quantum standardization process. Our approach uses a polynomial representation of the field elements together with mechanisms to keep the coefficients within bounds during the arithmetic operations. We present timings and comparisons for SIKEp503 and suggest a novel 736-bit prime that offers a 1.17× speedup compared to SIKEp751 for a similar level of security.

On Polynomial Modular Number Systems over $ \mathbb{Z}/{p}\mathbb{Z} $

“…Since the definition of the PMNS representation system, all the research focused on the polynomial E(X) = X n − λ because the external reduction can be efficiently performed when λ is "small" (often a power of 2 to use logical operator) [6,48,26,25,22,23,14,18,49]. Now, from proposition 4.2, we know that the size of the coefficients used in the PMNS representation system depends on the parameter s which in turn depends on the coefficients of the polynomial E(X) since s = M T ∞ where M is the (2n−1)×n matrix whose rows are the coefficients of X i mod E(X).…”

“…Moreover, it offers also competitive timings on an ARM v8 architecture or a STM32F4 board. In [14], the authors extend the AMNS representation system to F p k and show how it can be used in order to improve the performances of SIKE [35], one of the alternate KEM candidate of the NIST post-quantum standardization process [46]. A first hardware implementation of the AMNS is described in [17].…”

PMNS for Efficient Arithmetic and Small Memory Cost

“…The optimisation of the internal reduction is discussed in [28]. Practical use of the PMNS in the area of cryptography is described in [17], [16], [11], [13]. The redundancy of this representation system is used to protect modular operations in cryptographic protocols [14], [27].…”

“…• in [13], the efficiency has been confirmed in the MPHELL library for other platforms (Armv8, STMF32F4), • in [11], the authors generalized the representation system to the field F p k and improved this way the performances of the SIKE protocol [3], one of the alternate KEM candidate of the NIST post-quantum standardization process [2].…”

An Alternative Approach to Polynomial Modular Number System Internal Reduction

“…Two approaches have been proposed to perform it based either on Barrett modular reduction algorithm [3] or Montgomery's one [14]. During the past few years improvements have been made on the implementation [10], generation [4], [8], randomization [9] and generalization [6], [11] of PMNS in various contexts. One interesting common feature between all those works is that they all perform the internal reduction step using the Montgomery-like approach.…”