Euclidean addition chains scalar multiplication on curves with efficient endomorphism

Abstract

To cite this version:Fangan-Yssouf Dosso, Fabien Herbaut, Nicolas Méloni, Pascal Véron. Euclidean addition chains scalar multiplication on curves with efficient endomorphism. Journal of Cryptographic Engineering, Springer, In press, <10.1007/s13389-018-0190-0>. <hal-01794402>Noname manuscript No. (will be inserted by the editor) Euclidean addition chains scalar multiplication on curves with efficient endomorphismThis is a pre-print of an article published in "Journal of Cryptographic Engineering". The final authenticated version is available online at: https://doi.org/10.1007/s13389-018-0190-0.the date of receipt and acceptance should be inserted later Abstract Random Euclidean addition chain generation has proven to be an efficient low memory and SPA secure alternative to standard ECC scalar multiplication methods in the context of fixed base point [21]. In this work, we show how to generalize this method to random point scalar multiplication on elliptic curves with an efficiently computable endomorphism. In order to do so we generalize results from [21] on the relation of random Euclidean chains generation and elliptic curve point distribution obtained from those chains. We propose a software implementation of our method on various platforms to illustrate the impact of our approach. For that matter, we provide a comprehensive study of the practical computational cost of the modular multiplication when using Java and C standard libraries developed for the arithmetic over large integers.

Compact Variable-base ECC Scalar Multiplication using Euclidean Addition Chains

“…According to (Dosso et al, 2018), this approach fits well with the GLV-like context of an elliptic curve endowed with an efficiently computable endomorphism. Indeed, the method described in (Dosso et al, 2018) leads to a simple and compact way to compute the elliptic curve scalar multiplication kP (ECSM) for a Diffie-Hellman key exchange protocol. The main drawback of this method is that it requires the use of a larger curve as compared to other methods for the same security level.…”

“…The aim of this paper is to improve the method described in Section 4 of (Dosso et al, 2018). First, we obtain an asymptotic reduction of 7.97% of the size of the base field required.…”

“…Next, we modify the original ZADD algorithm to save one register over F p . For practical usage, we obtain a reduction of 21% of the memory used in the method described in (Dosso et al, 2018).…”

“…In comparison with the method proposed in (Dosso et al, 2018), the progress lies in the restriction of the random generation to a subset S of all Euclidean addition chains. Call the chaining (u, v) → (u, u+v) a small step and the chaining (u, v) → (v, u+ v) a big step.…”

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The random generation of Euclidean addition chains fits well with a GLV context (Dosso et al, 2018) and provides a method with decent performance despite the growth of the base field required to get the same level of security. The aim of this paper is to reduce the size of the base field required.

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