Side Channel Attack Resistance of the Elliptic Curve Point Multiplication using Eisenstein Integers

Thiers, Johann-Philipp; Safieh, Malek; Freudenberger, Jürgen

doi:10.1109/icce-berlin50680.2020.9352202

Cited by 2 publications

(1 citation statement)

References 20 publications

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“…Eisenstein integers are complex numbers of the form x = a + bω, where ω = 1 2 • 1 + √ −3 , and Eisenstein integer fields can be constructed for primes of the form p mod 6 = 1. An elliptic curve point multiplication using Eisenstein integers was considered in [49] showing similar properties as the point multiplication over Gaussian integers. Up to now no efficient modulo operation for Eisenstein integers is known.…”

Section: Discussionmentioning

confidence: 99%

Montgomery Reduction for Gaussian Integers

Safieh

Freudenberger

2021

Cryptography

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Modular arithmetic over integers is required for many cryptography systems. Montgomery reduction is an efficient algorithm for the modulo reduction after a multiplication. Typically, Montgomery reduction is used for rings of ordinary integers. In contrast, we investigate the modular reduction over rings of Gaussian integers. Gaussian integers are complex numbers where the real and imaginary parts are integers. Rings over Gaussian integers are isomorphic to ordinary integer rings. In this work, we show that Montgomery reduction can be applied to Gaussian integer rings. Two algorithms for the precision reduction are presented. We demonstrate that the proposed Montgomery reduction enables an efficient Gaussian integer arithmetic that is suitable for elliptic curve cryptography. In particular, we consider the elliptic curve point multiplication according to the randomized initial point method which is protected against side-channel attacks. The implementation of this protected point multiplication is significantly faster than comparable algorithms over ordinary prime fields.

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Section: Discussionmentioning

confidence: 99%

Montgomery Reduction for Gaussian Integers

Safieh

Freudenberger

2021

Cryptography

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A Compact Coprocessor for the Elliptic Curve Point Multiplication over Gaussian Integers

2020

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This work presents a new concept to implement the elliptic curve point multiplication (PM). This computation is based on a new modular arithmetic over Gaussian integer fields. Gaussian integers are a subset of the complex numbers such that the real and imaginary parts are integers. Since Gaussian integer fields are isomorphic to prime fields, this arithmetic is suitable for many elliptic curves. Representing the key by a Gaussian integer expansion is beneficial to reduce the computational complexity and the memory requirements of secure hardware implementations, which are robust against attacks. Furthermore, an area-efficient coprocessor design is proposed with an arithmetic unit that enables Montgomery modular arithmetic over Gaussian integers. The proposed architecture and the new arithmetic provide high flexibility, i.e., binary and non-binary key expansions as well as protected and unprotected PM calculations are supported. The proposed coprocessor is a competitive solution for a compact ECC processor suitable for applications in small embedded systems.

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Side Channel Attack Resistance of the Elliptic Curve Point Multiplication using Eisenstein Integers

Cited by 2 publications

References 20 publications

Montgomery Reduction for Gaussian Integers

Montgomery Reduction for Gaussian Integers

A Compact Coprocessor for the Elliptic Curve Point Multiplication over Gaussian Integers

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