SPA Resistant Scalar Multiplication Based on Addition and Tripling Indistinguishable on Elliptic Curve Cryptosystem

Liu, Shuanggen; Yao, Huatong; Wang, Xu An

doi:10.1109/3pgcic.2015.20

Cited by 6 publications

(4 citation statements)

References 9 publications

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“…As an alternative to the binary expansion key representation, τ-adic expansions of the integer k with a non-binary basis τ were proposed to speed up the point multiplication (PM) and to improve the resistance against attacks [38,[43][44][45][46][47]. The τ-adic expansion results in the point multiplication…”

Section: Elliptic Curve Point Multiplicationmentioning

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: Elliptic Curve Point Multiplicationmentioning

confidence: 99%

Montgomery Reduction for Gaussian Integers

Safieh

Freudenberger

2021

Cryptography

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“…As proposed in [25], the non-binary expansions are beneficial to harden ECC implementations against attacks based on power analysis. The products κ i P(x, y) can be precomputed and stored in memory.…”

Section: Algorithm 2 τ-Adic Expansion Of Integers or Gaussian Integersmentioning

confidence: 99%

“…This method requires two different point operations, the point addition and the point doubling operation. Alternatively, τ-adic expansions of the integer k with a non-binary basis τ were proposed to speed up the point multiplication (PM) and to improve the resistance against attacks [20][21][22][23][24][25]. This τ-adic expansion results in the point multiplication kP = l−1 ∑ i=0 κ i τ i P = τ(.…”

Section: Introductionmentioning

confidence: 99%

“…In [26,28], arithmetic over binary extension field is considered, where the squaring operation is very efficient [29,30]. It was demonstrated in [22,24,25,[31][32][33][34] that non-binary expansions are beneficial to harden ECC implementations against attacks (SCA). Hardware implementations of the elliptic curve point multiplication are prone to SCA.…”

Section: Introductionmentioning

confidence: 99%

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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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