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

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

doi:10.1109/zinc50678.2020.9161769

Cited by 4 publications

(14 citation statements)

References 23 publications

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“…From the estimate (11) follows that it is advantageous to represent the key as a Gaussian integer. For any Gaussian integer ring x ∈ G n the absolute value |x| is bounded by |x| ≤ √ n/2 [41]. Hence, we can estimate the maximum number of digits l for the τ-adic expansion of a key k ∈ G n as…”

Section: Elliptic Curve Point Multiplication For τ-Adic Expansionmentioning

confidence: 99%

“…In this section, we discuss the resistance of the PM implementation against SCA. In particular, we consider the key expansion algorithm from [41] which improves the resistance against TA and SPA. We demonstrate that this key expansion employing Gaussian integers reduces the memory requirements and computational complexity compared with other τ-adic expansions.…”

Section: Resistance Against Side-channel Attacksmentioning

confidence: 99%

“…To protect the PM against such attacks, Algorithm 4 was recommended in [41] for the key expansions. This algorithm replaces all values κ i = 0 by κ i = τ.…”

Section: Resistance Against Side-channel Attacksmentioning

confidence: 99%

“…In particular, we consider the PM for Gaussian integers, where both the key as well as τ are Gaussian integers. The material in this paper was presented in part at the Zooming Innovation in Consumer Electronics International Conference 2020 [41]. In [41] a new algorithm for the τ-adic expansion of the key k was presented.…”

Section: Introductionmentioning

confidence: 99%

“…The material in this paper was presented in part at the Zooming Innovation in Consumer Electronics International Conference 2020 [41]. In [41] a new algorithm for the τ-adic expansion of the key k was presented. The resulting expansion increases the robustness against TA and SPA and reduces the required memory size and the computational complexity for a protected PM.…”

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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Section: Elliptic Curve Point Multiplication For τ-Adic Expansionmentioning

confidence: 99%

Section: Resistance Against Side-channel Attacksmentioning

confidence: 99%

“…To protect the PM against such attacks, Algorithm 4 was recommended in [41] for the key expansions. This algorithm replaces all values κ i = 0 by κ i = τ.…”

Section: Resistance Against Side-channel Attacksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Thiers

Safieh

Freudenberger

2020

2020 IEEE 10th International Conference on Consumer Electronics (ICCE-Berlin)

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

Cited by 4 publications

References 23 publications

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

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

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

Montgomery Reduction for Gaussian Integers

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