Modular Number Systems: Beyond the Mersenne Family

Bajard, Jean-Claude; Imbert, Laurent; Plantard, Thomas

doi:10.1007/978-3-540-30564-4_11

Cited by 35 publications

(50 citation statements)

References 6 publications

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“…they do not take into account the possible special form of the modulus. When implementing ECC/HECC algorithms, it is a good idea to use primes that allow fast modular arithmetic, such as those recommended by the NIST [39], the SEC Group [43], or more generally the primes belonging to what Bajard et al called the Mersenne family [42,11,5]. In these cases, the multiplication becomes much more efficient than the inversion.…”

Section: Elliptic Curve Cryptographymentioning

confidence: 99%

The double-base number system and its application to elliptic curve cryptography

2007

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Abstract. We describe an algorithm for point multiplication on generic elliptic curves, based on a representation of the scalar as a sum of mixed powers of 2 and 3. The sparseness of this so-called double-base number system, combined with some efficient point tripling formulae, lead to efficient point multiplication algorithms for curves defined over both prime and binary fields. Side-channel resistance is provided thanks to side-channel atomicity.

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Section: Elliptic Curve Cryptographymentioning

confidence: 99%

The double-base number system and its application to elliptic curve cryptography

2007

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“…In this section, we investigate if Algorithm 3 can be applied also in modular number systems (MNS) proposed in [3,4] by Bajard et. al.…”

Section: Application To Modular Number Systems?mentioning

confidence: 99%

“…In Section 7, we consider the applications of LWPFIs. In Section 8, we consider applying our coefficient reduction method to modular number systems [3,4].…”

Section: Introductionmentioning

confidence: 99%

Montgomery Reduction Algorithm for Modular Multiplication Using Low-Weight Polynomial Form Integers

Chung

Hasan

2007

18th IEEE Symposium on Computer Arithmetic (ARITH '07)

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Abstract. We extend low-weight polynomial form integers (LWPFIs) presented in [5]. An LWPFI p is an integer expressed as a degree-l, monic polynomial such that p = t l + f l−1 t l + · · · + f1t + f0, where t can be any positive integer. In [5], fi's are limited to 0 and ±1, but here we let |fi| ≤ ξ for some small positive integer ξ. In modular multiplication based on LWPFI, elements in Zp are expressed in polynomial in t and multiplication is performed in Z[t]/f (t). The coefficients must be reduced for subsequent modular multiplications. In [5], a coefficient reduction algorithm based on a division algorithm derived from the Barrett reduction algorithm is presented. In this report, we present a coefficient reduction algorithm based on the Montgomery reduction algorithm and its detailed analysis results. Bounds on the input and output of our coefficient reduction algorithm is carefully analyzed. We give conditions for eliminating the final subtractions at the end of the Montgomery reduction algorithm. In addition, we present efficient modular addition and subtraction methods using LWPFI moduli.

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“…We introduce a novel binary field representation, the Double Polynomial System (DPS), inspired from AMNS number system of Bajard et al (J.-C. Bajard, 2005).…”

Section: Dps Representationmentioning

confidence: 99%

Subquadratic Binary Field Multiplier in Double Polynomial System

Giorgi¹,

Nègre²,

Plantard³

2007

Proceedings of the Second International Conference on Security and Cryptography

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We propose a new space efficient operator to multiply elements lying in a binary field F 2 k . Our approach is based on a novel system of representation called Double Polynomial System which set elements as a bivariate polynomials over F 2 . Thanks to this system of representation, we are able to use a Lagrange representation of the polynomials and then get a logarithmic time multiplier with a space complexity of O(k 1:31 ) improving previous best known method. Disciplines Physical Sciences and Mathematics Publication DetailsGiorgi, P., Negre, C. & Plantard, T. (2007) Abstract:We propose a new space efficient operator to multiply elements lying in a binary field F 2 k . Our approach is based on a novel system of representation called Double Polynomial System which set elements as a bivariate polynomials over F 2 . Thanks to this system of representation, we are able to use a Lagrange representation of the polynomials and then get a logarithmic time multiplier with a space complexity of O(k 1.31 ) improving previous best known method.

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Modular Number Systems: Beyond the Mersenne Family

Cited by 35 publications

References 6 publications

The double-base number system and its application to elliptic curve cryptography

The double-base number system and its application to elliptic curve cryptography

Montgomery Reduction Algorithm for Modular Multiplication Using Low-Weight Polynomial Form Integers

Subquadratic Binary Field Multiplier in Double Polynomial System

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