Speeding Up Bipartite Modular Multiplication

Knežević, Miroslav; Vercauteren, Fréderik; Verbauwhede, Ingrid

doi:10.1007/978-3-642-13797-6_12

Cited by 15 publications

(13 citation statements)

References 12 publications

(18 reference statements)

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“…Some computations (and storage) can be avoided when primes of the form p = 2 α (2 β − γ) − 1 are used for positive integers α, β and γ (cf. [40,38,1,32,15]). When the prime p is two bits short of a multiple of the word size w (i.e.…”

Section: Modular Arithmetic -Choosing Primesmentioning

confidence: 99%

Selecting elliptic curves for cryptography: an efficiency and security analysis

et al. 2015

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We select a set of elliptic curves for cryptography and analyze our selection from a performance and security perspective. This analysis complements recent curve proposals that suggest (twisted) Edwards curves by also considering the Weierstrass model. Working with both Montgomery-friendly and pseudo-Mersenne primes allows us to consider more possibilities which help to improve the overall efficiency of base field arithmetic. Our Weierstrass curves are backwards compatible with current implementations of prime order NIST curves, while providing improved efficiency and stronger security properties. We choose algorithms and explicit formulas to demonstrate that our curves support constant-time, exception-free scalar multiplications, thereby offering high practical security in cryptographic applications. Our implementation shows that variable-base scalar multiplication on the new Weierstrass curves at the 128-bit security level is about 1.4 times faster than the recent implementation record on the corresponding NIST curve. For practitioners who are willing to use a different curve model and sacrifice a few bits of security, we present a collection of twisted Edwards curves with particularly efficient arithmetic that are up to 1.42, 1.26 Michael Naehrig E-mail: mnaehrig@microsoft.com and 1.24 times faster than the new Weierstrass curves at the 128-, 192-and 256-bit security levels, respectively. Finally, we discuss how these curves behave in a realworld protocol by considering different scalar multiplication scenarios in the transport layer security (TLS) protocol. The proposed curves and the results of the analysis are intended to contribute to the recent efforts towards recommending new elliptic curves for Internet standards.

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Section: Modular Arithmetic -Choosing Primesmentioning

confidence: 99%

Selecting elliptic curves for cryptography: an efficiency and security analysis

et al. 2015

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“…A RPS based Montgomery multiplication algorithm has been described in [22]. However, there is no such algorithm for the BA-P. We cite [15,16,19] and the references therein. The mod polynomials that have been used are of the type u a and hence they don't lend themselves to RPS.…”

Section: A New Barrett Algorithm For Polynomialsmentioning

confidence: 99%

“…In [15] the authors present a digit-serial multiplication in GF(2 N ) based on Barrett modular reduction. An improved version of digit-serial multiplication algorithm is described in [16]. Other aspects such as avoiding the pre-computation phase have also been explored [19].…”

Section: Introductionmentioning

confidence: 99%

New residue arithmetic based Barrett algorithms: Modular polynomial computations

Garg

Xiao

2017

2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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In this paper, we derive a new computational algorithm for Barrett technique for modular polynomial multiplication, termed BA-P. BA-P is then applied to a new residue arithmetic based Barrett algorithm for modular polynomial multiplication (BA-MPM). The focus of the work is an algorithm that carries out the entire computation using only modular arithmetic without conversion to large degree polynomials. There are several parts to this work. First, we set up a new BA-P using polynomials other than u  . Second, residue arithmetic based BA-MPM is described. A complete mathematical framework is described including proofs of the steps in the computations and the validity of results. Third, we present a computational procedure for BA-MPM. Fourth, the BA-MPM is used as a basis for algorithms for modular polynomial exponentiation (MPE). Applications are in areas of signal security and cryptography.

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“…For instance, the precomputation of μ can be avoided when − p −1 ≡ ±1 (mod r ), which also avoids computing a multiplication by μ for every iteration inside the Montgomery multiplication routine. This technique has been suggested in [1,36,40] as well. When μ is small, e.g., μ = ±1, one could lower the cost of the multiplication of p with (μ · c 0 ) mod r by choosing the n − 1 most significant words of p in a similar fashion as for the generalized Mersenne primes: p/2 b = 2 s + i∈S i.…”

Section: Montgomery-friendly Primesmentioning

confidence: 99%

Fast Cryptography in Genus 2

et al. 2014

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In this paper, we highlight the benefits of using genus 2 curves in public-key cryptography. Compared to the standardized genus 1 curves, or elliptic curves, arithmetic on genus 2 curves is typically more involved but allows us to work with moduli of half the size. We give a taxonomy of the best known techniques to realize genus 2-based cryptography, which includes fast formulas on the Kummer surface and efficient fourdimensional GLV decompositions. By studying different modular arithmetic approaches on these curves, we present a range of genus 2 implementations. On a single core of an Intel Core i7-3520M (Ivy Bridge), our implementation on the Kummer surface breaks the 125 thousand cycle barrier which sets a new software speed record at the 128-bit security level for constant-time scalar multiplications compared to all previous genus 1 and genus 2 implementations.

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Speeding Up Bipartite Modular Multiplication

Cited by 15 publications

References 12 publications

Selecting elliptic curves for cryptography: an efficiency and security analysis

Selecting elliptic curves for cryptography: an efficiency and security analysis

New residue arithmetic based Barrett algorithms: Modular polynomial computations

Fast Cryptography in Genus 2

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