Parallel Montgomery Multiplication in GF (2^k) Using Trinomial Residue Arithmetic

Bajard, Jean-Claude; Imbert, Laurent; Jullien, G.A.

doi:10.1109/arith.2005.34

Cited by 24 publications

(22 citation statements)

References 21 publications

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“…The second type are the subquadratic space complexity designs which require smaller number of gates for their implementation (O(n δ ), δ < 2), cf. [2], [11], [15], [1]. The latter presents practical architectures for hardware implementation of large field sizes, specially used in elliptic curve cryptographic applications [7], [10] (163 < n < 571).…”

Section: Introductionmentioning

confidence: 99%

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Block Recombination Approach for Subquadratic Space Complexity Binary Field Multiplication Based on Toeplitz Matrix-Vector Product

Hasan

Méloni

Namin

et al. 2012

IEEE Trans. Comput.

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! AbstractIn this paper, we present a new method for parallel binary finite field multiplication which results in subquadratic space complexity. The method is based on decomposing the building blocks of Fan-Hasan subquadratic Toeplitz matrix-vector multiplier. We reduce the space complexity of their architecture by recombining the building blocks. In comparison to other similar schemes available in the literature, our proposal presents a better space complexity while having the same time complexity. We also show that block recombination can be used for efficient implementation of the GHASH function of Galois Counter Mode (GCM).

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Section: Introductionmentioning

confidence: 99%

“…Another well-known algorithm is the Winograd short convolution algorithm [17] used for the same purpose in [15]. Chinese Reminder Theorem (CRT) is another example of such algorithms that results in subquadratic complexity multipliers [1].…”

Section: Introductionmentioning

confidence: 99%

Block Recombination Approach for Subquadratic Space Complexity Binary Field Multiplication Based on Toeplitz Matrix-Vector Product

Hasan

Méloni

Namin

et al. 2012

IEEE Trans. Comput.

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“…Horner's rule is a basic and widely used method for computing polynomials, and is used in numerous complex applications [13,14,15,16]. It works by transforming the polynomial into a series of multiply-add operations.…”

Section: Horner's Rulementioning

confidence: 99%

Square-rich fixed point polynomial evaluation on FPGAs

Xu¹,

Fahmy

McLoughlin

2014

Proceedings of the 2014 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays

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Polynomial evaluation is important across a wide range of application domains, so significant work has been done on accelerating its computation. The conventional algorithm, referred to as Horner's rule, involves the least number of steps but can lead to increased latency due to serial computation. Parallel evaluation algorithms such as Estrin's method have shorter latency than Horner's rule, but achieve this at the expense of large hardware overhead. This paper presents an efficient polynomial evaluation algorithm, which reforms the evaluation process to include an increased number of squaring steps. By using a squarer design that is more efficient than general multiplication, this can result in polynomial evaluation with a 57.9% latency reduction over Horner's rule and 14.6% over Estrin's method, while consuming less area than Horner's rule, when implemented on a Xilinx Virtex 6 FPGA. When applied in fixed point function evaluation, where precision requirements limit the rounding of operands, it still achieves a 52.4% performance gain compared to Horner's rule with only a 4% area overhead in evaluating 5 th degree polynomials.

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“…cryptography, see the survey [45] and [40,41,42,43,40]. For a general and recent reference on elliptic cryptography, see [99].…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Dynamical directions in numeration

Barat¹,

Berthé²,

Liardet³

et al. 2006

Ann. inst. Fourier

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Parallel Montgomery Multiplication in GF (2^k) Using Trinomial Residue Arithmetic

Cited by 24 publications

References 21 publications

Block Recombination Approach for Subquadratic Space Complexity Binary Field Multiplication Based on Toeplitz Matrix-Vector Product

Block Recombination Approach for Subquadratic Space Complexity Binary Field Multiplication Based on Toeplitz Matrix-Vector Product

Square-rich fixed point polynomial evaluation on FPGAs

Dynamical directions in numeration

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