A Non-redundant and Efficient Architecture for Karatsuba-Ofman Algorithm

Chang, Nam Su; Kim, Chang Han; Lim, Jongsik

doi:10.1007/11556992_21

Cited by 8 publications

(5 citation statements)

References 9 publications

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“…Because of applications in a variety of areas, such as cryptography and coding theory, new techniques for improving polynomial multiplication have been presented in numerous papers, e.g., [1,4,5,7,8,[13][14][15][16][17][18]20,[23][24][25]27,28]. For cryptographic applications, arithmetic in the binary extension field F 2 n is often used and, of the basic operations in F 2 n , multiplication contributes most to the total number of bit operations.…”

Section: Introductionmentioning

confidence: 99%

Some new results on binary polynomial multiplication

Cenk

Hasan

2015

J Cryptogr Eng

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This paper presents several methods for reducing the number of bit operations for multiplication of polynomials over the binary field. First, a modified Bernstein's 3-way algorithm is introduced, followed by a new 5-way algorithm. Next, a new 3-way algorithm that improves asymptotic arithmetic complexity compared to Bernstein's 3-way algorithm is introduced. This new algorithm uses three multiplications of one-third size polynomials over the binary field and one multiplication of one-third size polynomials over the finite field with four elements. Unlike Bernstein's algorithm, which has a linear delay complexity with respect to input size, the delay complexity of the new algorithm is logarithmic. The number of bit operations for the multiplication of polynomials over the finite field with four elements is also computed. Finally, all these new results are combined to obtain improved complexities.

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Section: Introductionmentioning

confidence: 99%

Some new results on binary polynomial multiplication

Cenk

Hasan

2015

J Cryptogr Eng

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“…Of course, Karatsuba's method saves many bit operations at this size. See, e.g., [4], [42], [18], [48], [45], [41], [8], [9], and [21]. Bernstein's Karatsuba/Toom combination in [9] multiplies 256-bit polynomials using only about 133 · 256 bit operations.…”

Section: New Speeds For Binary-fieldmentioning

confidence: 99%

Faster Binary-Field Multiplication and Faster Binary-Field MACs

Bernstein

Chou

2014

Lecture Notes in Computer Science

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Abstract. This paper shows how to securely authenticate messages using just 29 bit operations per authenticated bit, plus a constant overhead per message. The authenticator is a standard type of "universal" hash function providing information-theoretic security; what is new is computing this type of hash function at very high speed.At a lower level, this paper shows how to multiply two elements of a field of size 2 128 using just 9062 ≈ 71 · 128 bit operations, and how to multiply two elements of a field of size 2 256 using just 22164 ≈ 87 · 256 bit operations. This performance relies on a new representation of field elements and new FFT-based multiplication techniques. This paper's constant-time software uses just 1.89 Core 2 cycles per byte to authenticate very long messages. On a Sandy Bridge it takes 1.43 cycles per byte, without using Intel's PCLMULQDQ polynomialmultiplication hardware. This is much faster than the speed records for constant-time implementations of GHASH without PCLMULQDQ (over 10 cycles/byte), even faster than Intel's best Sandy Bridge implementation of GHASH with PCLMULQDQ (1.79 cycles/byte), and almost as fast as state-of-the-art 128-bit prime-field MACs using Intel's integermultiplication hardware (around 1 cycle/byte).

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“…Thus, its complexity is 2n 2 +O(n). Many researchers have tried to improve this algorithm, following two main directions: (1) provide a better asymptotic estimation [34,16,35,24]; (2) reduce the effective number of bit operations [5,12,14,22,21].…”

Section: Introductionmentioning

confidence: 99%

“…Their aim is to improve bounds published in literature for specific value (small) of n, and these improvements that are not visible in the asymptotics. Consequently, a number of papers tries to reduce the effective number of bit operations [34,16,35,24]. As far as we know, the best explicit upper bounds for the polynomial multiplication appear in [6,19,21,12].…”

Section: Introductionmentioning

confidence: 99%

Polynomial multiplication over binary finite fields: new upper bounds

2019

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When implementing a cryptographic algorithm, efficient operations have high relevance both in hardware and software. Since a number of operations can be performed via polynomial multiplication, the arithmetic of polynomials over finite fields plays a key role in real-life implementations -e.g. accelerating cryptographic and cryptanalytic software (preand post-quantum) [18]. One of the most interesting paper that addressed the problem has been published in 2009. In [5], Bernstein suggests to split polynomials into parts and presents a new recursive multiplication technique which is faster than those commonly used. In order to further reduce the number of bit operations [6] required to multiply n-bit polynomials, researchers adopt different approaches. In [19] a greedy heuristic has been applied to linear straight-line sequences listed in [6]. In 2013, D'angella, Schiavo and Visconti [21] skip some redundant operations of the multiplication algorithms described in [5]. In 2015, Cenk, Negre and Hasan [12] suggest new multiplication algorithms. In this paper, (a) we present a "k-1"level Recursion algorithm that can be used to reduce the effective number of bit operations required to multiply n-bit polynomials; and (b) we use algebraic extensions of F2 combined with Lagrange interpolation to improve the asymptotic complexity.

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A Non-redundant and Efficient Architecture for Karatsuba-Ofman Algorithm

Cited by 8 publications

References 9 publications

Some new results on binary polynomial multiplication

Some new results on binary polynomial multiplication

Faster Binary-Field Multiplication and Faster Binary-Field MACs

Polynomial multiplication over binary finite fields: new upper bounds

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