Based on Toeplitz matrix-vector products and coordinate transformation techniques, we present a new scheme for subquadratic space complexity parallel multiplication in GF(2 n) using the shifted polynomial basis. Both the space complexity and the asymptotic gate delay of the proposed multiplier are better than those of the best existing subquadratic space complexity parallel multipliers. For example, with n being a power of 2 and 3, the space complexity is about 8 % and 10 % better, while the asymptotic gate delay is about 33 % and 25 % better, respectively. Another advantage of the proposed matrix-vector product approach is that it can also be used to design subquadratic space complexity polynomial, dual, weakly dual and triangular basis parallel multipliers. To the best of our knowledge, this is the first time that subquadratic space complexity parallel multipliers are proposed for dual, weakly dual and triangular bases. A recursive design algorithm is also proposed for efficient construction of the proposed subquadratic space complexity multipliers. This design algorithm can be modified for the construction of most of the subquadratic space complexity multipliers previously reported in the literature.

We present efficient squaring formulae based on the Toom-Cook multiplication algorithm. The latter always requires at least one non-trivial constant division in the interpolation step. We show such non-trivial divisions are not needed in the case two operands are equal for three, four and five-way squarings. Our analysis shows that our 3-way squaring algorithms have much less overhead than the best known 3-way Toom-Cook algorithm. Our experimental results show that one of our new 3-way squaring methods performs faster than mpz_mul() in GNU multiple precision library (GMP) for squaring integers of approximately 2400–6700 bits on Pentium IV Prescott 3.2GHz. For squaring in Z[x], our 3-way squaring algorithms are much superior to other known squaring algorithms for small input size. In addition, we present 4-way and 5-way squaring formulae which do not require any constant divisions by integers other than a power of 2. Under some reasonable assumptions, our 5-way squaring formula is faster than the recently proposed Montgomery’s 5-way Karatsuba-like formulae. Keywords: Squaring, Karatsuba algorithm, Toom-

We propose the first general multiplication algorithm in GF(2 k) with a subquadratic area complexity of O(k 8/5) = O(k 1.6). Using the Chinese Remainder Theorem, we represent the elements of GF(2 k); i.e. the polynomials in GF(2)[X] of degree at most k − 1, by their remainder modulo a set of n pairwise prime trinomials, T1,..., Tn, of degree d and such that nd ≥ k. Our algorithm is based on Montgomery’s multiplication applied to the ring formed by the direct product of the trinomials.

We describe how a recently proposed way to split input operands allows for fast VLSI implemen-tations of GF(2)[x] Karatsuba-Ofman multipliers. The XOR gate delay of the proposed multiplier is better than that of previous Karatsuba-Ofman multipliers. For example, it is reduced by about 33 % and 25 % for n = 2 i and n = 3 i (i> 1), respectively. Index Terms Karatsuba algorithm, Karatsuba-Ofman algorithm, polynomial multiplication, subquadratic space complexity multiplier, finite field. I.

Based on a recently proposed Toeplitz matrix-vector product approach, a subquadratic computational complexity scheme is presented for multiplications in binary extended finite fields using Type I and II optimal normal bases. basis. Index Terms Finite field, subquadratic computational complexity multiplication, normal basis, optimal normal

Abstract We propose the first general multiplication algorithm in GF(2k) with a subquadratic area complexity of O(k8/5) = O(k1.6). Using the Chinese Remainder Theorem, we represent the elements of GF(2k); i.e. the polynomials in GF(2)[X] of degree at most k- 1, by their remainder modulo a set of n pairwise prime trinomials, T1,..., Tn, of degree d and such that nd> = k. Our algorithm is based on Montgomery's multiplication applied to the ring formed by the direct product of the trinomials.

We show that multiplication complexities of n-term Karatsuba-Like formulae of GF (2)[x] (7 < n < 19) presented in the above paper can be further improved using the Chinese Remainder Theorem and the construction multiplication modulo (x − ∞) w. Index Terms Karatsuba algorithm, polynomial multiplication, finite field.

In 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).

Abstract. Following the previous work by Bajard-Didier-Kornerup, Mc-Laughlin, Mihailescu and Bajard-Imbert-Jullien, we present an algorithm for modular polynomial multiplication that implements the Montgomery algorithm in a residue basis; here, as in Bajard et al.’s work, the moduli are trinomials over F2. Previous work used a second residue basis to perform the final division. In this paper, we show how to keep the same residue basis, inspired by l’Hospital rule. Additionally, applying a divideand-conquer approach to the Chinese remaindering, we obtain improved estimates on the number of additions for some useful degree ranges.

Abstract. Finitefieldarithmetichasreceivedaconsiderableattentioninthecurrentcryptographic applications. Many researchers have focused on finite field multiplication due to its importance in various cryptographic operations. Moreover, finite field multiplication can be considered as a cornerstone for elliptic curve cryptosystems. Fan and Hasan [1] introduced a new sub-quadratic computational complexity approach for finite field multiplication. It is based on Toeplitz matrix-vector products. In this paper we consider efficient implementation of this approach on general purpose processors usingType II Optimal Normal Basis (ONB II). To this end, a memory and time efficient implementation scheme is proposed for the Fan and Hasan approach. Also, in this paper we provide a modified version of the best quadratic complexity multiplication algorithm due to Reyhani-Masoleh [2]. The proposed modification reduces the number of OR and SHIFT instructions by 50% and the number of AND instructions by about 25%. We simulate the implementation on three different architectures and present the results. Furthermore, we present an idea to fully parallelize the implementation of the Fan and Hasan scheme.