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—Based on Toeplitz matrix-vector products and coordinate transformation techniques, we present a new scheme for subquadratic space complexity parallel multiplication in GF ð2 n Þ 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… (More)

ÐBecause of their shorter key sizes, cryptosystems based on elliptic curves are being increasingly used in practical applications. A special class of elliptic curves, namely, Koblitz curves, offers an additional, but crucial, advantage of considerably reduced processing time. In this article, power analysis attacks are applied to cryptosystems that use… (More)

—For cryptographic applications, normal bases have received considerable attention, especially for hardware implementation. In this article, we consider fast software algorithms for normal basis multiplication over the extended binary field GFð2 m Þ. We present a vector-level algorithm which essentially eliminates the bit-wise inner products needed in the… (More)

—Representing the field elements with respect to the polynomial (or standard) basis, we consider bit parallel architectures for multiplication over the finite field GF ð2 m Þ. In this effect, first we derive a new formulation for polynomial basis multiplication in terms of the reduction matrix Q. The main advantage of this new formulation is that it can be… (More)

This article presents simple and highly regular architectures for finite field multipliers using a redundant representation. The basic idea is to embed a finite field into a cyclotomic ring which has a basis with the elegant multiplicative structure of a cyclic group. One important feature of our architectures is that they provide area-time trade-offs which… (More)

In this paper we propose to take one step back in the use of double base number systems for elliptic curve point scalar multiplication. Using a modified version of Yao's algorithm, we go back from the popular double base chain representation to a more general double base system. Instead of representing an integer k as P n i=1 2 b i 3 t i where (bi) and (ti)… (More)

ÐThe Massey-Omura multiplier of qp
P m uses a normal basis and its bit parallel version is usually implemented using m identical combinational logic blocks whose inputs are cyclically shifted from one another. In the past, it was shown that, for a class of finite fields defined by irreducible all-one polynomials, the parallel Massey-Omura multiplier had… (More)

A high performance elliptic curve coprocessor is developed, which is optimized for a binary field recommended by NIST. The architecture uses a field multiplier capable of performing a field multiplication over the extension field with degree 163 in 0.060 /spl mu/sec. The coprocessor uses Lopez and Dahab's projective coordinate system and is optimized… (More)