A VLSI algorithm for division in GF(2m) with the canonical basis representation is proposed. It is based on the extended Binary GCD algorithm for GF(2m), and performs division through iteration of simple operations, such as shifts and bitwise exclusive-OR operations. A divider in GF(2m) based on the algorithm has a linear array structure with a bit-slice feature and carries out division in 2m clock cycles. The amount of hardware of the divider is proportional to m and the depth is a constant independent of m.

A new approach on designing a finite field multiplier architecture is proposed. The proposed architecture trades reduction in the number of clock cycles with resources. This architecture features high performance, simple structure, scalability and independence on the choice of the finite field, and can be used in high security cryptographic applications such as elliptic curve crypto-systems in large prime Galois Fields (GF(2m)).

A new elliptic curve scalar multiplication algorithm is proposed. The algorithm offers about twice the throughput of some conventional OEF-base algorithms because it combines the Frobenius map with the table reference method based on base-φ expansion. Furthermore, since this algorithm suits conventional computational units such as 16, 32 and 64 bits, its base field Fpm is expected to enhance elliptic curve operation efficiency more than Fq (q is a prime) or F2n.

The concept of a basis matrix is introduced to investigate the trade-off between complexity and storage for multiplication in a finite field. The effect on the storage requirements of using polynomial and normal bases for element representation is also considered.

Polynomials are called permutation polynomials if they induce bijective functions. This paper investigates algebraic properties of permutation polynomials over a finite field, especially properties associated with permutation cycles. A permutation polynomial has a simple structure but good randomness properties suitable for applications. The cycle structure of permutations are considered to be related to randomness. We investigate the algebraic structure from the viewpoint of randomness. First we show the relationship between polynomials and permutations using a matrix equation. Then, we give a general form of a permutation polynomial corresponding to a product C1C2Ck of pairwise disjoint cycles. Finally, permutation polynomials with fixed points -or with 2, 3 and 4-cycles -and their compositions are given together with distribution of degree of the permutation polynomials.

This paper presents a new method to derive the phase difference between n-tuples of an m-sequence over GF(p) of period pn-1. For the binary m-sequence of the characteristic polynomial f(x)=xn+xd+1 with d=1,2c or n-2c, the explicit formulas of the phase difference from the initial n-tuple are efficiently derived by our method for specific n-tuples such as that consisting of all 1's and that cosisting of one 1 and n-1 0's, although the previously known formula exists only for that consisting of all 1's.

The competing demands of speed and fault tolerance in finite field Fourier transform implementations have been optimally balanced here by using the chord property in finite field.