Joseph P. Brennan

Low complexity multiplication in a finite field using ring representation

Elements of a finite field, GF(2/sup m/), are represented as elements in a ring in which multiplication is more time efficient. This leads to faster multipliers with a modest increase in the number of XOR and AND gates needed to construct the multiplier. Such multipliers are used in error control coding and cryptography. We consider rings modulo trinomials and 4-term polynomials. In each case, we show that our multiplier is faster than multipliers over elements in a finite field defined by irreducible pentanomials. These results are especially significant in the field of elliptic curve cryptography, where pentanomials are used to define finite fields. Finally, an efficient systolic implementation of a multiplier for elements in a ring defined by x/sup n/+x+1 is presented.

Montgomery multiplication over rings

Abstract Montgomery multiplication of two elements a and b of a finite field F q is defined as abr - 1 where r is a fixed field element in F q × . In this paper we define Montgomery multiplication of elements a ( x ) and b ( x ) in a polynomial ring modulo the ideal generated by a reducible polynomial f ( x ) . We then show that Montgomery multiplication over a field represented by a polynomial ring modulo an irreducible pentanomial can be performed more efficiently in terms of time delay by embedding the field in a quotient of a polynomial ring modulo a reducible trinomial. The trinomial has a degree that is slightly higher than that of the pentanomial, thereby increasing the number of gates in the multiplier by a small amount.

Sequences that Preserve the Homological Degree

The homological degree is a cohomological degree generalizing the classical degree function of a module. Given a finitely generated graded module M, over a standard graded ring A, this note is concerned with the question of when, for an element x in the augmentation ideal of A that is not in any associated prime of M other than the augmentation ideal itself, the homological degree of M is equal to the homological degree of M/xM. This question is answered when the dimension of M is one or two.

An extension of a theorem of Frobenius and Stickelberger to modules of projective dimension one over a factorial domain

Abstract Let R be a Cohen–Macaulay ring. A quasi-Gorenstein R-module is an R-module such that the grade of the module and the projective dimension of the module are equal and the canonical module of the module is isomorphic to the module itself. After discussing properties of finitely generated quasi-Gorenstein modules, it is shown that this definition allows for a characterization of diagonal matrices of maximal rank over a Cohen–Macaulay factorial domain R extending a theorem of Frobenius and Stickelberger to modules of projective dimension 1 over a commutative factorial Cohen–Macaulay domain.