Ronald C. Mullin

Applications of finite fields

1 Introduction to Finite Fields and Bases.- 2 Factoring Polynomials over Finite Fields.- 3 Construction of Irreducible Polynomials.- 4 Normal Bases.- 5 Optimal Normal Bases.- 6 The Discrete Logarithm Problem.- 7 Elliptic Curves over Finite Fields.- 8 Elliptic Curve Cryptosystems.- 9 Introduction to Algebraic Geometry.- 10 Codes From Algebraic Geometry.- Appendix - Other Applications.

Optimal normal bases in GF( p n )

Abstract In this paper the use of normal bases for multiplication in the finite fields GF(pn) is examined. We introduce the concept of an optimal normal basis in order to reduce the hardware complexity of multiplying field elements. Constructions for these bases in GF(2n) and extensions of the results to GF(pn) are presented. This work has applications in crytography and coding theory since a reduction in the complexity of multiplying and exponentiating elements of GF(2n) is achieved for many values of n, some prime.

An implementation of elliptic curve cryptosystems over F/sub 2/155

The authors describe a VLSI Galois field processor and how it can be applied to the implementation of elliptic curve groups. They demonstrate the feasibility of constructing very fast, and very secure, public key systems with a relatively simple device, and the possibility of putting such a system on a smart card. The registers necessary to implement the elliptic curve system will require less than 1 mm/sup 2/ (or less than 4%) of the area available on the card. >

An implementation for a fast public-key cryptosystem

In this paper we examine the development of a high-speed implementation of a system to perform exponentiation in fields of the form GF(2n). For sufficiently large n, this device has applications in public-key cryptography. The selection of representation and observations on the structure of multiplication have led to the development of an architecture which is of low complexity and high speed. A VLSI implementation has being fabricated with measured throughput for exponentiation for cryptographic purposes of approximately 300 kilobits per second.

The enumeration of c-nets via quadrangulations

Abstract A counting procedure for simple quadrangulations is established. Using the technique of counting simple quadrangulations together with a one-to-one correspondence between simple quadrangulations and c-nets, the enumeration of c-nets with i+1 vertices and j+1 faces is accomplished.

Arithmetic operations inGF(2m)

This article is concerned with various arithmetic operations inGF(2m). In particular we discuss techniques for computing multiplicative inverses and doing exponentiation. The method used for exponentiation is highly suited to parallel computation. All methods achieve much of their efficiency from exploiting a normal basis representation in the field.

On the existence of frames

This paper deals with two topics, namely, frames and pairwise balanced designs (PBDs). Frames, which were introduced by W.D. Wallis for the construction of (skew) Room squares, are shown to exist for most orders congruent to 1 (mod 4). This result relies heavily on the existence of PBDs since the set F = {v | there is a frame of order v] is shown to be PBD-closed. By employing a generalization of the usual recursive construction for PBDs, it is shown that B{5, 9, 13, 17}@?B{5, 9, 13}@?{69, 77, 97, 137, 237, 277, 317, 377, 569}@?{n | n @? 1 (mod 4), n>0}@?{29, 33, 49, 57, 93, 129, 133}, where B(K) denotes the set of orders of PBDs of index one having block-sizes from the set K. Frames of orders 5, 9, 13 and 17 are exhibited which immediately implies that F@?B{5, 9, 13, 17}. D.R. Stinson and W.D. Wallis have shown that {29, 49}@?F. Thus there is a frame of order @u for every positive integer @u congruent to 1 (mod 4) with the possible exceptions of @u @e {33, 57, 93, 133}.

Pairwise Balanced Designs with Consecutive Block Sizes

This paper deals with existence for pairwise balanced designs with block sizes 5,6 and 7, block sizes 6,7 and 8 and block sizes 7,8 and 9 and some consequences of these results.

Covering triples by quadruples: an asymptotic solution

Let C(3, 4, n) be the minimum number of four-element subsets (called blocks) of an n-element set, X, such that each three-element subset of X is contained in at least one block. Let L(3, 4, n) = ⌜n4⌜n−13⌜n−22⌝⌝⌝. Schoenheim has shown that C(3, 4, n) ⩾ L(3, 4, n). The construction of Steiner quadruple systems of all orders n≡2 or 4 (mod 6) by Hanani (Canad. J. Math. 12 (1960), 145–157) can be used to show that C(3, 4, n) = L(3, 4, n) for all n ≡ 2, 3, 4 or 5(od 6) and all n ≡ 1 (mod 12). The case n ≡ 7 (mod 12) is made more difficult by the fact that C(3, 4, 7) = L(3, 4, 7) + 1 and until recently no other value for C(3, 4, n) with n≡7 (mod 12) was known. In 1980 Mills showed by construction that C(3, 4, 499) = L(3, 4, 499). We use this construction and some recursive techniques to show that C(3, 4, n) = L(3, 4, n) for all n ⩾ 52423. We also show that if C(3, 4, n) = L(3, 4, n) for n = 31, 43, 55 and if a certain configuration on 54 points exists then C(3, 4, n) = L(3, 4, n) for all n ≠ 7 with the possible exceptions of n = 19 and n = 67. If we assume only C(3, 4, n) = L(3, 4, n) for n = 31 and 43 we can deduce that C(3, 4, n) = L(3, 4, n) for all n ≠ 7 with the possible exceptions of n ϵ {19, 55, 67, 173, 487}.

Quintessential pairwise balanced designs

The existence of pairwise balanced designs with block sizes from a set K is studied. The spectrum of orders for which such PBDs exist is determined when {5}⊂K⊆{5,6,7,8,9}, with relatively few possible exceptions in each case.