Joel V. Brawley

SOME CRYPTOGRAPHIC APPLICATIONS OF PERMUTATION POLYNOMIALS

Abstract : This paper discusses various ways to use permutation polynomials, defined on a finite field and more generally rational permutation functions, to construct cryptographic systems of a general mathematical nature. (Author)

Scalar polynomial functions on the n×n matrices over a finite field

Let F=GF(q) denote the finite field of order q, and let ƒ(x)ϵF[x]. Then f(x) defines, via substitution, a function from Fn×n, the n×n matrices over F, to itself. Any function ƒ:Fn×n → Fn×n which can be represented by a polynomialf(x)ϵF[x] is called a scalar polynomial function on Fn×n. After first determining the number of scalar polynomial functions on Fn×n, the authors find necessary and sufficient conditions on a polynomial ƒ(x) ϵ F[x] in order that it defines a permutation of (i) Dn, the diagonalizable matrices in Fn×n, (ii)Rn, the matrices in Fn×n all of whose roots are in F, and (iii) the matric ring Fn×n itself. The results for (i) and (ii) are valid for an arbitrary field F.

Involutory matrices over finite commutative rings

Abstract An n × n matrix A is called involutory iff A2=In, where In is the n × n identity matrix. This paper is concerned with involutory matrices over an arbitrary finite commutative ring R with identity and with the similarity relation among such matrices. In particular the authors seek a canonical set C with respect to similarity for the n × n involutory matrices over R—i.e., a set C of n × n involutory matrices over R with the property that each n × n involutory matrix over R is similar to exactly on matrix in C . Because of the structure of finite commutative rings and because of previous research, they are able to restrict their attention to finite local rings of characteristic a power of 2, and although their main result does not completely specify a canonical set C for such a ring, it does solve the problem for a special class of rings and shows that a solution to the general case necessarily contains a solution to the classically unsolved problem of simultaneously bringing a sequence A1,…,Av of (not necessarily involutory) matrices over a finite field of characteristic 2 to canonical form (using the same similarity transformation on each Ai). (More generally, the authors observe that a theory of similarity fot matrices over an arbitrary local ring, such as the well-known rational canonical theory for matrices over a field, necessarily implies a solution to the simultaneous canonical form problem for matrices over a field.) In a final section they apply their results to find a canonical set for the involutory matrices over the ring of integers modulo 2m and using this canonical set they are able to obtain a formula for the number of n × n involutory matrices over this ring.

Power Sums of Matrices over a Finite Field. II.

Abstract : In a previous paper the authors found necessary and sufficient conditions on a polynomial with coefficients in a finite field in order that it represent a permutation of that field. The present report expands the original attempts at solving that problem.

In Memory of Jack Levine (1907–2005)

Abstract This article is a tribute to the life and work of the mathematician and cryptographer Jack Levine (1907–2005). It includes a brief summary of his research related to cryptography, a short biography of his life, remembrances of him by his students and colleagues, lists of his publications and his books related to cryptography, and lists of his PhD and MS students with their thesis titles.

A Note on Polynomial Matrix Functions Over a Finite Field.

Abstract Let F n denote the ring of n × n matrices over the finite field F =GF( q ) and let A ( x )= A N x N + ⋯+ A 1 x + A 0 ϵF n [ x ]. A function ƒ:F n →F n is called a right polynomial function iff there exists an A ( x ) ϵF n [ x ] such that ƒ(B)=A N B N +⋯+A 1 B+ A 0 for every BϵF n . This paper obtains unique representations for and determines the number of right polynomial functions.

On distribution by rank of bases for vector spaces of matrices over a finite field

Abstract Let F m × n q denote the vector space of all m × n matrices over the finite field F q of order q , and let B =(A 1 ,A 2 ,…,A mn ) denote an ordered basis for F m × n q . If the rank of A i is r i , i =1,2,…, mn , then B is said to have rank ( r 1 , r 2 ,…, r mn ), and the number of ordered bases of F m x n q with rank ( r 1 , r 2 ,…, r mn is denoted by N q ( r 1 , r 2 ,…, r mn ). This paper determines formulas for the numbers N q ( r 1 , r 2 ,…, r mn ) for the case m = n =2, q arbitrary, and while some of the techniques of the paper extend to arbitrary m and n , the general formulas for the numbers N q ( r 1 , r 2 ,…, r mn ) seem quite complicated and remain unknown. An idea on a possible computer attack which may be feasible for low values of m and n is also discussed.