Gary McGuire

New families of quadratic almost perfect nonlinear trinomials and multinomials

We introduce two new infinite families of APN functions, one on fields of order 2^2^k for k not divisible by 2, and the other on fields of order 2^3^k for k not divisible by 3. The polynomials in the first family have between three and k+2 terms, the second familys polynomials have three terms.

A few more quadratic APN functions

We present an infinite family of quadrinomial APN functions on GF(2n) where n is divisible by 3 but not 9. The family contains inequivalent functions, obtained by setting some coefficients equal to 0. We also discuss the inequivalence proof (by computation) which shows that these functions are new.

On three weights in cyclic codes with two zeros

We show that the dual code of the binary cyclic code of length 2^m-1 with two zeros @a,@a^d cannot have three weights in the case that m is even and d=0(mod3). The proof involves the partial calculation of a coset weight distribution.

Construction of bent functions from near-bent functions

We give a construction of bent functions in dimension 2m from near-bent functions in dimension 2m-1. In particular, we give the first ever examples of non-weakly-normal bent functions in dimensions 10 and 12, which demonstrates the significance of our construction.

The weight distributions of cyclic codes with two zeros and zeta functions

We consider the weight distribution of the binary cyclic code of length 2^n-1 with two zeros @a^a,@a^b. Our proof gives information in terms of the zeta function of an associated variety. We carry out an explicit determination of the weight distribution in two cases, for the cyclic codes with zeros @a^3,@a^5 and @a,@a^1^1. These are the smallest cases of two infinite families where finding the weight distribution is an open problem. Finally, an interesting application of our methods is that we can prove that these two codes have the same weight distribution for all odd n.

Cyclic codes over Z/sub 4/, locator polynomials, and Newton's identities

Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock (1972) and Preparata (1968) codes that can be very simply constructed as binary images, under the Gray map, of linear codes over Z/sub 4/ that are defined by means of parity checks involving Galois rings. This paper describes how Fourier transforms on Galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes. These methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of Z/sub 4/. Linear codes with length 2/sup m/ (m, odd) and size 2(2/sup m+1/-5m-2). The Gray image of the code of length 32 is the best (64, 2/sup 37/) code that is presently known. This paper also determines the exact minimum Lee distance of the linear codes over Z/sub 4/ that are obtained from the extended binary two- and three-error-correcting BCH codes by Hensel lifting. The Gray image of the Hensel lift of the three-error-correcting BCH code of length 32 is the best (64, 2/sup 32/) code that is presently known. This code also determines an extremal 32-dimensional even unimodular lattice.

On the Function Field Sieve and the Impact of Higher Splitting Probabilities

In this paper we propose a binary field variant of the Joux-Lercier medium-sized Function Field Sieve, which results not only in complexities as low as \(L_{q^n}(1/3,(4/9)^{1/3})\) for computing arbitrary logarithms, but also in an heuristic polynomial time algorithm for finding the discrete logarithms of degree one and two elements when the field has a subfield of an appropriate size. To illustrate the efficiency of the method, we have successfully solved the DLP in the finite fields with 21971 and 23164 elements, setting a record for binary fields.

Determining the nonlinearity of a new family of APN functions

We compute the Walsh spectrum and hence the nonlinearity of a new family of quadratic multi-term APN functions. We show that the distribution of values in the Walsh spectrum of these functions is the same as the Gold function.

Gold and Kasami-Welch functions, quadratic forms, and bent functions

We use elementary facts about quadratic forms in characteristic 2 to evaluate the sign of some Walsh transforms in terms of a Jacobi symbol. These results are applied to the Walsh transforms of the Gold and Kasami-Welch functions. We prove that the Gold functions yield bent functions when restricted to certain hyperplanes. We also use the sign information to determine the dual bent function.

Construction of a (64, 2 ^{ 37} , 12) Codevia Galois Rings

Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock and Preparata codes, which exist for all lengths 4m ≥ 16. At length 16 they coincide to give the Nordstrom-Robinson code. This paper constructs a nonlinear (64, 237, 12) code as the binary image, under the Gray map, of an extended cyclic code defined over the integers modulo 4 using Galois rings. The Nordstrom-Robinson code is defined in this same way, and like the Nordstrom-Robinson code, the new code is better than any linear code that is presently known.