There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Almost bent functions oppose an optimum resistance to linear and differential cryptanalysis. We present basic properties of almost bent functions; particularly we give an upper bound on the degree. We develop the "coding theory" point of view for studying the existence of almost bent functions, showing explicitly the links with cyclic codes. We also give new characterizations of almost bent functions by means of associated Boolean functions.

/pdf/codes-bent-functions-and-permutations-suitable-for-des-4tcbdve4wi.pdf

Codes, Bent Functions and Permutations Suitable For DES-likeCryptosystems

To appear as a chapter of the volume " Boolean Methods and Models " ,

/pdf/vectorial-boolean-functions-for-cryptography-37xxxf2ncc.pdf

Vectorial Boolean Functions for Cryptography

Modular elliptic curves and fermat's last theorem

We give a new heuristic for all of the main terms in the integral moments of various families of primitive $L$-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical $L$-functions have a spectral interpretation, and that their value distributions can be modelled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

https://web.williams.edu/Mathematics/sjmiller/public_html/ntandrmt/handouts/conrey/ConreyFarmerKeatingRubinsteinSnaith_IntegralMomentsLFns.pdf

Integral moments of L-functions

We give a new heuristic for all of the main terms in the integral moments of various families of primitive L-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical L-functions have a spectral interpretation, and that their value distributions can be modeled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

The title refers to the fact, noted by Chebyshev in 1853, that primes congruent to 3 modulo 4 seem to predominate over those congruent to 1. We study this phenomenon and its generalizations. Assuming the Generalized Riemann Hypothesis and the Grand Simplicity Hypothesis (about the zeros of the Dirichlet L-function), we can characterize exactly those moduli and residue classes for which the bias is present. We also give results of numerical investigations on the prevalence of the bias for several moduli. Finally, we briefly discuss generalizations of the bias to the distribution to primes in ideal classes in number fields, and to prime geodesics in homology classes on hyperbolic surfaces.

Chebyshev's bias

By looking at the average behavior (n-level density) of the low-lying zeros of certain families of L-functions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups.

/pdf/low-lying-zeros-of-l-functions-and-random-matrix-theory-j0cb59nnc7.pdf

Low-lying zeros of L-functions and random matrix theory

We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly the autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions.

Michael O. Rubinstein

Papers

Integral moments of L-functions

Integral moments of L-functions

Chebyshev's bias

Low-lying zeros of L-functions and random matrix theory

Autocorrelation of Random Matrix Polynomials