50 research outputs found
Proof of a Conjectured Three-Valued Family of Weil Sums of Binomials
We consider Weil sums of binomials of the form , where is a finite field, is
the canonical additive character, , and .
If we fix and and examine the values of as runs
through , we always obtain at least three distinct values unless
is degenerate (a power of the characteristic of modulo ).
Choices of and for which we obtain only three values are quite rare and
desirable in a wide variety of applications. We show that if is a field of
order with odd, and with , then
assumes only the three values and . This
proves the 2001 conjecture of Dobbertin, Helleseth, Kumar, and Martinsen. The
proof employs diverse methods involving trilinear forms, counting points on
curves via multiplicative character sums, divisibility properties of Gauss
sums, and graph theory.Comment: 19 page
Mathematical aspects of the design and security of block ciphers
Block ciphers constitute a major part of modern symmetric cryptography. A mathematical analysis is necessary to ensure the security of the cipher. In this thesis, I develop several new contributions for the analysis of block ciphers. I determine cryptographic properties of several special cryptographically interesting mappings like almost perfect nonlinear functions. I also give some new results both on the resistance of functions against differential-linear attacks as well as on the efficiency of implementation of certain block ciphers
A conjecture about Gauss sums and bentness of binomial Boolean functions
In this note, the polar decomposition of binary fields of even extension degree is used to reduce the evaluation of the Walsh transform of binomial Boolean functions to that of Gauss sums. In the case of extensions of degree four times an odd number, an explicit formula involving a Kloosterman sum is conjectured, proved with further restrictions, and supported by extensive experimental data in the general case. In particular, the validity of this formula is shown to be equivalent to a simple and efficient characterization for bentness previously conjectured by Mesnager
On the Typical Size and Cancelations Among the Coefficients of Some Modular Forms
We obtain a nontrivial upper bound for almost all elements of the sequences
of real numbers which are multiplicative and at the prime indices are
distributed according to the Sato--Tate density. Examples of such sequences
come from coefficients of several -functions of elliptic curves and modular
forms. In particular, we show that
for a set of of asymptotic density 1, where is the Ramanujan
function while the standard argument yields instead of
in the power of the logarithm. Another consequence of our result is that in the
number of representations of by a binary quadratic form one has slightly
more than square-root cancellations for almost all integers .
In addition we obtain a central limit theorem for such sequences, assuming a
weak hypothesis on the rate of convergence to the Sato--Tate law. For Fourier
coefficients of primitive holomorphic cusp forms such a hypothesis is known
conditionally assuming the automorphy of all symmetric powers of the form and
seems to be within reach unconditionally using the currently established
potential automorphy.Comment: The second version contains some improvements and extensions of
previous results, suggested by Maksym Radziwill, who is now a co-autho