38 research outputs found
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
Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method
We prove a bound for quintilinear sums of Kloosterman sums, with congruence
conditions on the "smooth" summation variables. This generalizes classical work
of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms
in applications, notably the dispersion method. As a consequence, assuming the
Riemann hypothesis for Dirichlet -functions, we prove a power-saving error
term in the Titchmarsh divisor problem of estimating .
Unconditionally, we isolate the possible contribution of Siegel zeroes, showing
it is always negative. Extending work of Fouvry and Tenenbaum, we obtain
power-saving in the asymptotic formula for ,
reproving a result announced by Bykovski\u{i} and Vinogradov by a different
method. The gain in the exponent is shown to be independent of if a
generalized Lindel\"of hypothesis is assumed
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
Curve-lifted codes for local recovery using lines
In this paper, we introduce curve-lifted codes over fields of arbitrary
characteristic, inspired by Hermitian-lifted codes over .
These codes are designed for locality and availability, and their particular
parameters depend on the choice of curve and its properties. Due to the
construction, the numbers of rational points of intersection between curves and
lines play a key role. To demonstrate that and generate new families of locally
recoverable codes (LRCs) with high availabilty, we focus on norm-trace-lifted
codes. In some cases, they are easier to define than their Hermitian
counterparts and consequently have a better asymptotic bound on the code rate.Comment: 22 pages. Comments welcom