Linear complexity over F_q and over F_{q^m} for linear recurring sequences

Since the \F_q-linear spaces \F_q^m and \F_{q^m} are isomorphic, an $m$-fold multisequence $\mathbf{S}$ over the finite field \F_q with a given characteristic polynomial f \in \F_q[x], can be identified with a single sequence $\mathcal{S}$ over \F_{q^m} with characteristic polynomial $f$. The linear complexity of $\mathcal{S}$, which we call the generalized joint linear complexity of $\mathbf{S}$, can be significantly smaller than the conventional joint linear complexity of $\mathbf{S}$. We determine the expected value and the variance of the generalized joint linear complexity of a random $m$-fold multisequence $\mathbf{S}$ with given minimal polynomial. The result on the expected value generalizes a previous result on periodic $m$-fold multisequences. Finally we determine the expected drop of linear complexity of a random $m$-fold multisequence with given characteristic polynomial $f$, when one switches from conventional joint linear complexity to generalized joint linear complexity

Generalized joint linear complexity of linear recurring multisequences

The joint linear complexity of multisequences is an important security measure for vectorized stream cipher systems. Extensive research has been carried out on the joint linear complexity of $N$-periodic multisequences using tools from Discrete Fourier transform. Each $N$-periodic multisequence can be identified with a single $N$-periodic sequence over an appropriate extension field. It has been demonstrated that the linear complexity of this sequence, the so called generalized joint linear complexity of the multisequence, may be considerably smaller than the joint linear complexity, which is not desirable for vectorized stream ciphers. Recently new methods have been developed and results of greater generality on the joint linear complexity of multisequences consisting of linear recurring sequences have been obtained. In this paper, using these new methods, we investigate the relations between the generalized joint linear complexity and the joint linear complexity of multisequences consisting of linear recurring sequences

A new class of three-weight linear codes from weakly regular plateaued functions

Linear codes with few weights have many applications in secret sharing schemes, authentication codes, communication and strongly regular graphs. In this paper, we consider linear codes with three weights in arbitrary characteristic. To do this, we generalize the recent contribution of Mesnager given in [Cryptography and Communications 9(1), 71-84, 2017]. We first present a new class of binary linear codes with three weights from plateaued Boolean functions and their weight distributions. We next introduce the notion of (weakly) regular plateaued functions in odd characteristic $p$ and give concrete examples of these functions. Moreover, we construct a new class of three-weight linear $p$-ary codes from weakly regular plateaued functions and determine their weight distributions. We finally analyse the constructed linear codes for secret sharing schemes.Comment: The Extended Abstract of this work was submitted to WCC-2017 (the Tenth International Workshop on Coding and Cryptography

Improved asymptotic bounds for codes using distinguished divisors of global function fields

For a prime power $q$, let $\alpha_q$ be the standard function in the asymptotic theory of codes, that is, $\alpha_q(\delta)$ is the largest asymptotic information rate that can be achieved for a given asymptotic relative minimum distance $\delta$ of $q$-ary codes. In recent years the Tsfasman-Vl\u{a}du\c{t}-Zink lower bound on $\alpha_q(\delta)$ was improved by Elkies, Xing, and Niederreiter and \"Ozbudak. In this paper we show further improvements on these bounds by using distinguished divisors of global function fields. We also show improved lower bounds on the corresponding function $\alpha_q^{\rm lin}$ for linear codes

New cubic self-dual codes of length 54, 60 and 66

We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length 3\ell with the algebraic approach of [9]. In particular, we improve the previous results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50 new [66, 33, 12] cubic self-dual codes. We conjecture that there exist no more binary cubic self-dual codes with length 54, 60 and 66.Comment: 8 page

Some Artin–Schreier type function fields over finite fields with prescribed genus and number of rational places

AbstractWe give existence and characterization results for some Artin–Schreier type function fields over finite fields with prescribed genus and number of rational places simultaneously

On the Parity of Power Permutations

CCBYSide-channel analysis (SCA) attacks and many countermeasures to foil these attacks have been the subject of a large body of research. Different masking schemes have been proposed as countermeasures, one of which is Threshold Implementation (TI), which carries proof of security against DPA even in the presence of glitches. At the same time, it requires a smaller area and uses much less randomness than the other secure masking methods. One of the methods to have an efficient TI of high degree S-boxes is the decomposition method. Our goal in this paper is to analyze the nonlinear components of symmetric cryptographic algorithms. To minimize the area of the protected implementation of cryptographic algorithms, we show the conditions to decompose the substitutions boxes, which are permutations, of high algebraic degree into the ones of lower degree. To find the conditions, we target the decomposition of permutations into quadratic or cubic permutations by considering the power permutations and their parities, which help us determine whether the higher degree permutations are decomposable power permutations or not. Finally, the decomposition results about the finite fields and corresponding lower degree power permutations are presented