Linear Complexity and Polynomial Degree of a Function Over a Finite Field (original) (raw)

Expansion complexity and linear complexity of sequences over finite fields

Cryptography and Communications, 2016

The linear complexity is a measure for the unpredictability of a sequence over a finite field and thus for its suitability in cryptography. In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. We study the relationship between linear complexity and expansion complexity. In particular, we show that for purely periodic sequences both figures of merit provide essentially the same quality test for a sufficiently long part of the sequence. However, if we study shorter parts of the period or nonperiodic sequences, then we can show, roughly speaking, that the expansion complexity provides a stronger test. We demonstrate this by analyzing a sequence of binomial coefficients modulo p. Finally, we establish a probabilistic result on the behavior of the expansion complexity of random sequences over a finite field.

On the calculation of the linear complexity of periodic sequences

Contemporary Mathematics, 2010

Based on a result of Hao Chen in 2006 we present a general procedure how to reduce the determination of the linear complexity of a sequence over a finite field Fq of period un to the determination of the linear complexities of u sequences over Fq of period n. We apply this procedure to some classes of periodic sequences over a finite field Fq obtaining efficient algorithms to determine the linear complexity.

On the k-error linear complexity of binary sequences derived from polynomial quotients

Science China Information Sciences, 2015

We investigate the k-error linear complexity of p 2-periodic binary sequences defined from the polynomial quotients (including the well-studied Fermat quotients), which is defined by q p,w (u) ≡ u w − u wp p mod p with 0 ≤ q p,w (u) ≤ p − 1, u ≥ 0, where p is an odd prime and 1 ≤ w < p. Indeed, first for all integers k, we determine exact values of the k-error linear complexity over the finite field F 2 for these binary sequences under the assumption of 2 being a primitive root modulo p 2 , and then we determine their k-error linear complexity over the finite field F p for either 0 ≤ k < p when w = 1 or 0 ≤ k < p − 1 when 2 ≤ w < p. Theoretical results obtained indicate that such sequences possess 'good' error linear complexity.

A note on complete polynomials over finite fields and their applications in cryptography

Finite Fields and Their Applications, 2014

A recursive construction of complete mappings over finite fields is provided in this work. These permutation polynomials, characterized by the property that both f (x) ∈ F q [x] and its associated mapping f (x) + x are permutations, have an important application in cryptography in the construction of bent-negabent functions which actually leads to some new classes of these functions. Furthermore, we also provide a recursive construction of mappings over finite fields of odd characteristic, having an interesting property that both f (x) and f (x + c) + f (x) are permutations for every c ∈ F q. Both the multivariate and univariate representations are treated and some results concerning fixed points and the cycle structure of these permutations are given. Finally, we utilize our main result for the construction of so-called negabent functions and bent functions over finite fields.

On the linear complexity profile of some sequences derived from elliptic curves

Designs, Codes and Cryptography, 2015

For a given elliptic curve E over a finite field of odd characteristic and a rational function f on E we first study the linear complexity profiles of the sequences f (nG), n = 1, 2,. .. which complements earlier results of Hess and Shparlinski. We use Edwards coordinates to be able to deal with many f where Hess and Shparlinski's result does not apply. Moreover, we study the linear complexities of the (generalized) elliptic curve power generators f (e n G), n = 1, 2,. .. . We present large families of functions f such that the linear complexity profiles of these sequences are large.

Linear Complexity and Expansion Complexity of Some Number Theoretic Sequences

Lecture Notes in Computer Science, 2016

We study the predictability of some number theoretic sequences over finite fields and thus their suitability in cryptography. First we analyze the non-periodic binary sequence T = (tn) n≥0 with tn = 1 whenever n is the sum of three integer squares. We show that it has a large N th linear complexity, which is necessary but not sufficient for unpredictability. However, it also has a very small expansion complexity and thus is rather predictable. Next we prove that some linear combinations of p-periodic sequences of binomial coefficients modulo a prime p have a very small expansion complexity and are predictable despite of a high linear complexity. Finally, we consider the Legendre sequence and verify that it does not belong to this class of predictable sequences.

An Innovative Linear Complexity Computation for Cryptographic Sequences

13th International Conference on Computational Intelligence in Security for Information Systems (CISIS 2020), 2020

A simple algorithm to compute the linear complexity of binary sequences with period a power of 2 has been proposed. The algorithm exploits the fractal structure of the binomial representation in this kind of binary sequences. The application of the general algorithm to a particular family of cryptographic sequences (generalized sequences) improves its performance as decreases the amount of sequence to be processed.

Polynomial representations of the Diffie-Hellman mapping

Bulletin of the Australian Mathematical Society, 2001

We obtain lower bounds on the degrees of polynomials representing the Diffie-Hellman mapping (gx, gy) → gxy, where g is a primitive root of a finite field q of q elements. These bounds are exponential in terms of log q. In particular, these results can be used to obtain lower bounds on the parallel arithmetic complexity of breaking the Diffie-Hellman cryptosystem. The method is based on bounds of numbers of solutions of some polynomial equations.

Complexity Theoretic Aspects of Some Cryptographic Functions

Lecture Notes in Computer Science, 2003

In this work, we are interested in non-trivial upper bounds on the spectral norm of binary matrices M from {−1, 1} N×N . It is known that the distributed Boolean function represented by M is hard to compute in various restricted models of computation if the spectral norm is bounded from above by N 1−ε , where ε > 0 denotes a fixed constant. For instance, the size of a two-layer threshold circuit (with polynomially bounded weights for the gates in the hidden layer, but unbounded weights for the output gate) grows exponentially fast with n := log N . We prove sufficient conditions on M that imply small spectral norms (and thus high computational complexity in restricted models). Our general results cover specific cases, where the matrix M represents a bit (the least significant bit or other fixed bits) of a cryptographic decoding function. For instance, the decoding functions of the Pointcheval [9], the El Gamal , and the RSA-Paillier [2] cryptosystems can be addressed by our technique. In order to obtain our results, we make a detour on exponential sums and on spectral norms of matrices with complex entries. This method might be considered interesting in its own right.