Lower bounds on the linear complexity of the discrete logarithm in finite fields (original) (raw)

On the k-error linear complexity of cyclotomic sequences

Journal of Mathematical Cryptology, 2000

Exact values and bounds on the k-error linear complexity of p-periodic sequences which are constant on the cyclotomic classes are determined. This family of sequences includes sequences of discrete logarithms, Legendre sequences and Hall's sextic residue sequence.

On the reduction in multiplicative complexity achieved by the polynomial residue number system

IEEE Transactions on Signal Processing, 1992

The polynomial residue number system (PRNS) is known to reduce the complexity of polynomial m iltiplication from O(N2) to O (N). A new interpretation of this complexity reduction is given in the context of associative algt,bras over a finite field. The new point of view provides a clearer understanding of the Chinese remainder theorem.

Linear complexity of Ding generalized cyclotomic sequences

Journal of Shanghai University (English Edition), 2007

Minimal polynomials and linear complexity of binary Ding generalized cyclotomic sequences of order 2 with the two-prime residue ring Zpq are obtained by Bai in 2005. In this paper, we obtain linear complexity and minimal polynomials of all Ding generalized cyclotomic sequences. Our result shows that linear complexity of these sequences takes on the values p q and p q − 1 on our necessary and sufficient condition with probability 1/4 and the lower bound (p q − 1)/2 with probability 1/8. This shows that most of these sequences are good. We also obtained that linear complexity and minimal polynomials of these sequences are independent of their orders. This makes it no more difficult in choosing proper p and q.

Linear complexity of the discrete logarithm

Designs, Codes and Cryptography, 2003

We obtain new lower bounds on the linear complexity of several consecutive values of the discrete logarithm modulo a prime p. These bounds generalize and improve several previous results. Tel.: [61-(0)2] 9850 9585, Fax: [61-(0)2] 9850 9551.

On the linear complexity of bounded integer sequences over different moduli

Information Processing Letters, 2005

We give a relation between the linear complexity over the integers and over the residue rings modulo m of a bounded integer sequence. This relation can be used to obtain a variety of new results for several sequences widely studied in the literature. In particular we apply it to Sidelnikov sequences.

On the linear complexity of binary sequences derived from generalized cyclotomic classes modulo 2p

2019

The linear complexity of a sequence is an important parameter in its evaluation as a keystream cipher for cryptographic applications. Using of cyclotomic classes to construct sequences is an important method for designing sequences with high linear complexity. In this article, we study the linear complexity of generalized cyclotomic binary sequences of length 2npm. These sequences were constructed from new generalized cyclotomic classed prepared by X. Zeng at el. We investigate discrete Fourier transform of these sequences and define the sufficient conditions for the existence of sequences with high linear complexity. Key–Words: Binary sequences, linear complexity, cyclotomy, generalized cyclotomic sequence

Integer Complexity Generalizations in Various Rings

Cornell University - arXiv, 2022

In this paper, we investigate generalizations of the Mahler-Popkens complexity of integers. Specifically, we generalize to k-th roots of unity, polynomials over the naturals, and the integers mod m. In cyclotomic rings, we establish upper and lower bounds for integer complexity, investigate the complexity of roots of unity using cyclotomic polynomials, and introduce a concept of "minimality." In polynomials over the naturals, we establish bounds on the sizes of complexity classes and establish a trivial but useful upper bound. In the integers mod m, we introduce the concepts of "inefficiency", "resilience", and "modified complexity." In hopes of improving the upper bound on the complexity of the most complex element mod m, we also use graphs to visualize complexity in these finite rings.

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.