New results on periodic sequences with large k-error linear complexity

Linear complexity is an important cryptographic index of sequences. We study the linear complexity of $$p(q-1)$$ p ( q - 1 ) -periodic Legendre---Sidelnikov sequences, which combine the concepts of Legendre sequences and Sidelnikov sequences. We get lower and upper bounds on the linear complexity in different cases, and experiments show that the upper bounds can be attained. Remarkably, we associate the linear complexity of Legendre---Sidelnikov sequences with some famous primes including safe prime and Fermat prime. If $$2$$ 2 is a primitive root modulo $$\frac{q-1}{2}$$ q - 1 2 , and $$q$$ q is a safe prime greater than 7, the linear complexity is the period if $$p\equiv 3 \pmod 8$$ p ? 3 ( mod 8 ) ; $$p(q-1)-p+1$$ p ( q - 1 ) - p + 1 if $$p\equiv q \equiv 7 \pmod 8$$ p ? q ? 7 ( mod 8 ) , and $$p(q-1)-\frac{p-1}{2}$$ p ( q - 1 ) - p - 1 2 if $$p \equiv 7 \pmod 8, q \equiv 3 \pmod 8$$ p ? 7 ( mod 8 ) , q ? 3 ( mod 8 ) . If $$q$$ q is a Fermat prime, the linear complexity is the period if $$p \equiv 3 \pmod 8$$ p ? 3 ( mod 8 ) , and $$p(q-1)-q+2$$ p ( q - 1 ) - q + 2 if $$p \equiv 5 \pmod 8$$ p ? 5 ( mod 8 ) . It is very interesting that the Legendre---Sidelnikov sequence has maximal linear complexity and is balanced if we choose $$p=q$$ p = q to be some safe prime.

On error linear complexity of new generalized cyclotomic binary sequences of period p2

Counting Functions and Expected Values for the k-Error Linear Complexity

The linear complexity and k-error linear complexity of sequences are important measures of the strength of key-streams generated by stream ciphers. Based on the characters of the set of sequences with given linear complexity, people get the characterization of \(2^n\)-binary sequences with given k-error linear complexity for small k recently. In this paper, we put forward this study to get the distribution of linear complexity and k-error linear complexity of \(2^n\)-periodic binary sequences with fixed Hamming weight. First, we give the counting function of the number of \(2^n\)-periodic binary sequences with given linear complexity and fixed Hamming weight. Provide an asymptotic evaluation of this counting function when n gets large. Then we take a step further to study the distribution of \(2^n\)-periodic binary sequences with given 2-error linear complexity and fixed Hamming weight. Through an asymptotic analysis, we provide an estimate on the number of \(2^n\)-periodic binary sequences with given 2-error linear complexity and fixed Hamming weight.

Complexity measures for sequences, such as the linear complexity and the k-error linear complexity, play an important role in stream ciphers. This contribution studies the distribution of 1-error linear complexity of binary sequences with arbitrary prime period. For any odd prime N, the authors present all possible values of 1-error linear complexity of N-periodic binary sequences, and derive the exact formulas to count the number of N-periodic binary sequences with any given 1-error linear complexity.

The linear complexity and k-error linear complexity of a sequencehave been used as important measures for keystream strength, hencedesigning a sequence with high linear complexity and k-errorlinear complexity is a popular research topic in cryptography. In order to study k-error linear complexity of binarysequences with period 2n, a new tool called cube theory isdeveloped. In this paper, we first give ageneral decomposition approach to decompose a binary sequence with period 2n into some disjoint cubes. Second, a countingformula for m-cubes with the same linear complexity is derived, which is equivalent to the counting formula for k-error vectors. The counting formulaof 2n-periodic binary sequences which can be decomposedinto more than one cube is also investigated, which extends an important result by Etzion et al.

We study the linear complexity of sequences over the prime field \({\mathbb{F}_d}\) introduced by Sidel’nikov. For several classes of period length we can show that these sequences have a large linear complexity. For the ternary case we present exact results on the linear complexity using well known results on cyclotomic numbers. Moreover, we prove a general lower bound on the linear complexity profile for all of these sequences. The obtained results extend known results on the binary case. Finally we present an upper bound on the aperiodic autocorrelation.KeywordsSidel’nikov sequenceLinear complexityLinear complexity profileAperiodic autocorrelation

Sequences with an optimal autocorrelation property and with large linear complexity are needed in certain communication systems and cryptography. Tang and Ding obtained new classes of almost balanced binary sequences with an optimal autocorrelation value. In this reported work, it is explored as to when interleaved binary sequences of period 4 p obtained by Tang and Ding's technique from Hall sequences or Legendre and Hall sequences possess high linear complexity.

The linear complexity and the $ k $-error linear complexity of a binary sequence are important security measures for the security of the key stream. By studying binary sequences with the minimum Hamming weight, a new tool, named as the hypercube theory, is developed for $ p^n $-periodic binary sequences. In fact, the hypercube theory is based on a typical sequence decomposition and it is a very important tool for investigating the critical error linear complexity spectrum proposed by Etzion et al. To demonstrate the importance of hypercube theory, we first give a standard hypercube decomposition based on a well-known algorithm for computing linear complexity and show that the linear complexity of the first hypercube in the decomposition is equal to the linear complexity of the original sequence. Second, based on such decomposition, we give a complete characterization for the first decrease of the linear complexity for a $ p^n $-periodic binary sequence. This significantly improves the current existing results in literature. As to the importance of the hypercube, we finally derive a counting formula for the $ m $-hypercubes with the same linear complexity.

Following the idea of Xing et al., we investigate a general method for constructing families of pseudorandom sequences with low correlation and large linear complexity from elliptic curves over finite fields in this correspondence. With the help of the tool of exponential sums on elliptic curves, we study their periods, linear complexities, linear complexity profiles, distributions of r-patterns, periodic correlation, partial period distributions, and aperiodic correlation in detail. The results show that they have nice randomness.

Often in GNSS systems due to environmental issues the desired signal to be received may be degraded and attenuated due to multipath or some other reasons. Also the binary sequences used in GNSS systems are having low linear complexity and hence are easy to attack. To overcome above said problems it is essential to propose a method to generate binary sequences derived from chaotic real sequences which are found to have better odd correlation property than the GPS Ll C/A Signal and hence are more resistant to environment variations such as multipath or interference. The binary sequences generated from the proposed method are found to have very large linear complexity as compared with GPS Ll C/A Signals. This paper also presents the odd correlation properties of 32 GPS Ll C/A sequences, and compared with the odd correlation properties of the generated sequences in detail.

Binary sequences with optimal autocorrelation and large linear complexity have important applications in cryptography and communications. Very recently, a class of binary sequences of period 4p with optimal autocorrelation was proposed by interleaving four suitable Ding–Helleseth–Lam sequences (Des. Codes Cryptogr., https://doi.org/10.1007/s10623-017-0398-5 ), where p is an odd prime with $$p \equiv 1(\bmod 4)$$ . The objective of this paper is to determine the minimal polynomial and the linear complexity of this class of binary optimal sequences via a sequence polynomial approach. It turns out that this class of sequences has quite good linear complexity.

Comments on “a family of keystream generators with large linear complexity”

The k-error linear complexity(k-LC) of sequences is a very natural and useful generalization of the linear complexity(LC) which has been conveniently used as a measure of unpredictability of pseudorandom sequences, i.e., difficulty in recovering more of a sequence from a short, captured segment. However the effective method for computing the k-LC has been known only for binary sequences with period 2n (Stamp and Martin, 1993). This paper gives an alternative derivation of the Stamp-Martin algorithm. Our method can compute not only k-LC but also an error vector with Hamming weight ≤k which gives the k-LC.

Linear complexity and k-error linear complexity are the important measures for sequences in stream ciphers. This paper discusses the asymptotic behavior of the normalized k-error linear complexity $${L_{n,k}(\underline{s})/n}$$ of random binary sequences $${\underline{s}}$$ , which is based on one of Niederreiter's open problems. For k = n ?, where 0 ? ? ? 1/2 is a fixed ratio, the lower and upper bounds on accumulation points of $${L_{n,k}(\underline{s})/n}$$ are derived, which holds with probability 1. On the other hand, for any fixed k it is shown that $${\lim_{n\rightarrow\infty} L_{n,k}(\underline{s})/n = 1/2}$$ holds with probability 1. The asymptotic bounds on the expected value of normalized k-error linear complexity of binary sequences are also presented.