An algorithm for computing minimal bidirectional linear recurrence relations

We consider the problem of computing a linear recurrence relation (or equivalently a linear feedback shift register) of minimum order for a finite sequence over a field, with the additional requirement that not only the highest but also the lowest coefficient of the recurrence is nonzero. Such a recurrence relation can then be used to generate the sequence in both directions (increasing or decreasing order of indices), so we call it bidirectional. If the field is finite, a sequence is periodic if and only if it admits a bidirectional linear recurrence relation. For solving the above problem we propose an algorithm similar to the Berlekamp-Massey algorithm and prove its correctness. We describe the set of all solutions to this problem and show that if a sequence admits more than one linear recurrence relation then it admits a bidirectional one. We also prove some properties regarding the bidirectionality of the recurrences of the prefixes of the sequence

2-Adic Complexity of a Sequence Obtained from a Periodic Binary Sequence by Either Inserting or Deleting k Symbols within One Period

In this paper, we propose a method to get the lower bounds of the 2-adic complexity of a sequence obtained from a periodic sequence over GF(2) by either inserting or deleting k symbols within one period. The results show the variation of the distribution of the 2-adic complexity becomes as k increases. Particularly, we discuss the lower bounds when k respectively

Complexity measures for classes of sequences and cryptographic apllications

Pseudo-random sequences are a crucial component of cryptography, particularly in stream cipher design. In this thesis we will investigate several measures of randomness for certain classes of finitely generated sequences. We will present a heuristic algorithm for calculating the k-error linear complexity of a general sequence, of either finite or infinite length, and results on the closeness of the approximation generated. We will present an linear time algorithm for determining the linear complexity of a sequence whose characteristic polynomial is a power of an irreducible element, again presenting variations for both finite and infinite sequences. This algorithm allows the linear complexity of such sequences to be determined faster than was previously possible. Finally we investigate the stability of m-sequences, in terms of both k-error linear complexity and k-error period. We show that such sequences are inherently stable, but show that some are more stable than others

ANALYSIS OF SECURITY MEASURES FOR SEQUENCES

Stream ciphers are private key cryptosystems used for security in communication and data transmission systems. Because they are used to encrypt streams of data, it is necessary for stream ciphers to use primitives that are easy to implement and fast to operate. LFSRs and the recently invented FCSRs are two such primitives, which give rise to certain security measures for the cryptographic strength of sequences, which we refer to as complexity measures henceforth following the convention. The linear (resp. N-adic) complexity of a sequence is the length of the shortest LFSR (resp. FCSR) that can generate the sequence. Due to the availability of shift register synthesis algorithms, sequences used for cryptographic purposes should have high values for these complexity measures. It is also essential that the complexity of these sequences does not decrease when a few symbols are changed. The k-error complexity of a sequence is the smallest value of the complexity of a sequence obtained by altering k or fewer symbols in the given sequence. For a sequence to be considered cryptographically ‘strong’ it should have both high complexity and high error complexity values. An important problem regarding sequence complexity measures is to determine good bounds on a specific complexity measure for a given sequence. In this thesis we derive new nontrivial lower bounds on the k-operation complexity of periodic sequences in both the linear and N-adic cases. Here the operations considered are combinations of insertions, deletions, and substitutions. We show that our bounds are tight and also derive several auxiliary results based on them. A second problem on sequence complexity measures useful in the design and analysis of stream ciphers is to determine the number of sequences with a given fixed (error) complexity value. In this thesis we address this problem for the k-error linear complexity of 2n-periodic binary sequences. More specifically: 1. We characterize 2n-periodic binary sequences with fixed 2- or 3-error linear complexity and obtain the counting function for the number of such sequences with fixed k-error linear complexity for k = 2 or 3. 2. We obtain partial results on the number of 2n-periodic binary sequences with fixed k-error linear complexity when k is the minimum number of changes required to lower the linear complexity