634 research outputs found
Maximum-order complexity and -adic complexity
The -adic complexity has been well-analyzed in the periodic case. However,
we are not aware of any theoretical results on the th -adic complexity of
any promising candidate for a pseudorandom sequence of finite length or
results on a part of the period of length of a periodic sequence,
respectively. Here we introduce the first method for this aperiodic case. More
precisely, we study the relation between th maximum-order complexity and
th -adic complexity of binary sequences and prove a lower bound on the
th -adic complexity in terms of the th maximum-order complexity. Then
any known lower bound on the th maximum-order complexity implies a lower
bound on the th -adic complexity of the same order of magnitude. In the
periodic case, one can prove a slightly better result. The latter bound is
sharp which is illustrated by the maximum-order complexity of -sequences.
The idea of the proof helps us to characterize the maximum-order complexity of
periodic sequences in terms of the unique rational number defined by the
sequence. We also show that a periodic sequence of maximal maximum-order
complexity must be also of maximal -adic complexity
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
On the complexity of algebraic number I. Expansions in integer bases
Let be an integer. We prove that the -adic expansion of every
irrational algebraic number cannot have low complexity. Furthermore, we
establish that irrational morphic numbers are transcendental, for a wide class
of morphisms. In particular, irrational automatic numbers are transcendental.
Our main tool is a new, combinatorial transcendence criterion
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