We estimate multiplicative character sums over the integers with a fixed sum of binary digits and apply these results to study
the distribution of products of such integers in residues modulo a prime p. Such products have recently appeared in some cryptographic
algorithms, thus our results give some quantitative assurances of their pseudorandomness which is crucial for the security
of these algorithms.
Binary and quaternary sequences are the most important sequences in view of many practical applications. Any quaternary sequence
can be decomposed into two binary sequences and any two binary sequences can be combined into a quaternary sequence using
the Gray mapping. We analyze the relation between the measures of pseudorandomness for the two binary sequences and the measures
for the corresponding quaternary sequences, which were both introduced by Mauduit and Sárközy. Our results show that each
‘pseudorandom’ quaternary sequence corresponds to two ‘pseudorandom’ binary sequences which are ‘uncorrelated’.
We prove a bound on sums of products of multiplicative characters of shifted Fermat quotients modulo p. From this bound we derive results on the pseudorandomness of sequences of modular discrete logarithms of Fermat quotients
modulo p: bounds on the well-distribution measure, the correlation measure of order ℓ, and the linear complexity.
In this paper we look at the security of two block ciphers which were both claimed in the published literature to be secure
against differential crypt-analysis (DC). However, a more careful examination shows that none of these ciphers is very secure
against... differential cryptanalysis, in particular if we consider attacks with sets of differentials. For both these ciphers
we report new perfectly periodic (iterative) aggregated differential attacks which propagate with quite high probabilities.
The first cipher we look at is GOST, a well-known Russian government encryption standard. The second cipher we look at is
PP-1, a very recent Polish block cipher. Both ciphers were designed to withstand linear and differential cryptanalysis. Unhappily,
both ciphers are shown to be much weaker than expected against advanced differential attacks. For GOST, we report better and
stronger sets of differentials than the best currently known attacks presented at SAC 2000  and propose the first attack
ever able to distinguish 16 rounds of GOST from random permutation. For PP-1 we show that in spite of the fact, that its S-box
has an optimal theoretical security level against differential cryptanalysis , , our differentials are strong enough
to allow to break all the known versions of the PP-1 cipher.