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.
A high linear complexity profile is a desirable feature of sequences used for cryptographical purposes. For a given binary
sequence we estimate its linear complexity profile in terms of the correlation measure, which was introduced by Mauduit and
Srkzy. We apply this result to certain periodic sequences including Legendre sequences, Sidelnikov sequences and other sequences
related to the discrete logarithm.