On the correlation of binary sequences

In a series of papers Mauduit and Sárközy (partly with further coauthors) studied finite pseudorandom binary sequences. In particular, one of the most important applications of pseudorandomness is cryptography. If, e.g., we want to use a binary sequence EN∈{-1,+1}NEN∈{-1,+1}N (after transforming it into a bit sequence) as a key stream in the standard Vernam cipher [A. Menezes, P. van Oorschot, R. Vanstone, Handbook of Applied Cryptography, CRC Press, Boca Raton, 1997], then ENEN must possess certain pseudorandom properties. Does ENEN need to possess both small well-distribution measure and, for any fixed small k, small correlation measure of order k  ? In other words, if W(EN)W(EN) is large, resp. Ck(EN)Ck(EN) is large for some fixed small k  , then can the enemy utilize this fact to break the code? The most natural line of attack is the exhaustive search: the attacker may try all the binary sequences EN∈{-1,+1}NEN∈{-1,+1}N with large W(EN)W(EN), resp. large Ck(EN)Ck(EN), as a potential key stream. Clearly, this attack is really threatening only if the number of sequences EN∈{-1,+1}NEN∈{-1,+1}N with(i)large W(EN)W(EN), resp.(ii)large Ck(EN)Ck(EN)is “much less” than the total number 2N2N of sequences in {-1,+1}N{-1,+1}N, besides one needs a fast algorithm to generate the sequences of type (i), resp. (ii).The case (i) is easy, thus, for the sake of completeness, here we just present an estimate for the number of sequences ENEN with large W(EN)W(EN).The case (ii), i.e., the case of large correlation is much more interesting: this case will be studied in Section 2.In Section 3 we will sharpen the results of Section 2 in the special case when the order of the correlation is 2.Finally, in Section 4 we will study a lemma, which plays a crucial role in the estimation of the correlation in some of the most important constructions of pseudorandom binary sequences.