Open AccessBinary sequences and lattices constructed by discrete logarithms
Author(s) -
Yuchan Qi,
AUTHOR_ID,
Huaning Liu
Publication year - 2022
Publication title -
aims mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.329
H-Index - 15
ISSN - 2473-6988
DOI - 10.3934/math.2022259
Subject(s) - pseudorandomness , pseudorandom number generator , pseudorandom binary sequence , mathematics , binary number , pseudorandom generator theorem , pseudorandom generator , discrete logarithm , discrete mathematics , logarithm , dimension (graph theory) , modulo , combinatorics , algorithm , arithmetic , public key cryptography , computer science , encryption , mathematical analysis , operating system
In 1997, Mauduit and Sárközy first introduced the measures of pseudorandomness for binary sequences. Since then, many pseudorandom binary sequences have been constructed and studied. In particular, Gyarmati presented a large family of pseudorandom binary sequences using the discrete logarithms. Ten years later, to satisfy the requirement from many applications in cryptography (e.g., in encrypting "bit-maps'' and watermarking), the definition of binary sequences is extended from one dimension to several dimensions by Hubert, Mauduit and Sárközy. They introduced the measure of pseudorandomness for this kind of several-dimension binary sequence which is called binary lattices. In this paper, large families of pseudorandom binary sequences and binary lattices are constructed by both discrete logarithms and multiplicative inverse modulo $ p $. The upper estimates of their pseudorandom measures are based on estimates of either character sums or mixed exponential sums.