AIMS Mathematics (Jan 2022)

Binary sequences and lattices constructed by discrete logarithms

  • Yuchan Qi,
  • Huaning Liu

DOI
https://doi.org/10.3934/math.2022259
Journal volume & issue
Vol. 7, no. 3
pp. 4655 – 4671

Abstract

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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.

Keywords