Interleaved sequences of geometric sequences binarized with Legendre symbol of two types
Kazuyoshi Tsuchiya, Yasuyuki Nogami, Satoshi Uehara

TL;DR
This paper introduces interleaved sequences of two types of geometric sequences based on Legendre symbols, which achieve balance, doubled period, and favorable correlation and complexity properties for cryptographic pseudorandom number generation.
Contribution
It presents a novel interleaving method for geometric sequences of two types that enhances their statistical and cryptographic properties.
Findings
Interleaved sequences have the balance property.
Interleaving doubles the period of geometric sequences.
Sequences exhibit good correlation and high linear complexity.
Abstract
A pseudorandom number generator is widely used in cryptography. A cryptographic pseudorandom number generator is required to generate pseudorandom numbers which have good statistical properties as well as unpredictability. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a binary sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol. They showed the geometric sequences have good properties for the period, periodic autocorrelation and linear complexity. However, the geometric sequences do not have the balance property. In this paper, we introduce geometric…
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