A pseudo-random sequence generation scheme based on RNS and permutation polynomials

Abstract

Long period pseudo-random sequence plays an important role in modern information processing systems. Base on residue number system (RNS) and permutation polynomials over finite fields, a pseudorandom sequence generation scheme is proposed in this paper. It extends several short period random sequences to a long period pseudo-random sequence by using RNS. The short period random sequences are generated parallel by the iterations of permutation polynomials over finite fields. Due to the small dynamic range of each iterative calculation, the bit width in hardware implementation is reduced. As a result, we can use full look-up table (LUT) architecture to achieve high-speed sequence output. The methods to find proper permutation polynomials to generate long period sequences and the optimization algorithm of Chinese remainder theorem (CRT) mapping are also proposed in this paper. The period of generated pseudorandom sequence can exceed 2100 easily based on common used field programmable gate array (FPGA) chips. Meanwhile, this scheme has extensive freedom in choosing permutation polynomials. For example, 10905 permutation polynomials meet the long period requirement over the finite field F q with q ≢ 1(mod 3) and q ⩽ 503. The hardware implementation architecture is simple and multiplier free. Using Xilinx XC7020 FPGA chip, we implement a sequence generator with the period over 250, which only costs 20 18kb-BRAMs (block RAM) and a small amount of logics. And the speed can reach 449.236 Mbps. The National Institute of Standards and Technology (NIST) test results show that the sequence has good random properties.