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

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Science China(Information Sciences) 2018年第8期61卷 183-194页

作者： Shang MA Jianfeng LIU Zeguo YANG Yan ZHANG Jianhao HU National Key Laboratory of Science and Technology on Communications University of Electronic Science and Technology of China

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) ***, this scheme has extensive freedom in choosing permutation polynomials. For example, 10905 permutation polynomials meet the long period requirement over the finite field Fq 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 18 kb-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.

关键词： pseudo-random sequence residue number system(RNS) permutation polynomial high speed long period field programmable gate array(FPGA)

A proof of maximum contention-free property of interleavers for Turbo codes using permutation polynomials over integer rings

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Journal of Zhejiang University-Science A(Applied Physics & Engineering) 2007年第1期8卷 24-27页

作者： MA Xin-rui XU You-yun ZHANG Le Department of Electronic Engineering Shanghai Jiao Tong University Shanghai 200240 China Institute of Communication Engineering PLA University of Science and Technology Nanjing 210007 China

It is well known that interleavers play a critical role in Turbo coding/decoding schemes, and contention-free interleaver design has become a serious problem in the paraUelization of Turbo decoding, which is indispensable to meet the demands for high throughput and low latency in next generation mobile communication systems. This paper unveils the fact that interleavers based on permutation polynomials modulo N are contention-free for every window size W, a factor of the intedeaver length N, which, also called maximum contention-free interleavers.

关键词： Turbo codes Integer ring permutation polynomial Interleaver Maximum contention-free (MCF)

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permutation polynomials of ■

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Journal of Mathematical Research with Applications 2023年第2期43卷 166-174页

作者： Danyao WU Pingzhi YUAN School of Computer Science and Technology Dongguan University of Technology School of Mathematics South China Normal University

Let m be a positive integer and Fqr be a finite field with the characteristic of p. We prove that if p > m2-m and q ≡ 1(mod m), the polynomial x(1+(q-1)/m) + ax(a =≠ 0) is not a permutation polynomial over Fqr(r ≥ 2). And we verify that if q ≡ 1(mod 7) and p≠ 2, 3,then the polynomial x(1+(q-1)/7) + ax(a =≠ 0) is not a permutation polynomial over Fqr(r ≥ 2).

关键词： polynomial permutation polynomial finite field

On permutation polynomials in Two Variables over ■/p^2■

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Acta Mathematica Sinica,English Series 2009年第7期25卷 1191-1200页

作者： Qi Jiao WEI Qi Fan ZHANG College of Mathematics Sichuan University Chengdu 610064 P. R. China Chengdu University of Information Technology Chengdu 610059 P. R. China

We give a good characterization of the permutation polynomials in two variables over Z/p2Z.

关键词： permutation polynomial Witt polynomial Witt vector ring

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New permutation Reversed Dickson polynomials over Finite Fields

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Algebra Colloquium 2023年第1期30卷 111-120页

作者： Kaimin Cheng School of Mathematics and Information China West Normal University NanchongSichuan 637009China

Let p be an odd prime,and n,k be nonnegative *** Dn,k(1,x)be the reversed Dickson polynomial of the(k+1)-th *** this paper,by using Hermite's criterion,we study the permutational properties of the reversed Dickson polynomials Dn,k(1,x)over finite fields in the case of n=mp* with 0permutation polynomials over finite fields with characteristic p when n=2p^(s),or n=3p^(s),or n=4p^(s).

关键词： permutation polynomial reversed Dickson polynomial Hermite's criterion

On the Differential Uniformity and Nonlinearity of a Class of permutation Quadrinomials Over F_(2 2 m)

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Communications in Mathematical Research 2022年第2期38卷 223-245页

作者： Mengyu Hu Nian Li Xiangyong Zeng Hubei Key Laboratory of Applied Mathematics Faculty of Mathematics and StatisticsHubei UniversityWuhanChina

permutation polynomials with low differential uniformity and high nonlinearity are preferred in cryptographic systems. In 2018, Tu, Zeng and Helleseth constructed a new class of permutation quadrinomials over the finite field F_(2 2m) for an odd integer m. In this paper, we aim to investigate the differential uniformity and nonlinearity of this class of permutation polynomials so as to find 4-uniform permutation polynomials with high nonlinearity.

关键词： Differential uniformity finite field nonlinearity permutation polynomial

Zeros of Complete Symmetric polynomials over Finite Fields

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Journal of Systems Science & Complexity 2024年第4期37卷 1772-1788页

作者： CAO Wei School of Mathematics and Statistics Minnan Normal UniversityZhangzhou 363000China

Wan and Zhang(2021) obtained a nontrivial lower bound for the number of zeros of complete symmetric polynomials over finite fields,and proposed a problem whether their bound can be *** this paper,the author improves Wan-Zhang's bound from three *** proposed results are based on the estimates related to the number of certain permutations and the value sets of non-permutation polynomials associated to the complete symmetric *** the author believes that there are still possibilities to improve the bounds and hence Wan-Zhang's bound.

关键词： Complete symmetric polynomial finite field permutation polynomial value set

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On s-uniform property of compressing sequences derived from primitive sequences modulo odd prime powers

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Science China(Information Sciences) 2017年第5期60卷 84-95页

作者： Yupeng JIANG Qun-Xiong ZHENG Dongdai LIN State Key Laboratory of Information Security Institute of Information EngineeringChinese Academy of Sciences State Key Laboratory of Mathematical Engineering and Advanced Computing Zhengzhou Information Science and Technology Institute

Let Z/（p;） be the integer residue ring modulo pewith p an odd prime and e 2. We consider the suniform property of compressing sequences derived from primitive sequences over Z/（p;）. We give necessary and sufficient conditions for two compressing sequences to be s-uniform with α provided that the compressing map is of the form φ(x;, x;,..., x;) = g(x;) + η(x;, x;,..., x;), where g(x;) is a permutation polynomial over Z/（p） and η is an（e-1）-variable polynomial over Z/（p）.

关键词： compressing map linear recurring sequence primitive sequence permutation polynomial s uniform

A Class of New Quadratic Vectorial Bent Functions

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Chinese Journal of Electronics 2020年第5期29卷 873-879页

作者： PANG Tingting ZENG Xiangyong LI Nian XU Yunge The Faculty of Mathematics and Statistics Hubei Key Laboratory of Applied Mathematics Hubei University

A class of quadratic vectorial bent functions having the form F（x）=Trm~n(ax~(2s1+1))+Tr1~n(bx~(2s2+1)) is investigated,where n,m,s1,s2are positive integers and the coefficients a,b belong to the finite field F2~*** some discussions on the permutation property of certain linearized polynomials over F2~n,several classes of quadratic vectorial bent functions are presented for special cases of n,and it is also verified by computer that some vectorial bent functions proposed are extended affine inequivalent to all known quadratic vectorial bent functions.

关键词： Vectorial bent function permutation polynomial Extended affine equivalence Finite field