基于费马商的r元序列的迹表示
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摘要
基于费马商构造的伪随机序列均具有良好的密码学性质,本文根据有限域上迹函数理论及陪集理论,通过确定基于费马商构造的r元序列的离散傅里叶变换,研究得到该序列的迹函数表示.所给出的迹函数表示不仅对序列的工程实现有重要意义,而且对分析序列的其他伪随机性质提供了新的工具和方法.
Families of pseudorandom sequences derived from Fermat quotients possess good cryptographic properties.In this paper,based on the theory of trace function in the finite field and cosets,we firstly determine the discrete Fourier transform( DFT) of the r-ary sequences derived from Fermat quotients. Then from which we obtain the sequences' trace representation. The trace representation we determined plays an important role in the engineering realization of the sequences,and also provides a newtool for analyzing the pseudorandom properties of the sequences.
Families of pseudorandom sequences derived from Fermat quotients possess good cryptographic properties.In this paper,based on the theory of trace function in the finite field and cosets,we firstly determine the discrete Fourier transform( DFT) of the r-ary sequences derived from Fermat quotients. Then from which we obtain the sequences' trace representation. The trace representation we determined plays an important role in the engineering realization of the sequences,and also provides a newtool for analyzing the pseudorandom properties of the sequences.
引文
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[2]Golomb S W,Gong G.Signal designs with correlation:for w ireless communications,cryptography and radar applications[M].Cambridge University Press,2005.
[3]Dai Z,Gong G,Song H Y.Trace representation and linear complexity of binary e-th residue sequences[A].Proceedings of International Workshop on Coding and Cryptography(WCC2003)[C].France,2003.121-133.
[4]Dai Z,Gong G,Song H Y.Ye D.A trace representation of binary Jacobi sequences[J].Discrete M athematics,2009,309(6):1517-1527.
[5]Dai Z,Gong G,Song H Y,Ye D,et al.Trace representation and linear complexity of binary e-th pow er residue sequences of period P[J].IEEE Transactions on Information Theory,2011,57(3):1530-1547.
[6]杜小妮,陈智雄.关于Legendre序列迹表示的注记[J].电子学报,2011,39(4):869-871.Du Xiaoni,Chen Zhixiang.A note on trace representation of legendre sequences[J].Acta Electronica Sinica,2011,39(4):869-871.(in Chinese)
[7]Ostafe A,Shparlinski I E.Pseudorandomness and dynamics of Fermat quotients[J].SIAM Journal on Discrete M athematics,2011,25(1):50-71.
[8]Chen Z,Ostafe A,Winterhof A.Structure of pseudorandom numbers derived from Fermat quotients[A].International Workshop on the Arithmetic of Finite Fields[C].Springer Berlin Heidelberg,2010.73-85.
[9]Gomez D,Winterhof A.Multiplicative character sums of Fermat quotients and pseudorandom sequences[J].Periodica M athematica Hungarica,2012,64(2):161-168.
[10]Wu C,Chen Z,Du X.Binary threshold sequences derived from carmichael quotients w ith even numbers modulus[J].IEICE Transactions on Fundamentals of Electronics,Communications and Computer Sciences,2012,95(7):1197-1199.
[11]Chen Z,Hu L,Du X.Linear complexity of some binary sequences derived from Fermat quotients[J].China Communications,2012,9(2):105-108.
[12]Du X,Chen Z,Hu L.Linear complexity of binary sequences derived from Euler quotients w ith prime-pow er modulus[J].Information Processing Letters,2012,112(14):604-609.
[13]Chen Z,Du X.On the linear complexity of binary threshold sequences derived from Fermat quotients[J].Designs,Codes and Cryptography,2013,67(3):317-323.
[14]Chen Z.Trace representation and linear complexity of binary sequences derived from Fermat quotients[J].Science China Information Sciences,2014,57(11):1-10.
[15]Ye Z,Ke P,Zhang S,et al.Some notes on pseudorandom binary sequences derived from Fermat-Euler quotients[J].IEICE Transactions on Fundamentals of Electronics,Communications and Computer Sciences,2015,98(10):2199-2202.
[16]Udaya P,Siddiqi M U.Generalized GMW quadriphase sequences satisfying the Welch bound w ith equality[J].Applicable Algebra in Engineering,Communication and Computing,2000,10(3):203-225.
[17]Du X,Wu C,Wei W.An extension of binary threshold sequences from Fermat quotients[J].Advances in M athematics of Communications,2016,10(4):743-752.