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• WG SEQUENCES GENERATOR

Dobbertin, New cyclic difference sets with Springer parameters, preprint, August 1999. Workshop “ Difference sets, sequences and their correlation properties”, Bad Windsheim, August 3–14, 1998, Kluwer, Dordrecht, pp. Dobbertin, Kasami power functions, permutation polynomials and cyclic difference sets, Proceedings of the NATO-A.S.I. John Dillon, Multiplicative difference sets via additive characters, Designs, Codes and Cryptography. Xingong Chang, Zongduo Dai and Guang Gong, Some cryptographic properties of exponential functions, Advances in Cryptology-AsiaCrypt’94, Lecture Notes in Computer Science, No.

WG SEQUENCES GENERATOR

pseudo-random sequence (number) generator.It is shown that the WG transformations regarded as Boolean functions have large linear span (this concept will be defined in this paper) and high degree. A criterion for whether the WG transformations regarded as Boolean functions are r-resilient is derived. When the WG transformation are regarded as Boolean functions, it is proved that they have high nonlinearity. Thus this is the first type of pseudo-random sequences with good correlation and statistic properties, large linear span and efficient implementation. It is shown that the WG transformation sequences, in addition to the known 2-level auto correlation, have threelevel cross correlation with m-sequences, large linear ! span increasing exponentially with n and efficient implementation. The other is to exhibit the security property of the WG transformations regarded as Boolean functions. One is to present randomness or unpredictability of the WG transformation sequences. In this paper, we investigate a two-faced property of the WG transformation sequences for application in stream ciphers and pseudo-random number generators. Dillon first proved this result for odd n in 1998, and finally, Dobbertin and Dillon proved it for even n in 1999. Later on, No, Chung and Yun found another way to construct the WG sequences and verified their result for 5 ≤ n ≤ 23. These sequences were discovered by Golomb, Gong and Gaal in 1998 and verified for 5 ≤ n ≤ 20. Welch-Gong (WG) transformation sequences are binary sequences of period 2 n.