Cryptographic properties of the Welch-Gong transformation sequence generators

Welch-Gong (WG) transformation sequences are binary sequences of period 2(n) - 1 with two-level autocorrelation. These sequences were discovered by Golomb, Gong, and Gaal in 1998 and they verified the validity of their construction for 5 less than or equal to n less than or equal to 20. Later, No, Chung, and Yun found another way to construct the WG sequences and verified their result for 5 less than or equal to n less than or equal to 23. Dillon first proved this result for odd n in 1998, and, finally, Dobbertin and Dillon proved it for even n in 1999. In this paper, we investigate a two-faced property of the WG transformation sequences for application in stream ciphers and pseudorandom number generators. One is to present the randomness or unpredictability of the WG transformation sequences. The other is to exhibit the security properties of the WG transformations regarded as Boolean functions. In particular, we prove that the WG transformation sequences, in addition to the known two-level autocorrelation and three-level cross correlation with m-sequences, have the ideal 2-tuple distribution, and large linear span increasing exponentially with n. Moreover, it can be implemented efficiently. This is the first type of pseudorandom sequences with good correlation, statistic properties, large linear span, and efficient implementation. When WG transformations are regarded as Boolean functions, they have high nonlinearity. We derive a criterion for the Boolean representation of WG transformations to be r-resilient and show that they are at least I-resilient under some basis of the finite field GF (2(n)). An algorithm to find such bases is given. The degree and linear span of WG transformations are presented as well.