Some cryptographic properties of exponential functions

Welch-Gong (WG) transformation sequences are binary sequences of period 2 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 20. Later, No, Chung, and Yun found another way to construct the WG sequences and verified their result for 5 23. Dillon first proved this result for odd in 1998, and, finally, Dobbertin and Dillon proved it for even 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-sequences, have the ideal 2-tuple distribution, and large linear span increasing exponentially with. 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-resilient and show that they are at least 1-resilient under some basis of the finite field GF (2). An algorithm to find such bases is given. The degree and linear span of WG transformations are presented as well.

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“…Let be the WG transformation defined by (1). Then the linear span of any Boolean form of is equal to the linear span of , i.e.,…”

Section: B Existence and Construction Of -Resilient Wg Transformationsmentioning

confidence: 99%

“…Therefore, there is a connection among binary sequences with period , polynomial functions from GF to GF , and Boolean functions in variables. Chang, Dai, and Gong [1] tried to use this connection. They applied -sequences with three-level cross correlation to construct Boolean functions with maximal nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

Cryptographic properties of the Welch-Gong transformation sequence generators

Gong

Youssef

2002

IEEE Trans. Inform. Theory

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“…Remark 7. Chang, Dai and Gong [1] discussed how to construct Boolean functions with the maximal non-linearity in terms of binary m-sequences with threevalued cross correlation. In particular, they discussed the case T r(x r ) for some special choices of r. Gong and Golomb [10], pointed out that the monomial functions T r(x r ) are not secure when used as combining functions or filtering functions in stream cipher systems or block cipher modes, because they correspond to m-sequences.…”

Section: The Hadamard Transform Of F (X) and The Walsh Transform Of Fmentioning

confidence: 99%

“…Thus there is a connection among binary sequences with period 2 n − 1, polynomial functions from GF (2 n ) to GF (2) and Boolean functions in n variables. Chang, Dai and Gong [1] tried to use this connection. I.e., they applied m-sequences with three-level cross correlation to construct Boolean functions with the maximal non-linearity.…”

Section: Introductionmentioning

confidence: 99%

On Welch-Gong Transformation Sequence Generators

Gong

Youssef

2001

Selected Areas in Cryptography

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Welch-Gong (WG) transformation sequences are binary sequences of period 2 n − 1 with 2-level auto correlation. These sequences were discovered by Golomb, Gong and Gaal in 1998 and verified for 5 ≤ n ≤ 20. Later on, No, Chung and Yun found another way to construct the WG sequences and verified their result for 5 ≤ n ≤ 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 pseudo-random number generators. One is to present randomness or unpredictability of the WG transformation sequences. The other is to exhibit the security property of the WG transformations regarded as Boolean functions. 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. Thus this is the first type of pseudo-random sequences with good correlation and statistic properties, large linear span and efficient implementation. When the WG transformation are regarded as Boolean functions, it is proved that they have high nonlinearity. A criterion for whether the WG transformations regarded as Boolean functions are r-resilient is derived. 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.

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Progress and Prospect of Some Fundamental Research on Information Security in China

Feng

Wang

2006

J Comput Sci Technol

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Some cryptographic properties of exponential functions

Cited by 5 publications

References 3 publications

Cryptographic properties of the Welch-Gong transformation sequence generators

Cryptographic properties of the Welch-Gong transformation sequence generators

On Welch-Gong Transformation Sequence Generators

Progress and Prospect of Some Fundamental Research on Information Security in China

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