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Mathematics
Discrete Mathematics and Combinatorics

Symmetric Clusters in Hierarchy with Cryptographic Properties

Profile image of Jeffrey ZhengJeffrey Zheng

2018

https://doi.org/10.1007/978-981-13-2282-2_5
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28 pages

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Abstract

Symmetric Boolean functions play a key role in stream ciphers. Symmetric constructions provide core components in cryptographic applications. In this chapter, four meta symmetric clustering schemes (combination, crossing, variant and rotation) are organized in a hierarchy for n variables of 0-1 vectors in measuring phase spaces. Local counting properties in a cluster and global counting properties in a given level are formulated. From selected symmetric clusters, a number of various symmetric Boolean functions are formulated. Counting properties on symmetric clusters, vectors in selected clusters and special symmetric Boolean functions are listed. Four sets of symmetric Boolean functions are compared. Properties of symmetric clusters and Boolean functions are discussed. Main results are expressed in theorems and tables. Among four meta schemes, the variant scheme presents novel properties approximately with O n 2 /4 clusters on a 2D phase space different from other schemes: combinatorial O (n), crossing O (n/2) and rotation O (2 n /n) on 1D measuring phase spaces, respectively. The variant pseudorandom number generator is a similar approach on RC4 and HC128 stream ciphers using word-oriented 0-1 vectors. Further advanced researches and explorations on relevant optimal configurations are required.

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