Abstract
Output sequences of the cryptographic pseudo-random number generator, known as the generalized self-shrinking generator, are obtained self-decimating Pseudo-Noise (PN)-sequences with shifted versions of themselves. In this paper, we present three different representations of this family of sequences. Two of them, the p and G-representations, are based on the parameters p and G corresponding to shifts and binary vectors, respectively, used to compute the shifted versions of the original PN-sequence. In addition, such sequences can be also computed as the binary sum of diagonals of the Sierpinski’s triangle. This is called the B-representation. Characteristics and generalities of the three representations are analyzed in detail. Under such representations, we determine some properties of these cryptographic sequences. Furthermore, these sequences form a family that has a group structure with the bit-wise XOR operation.
Highlights
Most of the devices that form part of the Internet-of-Things (IoT) require cryptographic security features to prevent users from data losses and the risks related to an improper use of passwords
In [27], other authors presented an extension of the self-shrinking generator to the Galois field of pn elements with p a prime integer, that is, the p-ary Generalized Self-Shrinking Generator (p-generalized self-shrinking generator (GSSG))
As we show in this paper, group theory has applications in cryptography, since the set of output sequences of the generalized self-shrinking generator has the structure of an additive group and some of the properties of this family of sequences can be deduced as a consequence of this fact
Summary
Most of the devices that form part of the Internet-of-Things (IoT) require cryptographic security features to prevent users from data losses and the risks related to an improper use of passwords. Used to generate random number sequences for cryptographic applications, such as key and nonces generation, digital signatures, and IoT security. In [25], the authors studied the randomness of the family of sequences generated by the GSSG by means of several complete and powerful batteries of statistical tests and graphical tools They provided a useful vision of the behavior of such sequences and proved their suitability for cryptographic applications. In [27], other authors presented an extension of the self-shrinking generator to the Galois field of pn elements with p a prime integer, that is, the p-ary Generalized Self-Shrinking Generator (p-GSSG) They proved that the sequences generated by this new generator have large periods and good statistical properties. We give a binomial expression of these sequences, providing a relation among binomial coefficients, binary sequences and group theory
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