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An algorithm for judging and generating multivariate quadratic quasigroups over Galois fields

SPRINGERPLUS 2016年第1期5卷 1845页

作者： Zhang, Ying Zhang, Huisheng Dalian Maritime Univ Dept Math Dalian 116024 Peoples R China

As the basic cryptographic structure for multivariate quadratic quasigroup (MQQ) scheme, MQQ has been one of the latest tools in designing MQ cryptosystem. There have been several construction methods for MQQs in the literature, however, the algorithm for judging whether quasigroups of any order are MQQs over Galois fields is still lacking. To this end, the objective of this paper is to establish a necessary and sufficient condition for a given quasigroup of order p(kd) to be an MQQ over GF(p(k)). Based on this condition, we then propose an algorithm to justify whether or not a given quasigroup in the form of multiplication table of any order pkd is an MQQ over GF(p(k)), and generate the d boolean functions of the MQQ if the quasigroup is an MQQ. As a result, we can obtain all the MQQs over GF(p(k)) in theory using the proposed algorithm. Two examples are provided to illustrate the validity of our method.

关键词： Quasigroup Multivariate quadratic quasigroup vector-valued boolean functions Judging method Generating algorithm

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A GENERALIZED ALGORITHM FOR JUSTIFYING AND GENERATING BILINEAR MULTIVARIATE QUADRATIC QUASIGROUPS OVER GALOIS FIELDS

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INTERNATIONAL JOURNAL OF INNOVATIVE COMPUTING INFORMATION AND CONTROL 2017年第4期13卷 1415-1423页

作者： Zhang, Ying Dalian Maritime Univ Dept Math 1 Linghai Rd Dalian 116026 Peoples R China

As the basic cryptographic structure for MQQ scheme, multivariate quadratic quasigroup (MQQ) has been one of the latest tools in designing cryptographic system. However, it is limited to the operation over GF(2) and lacks the general understanding. In this paper, we propose a necessary and sufficient condition to verify whether a given quasigroup of any order pi'd is a bilinear MQQ over GF(pk), which shows checking whether an arbitrary quasigroup is bilinear MQQ is equivalent to solving a simple matrix equation. Based on this newly established condition, a deterministic algorithm is proposed to judge whether or not a given quasigroup is bilinear MQQ, and to generate bilinear MQQ if it is. Two examples are given to show the validity of our results.

关键词： Quasigroup Multivariate quadratic quasigroup vector-valued boolean functions Judging method Generating algorithm

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