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

作者：Zhang, Ying^*; Zhang, Huisheng

来源：SpringerPlus, 2016, 5(1): 1845.

DOI：10.1186/s40064-016-3525-2

• Quasigroup
• Multivariate quadratic quasigroup
• Vector-valued boolean functions
• Judging method
• Generating algorithm

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.

• 出版日期2016-10-22
• 单位大连海事大学