An algorithm for judging and generating multivariate quadratic quasigroups over Galois fields.
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 orderpkdto be an MQQ over [Formula: see text]. 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 orderpkdis an MQQ over [Formula: see text], and generate thedBoolean functions of the MQQ if the quasigroup is an MQQ. As a result, we can obtain all the MQQs over [Formula: see text] in theory using the proposed algorithm. Two examples are provided to illustrate the validity of our method.
Department of Mathematics, Dalian Maritime University, Dalian, 116024 China.
Ying
Y
Zhang
Department of Mathematics, Dalian Maritime University, Dalian, 116024 China.
Huisheng
H
Zhang
Generating algorithm; Judging method; Multivariate quadratic quasigroup; Quasigroup; Vector-valued boolean functions