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

Ying Zhang1, Huisheng Zhang1

  • 1Department of Mathematics, Dalian Maritime University, Dalian, 116024 China.

Springerplus
|November 8, 2016
PubMed
Summary
This summary is machine-generated.

This study establishes a condition to identify multivariate quadratic quasigroups (MQQs) over Galois fields. An algorithm is proposed to test quasigroups and generate MQ cryptosystems, enabling theoretical MQQ discovery.

Keywords:
Generating algorithmJudging methodMultivariate quadratic quasigroupQuasigroupVector-valued boolean functions

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Area of Science:

  • Cryptography
  • Abstract Algebra
  • Computational Mathematics

Background:

  • Multivariate Quadratic (MQ) cryptosystems are a significant area in post-quantum cryptography.
  • Multivariate Quadratic Quasigroup (MQQ) schemes are fundamental structures for designing MQ cryptosystems.
  • Existing literature lacks a general algorithm to determine if a quasigroup is an MQQ over Galois fields.

Purpose of the Study:

  • To establish a necessary and sufficient condition for a quasigroup of order p^n to be an MQQ over GF(p^n).
  • To develop an algorithm for verifying if a given quasigroup is an MQQ and generating its associated Boolean functions.
  • To provide a theoretical framework for obtaining all MQQs over GF(p^n).

Main Methods:

  • Derivation of a precise mathematical condition for MQQ identification.
  • Development of a computational algorithm based on the derived condition.
  • Application of the algorithm to quasigroups represented by multiplication tables.
  • Generation of d Boolean functions for verified MQQs.

Main Results:

  • A necessary and sufficient condition for a quasigroup to be an MQQ over GF(p^n) is established.
  • A novel algorithm is proposed to efficiently test quasigroups for the MQQ property.
  • The algorithm successfully generates the Boolean functions of MQQ schemes.
  • The method is validated through two illustrative examples.

Conclusions:

  • The proposed condition and algorithm provide a robust method for identifying and constructing MQQs over GF(p^n).
  • This work contributes to the theoretical understanding and practical design of MQ cryptosystems.
  • The algorithm facilitates the systematic generation of MQQs, advancing cryptographic research.