Evaluation of solving time for multivariate quadratic equation system using XL algorithm over small finite fields on GPU

Satoshi Tanaka, Chen Mou Cheng, Kouichi Sakurai

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

The security of multivariate public-key cryptography is largely determined by the complexity of solving multivariate quadratic equations over finite fields, a.k.a. the MQ problem. XL (eXtended Linearization) is an efficient algorithm for solving the MQ problem, so its running time is an important indicator for the complexity of solving the MQ problem. In this work, we implement XL on graphics processing unit (GPU) and evaluate its solving time for theMQ problem over several small finite fields, namely, GF(2), GF(3), GF(5), and GF(7). Our implementations can solve MQ instances of 74 equations in 37 unknowns over GF(2) in 36,972 s, 48 equations in 24 unknowns over GF(3) in 933 s, 42 equations in 21 unknowns over GF(5) in 347 s, as well as 42 equations in 21 unknowns over GF(7) in 387 s. Moreover, we can also solve the MQ instance of 48 equations in 24 unknowns over GF(7) in 34,882 s, whose complexity is about O(267) with exhaustive search.

Original languageEnglish
Title of host publicationMathematics and Computing - ICMC 2015
EditorsDebasis Giri, Ram N. Mohapatra, Dipanwita Roy Chowdhury
PublisherSpringer New York LLC
Pages349-361
Number of pages13
ISBN (Print)9788132224518
DOIs
Publication statusPublished - Jan 1 2015
Event2nd International Conference on Mathematics and Computing, ICMC 2015 - Haldia, India
Duration: Jan 5 2015Jan 10 2015

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume139
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Other

Other2nd International Conference on Mathematics and Computing, ICMC 2015
CountryIndia
CityHaldia
Period1/5/151/10/15

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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