Original title: Semigroup Structure of Sets of Solutions to Equation X^s = X^m
Authors: Porubský, Štefan
Document type: Research reports
Year: 2017
Language: eng
Series: Technical Report, volume: V-1247
Abstract: Using an idempotent semigroup approach we describe the semigroup and group structure of the set of solutions to equation X^m = X^s in successive steps over a periodic commutative semigroup, over multiplicative semigroups of factor rings of residually finite commutative rings and finally over multiplicative semigroups of factor rings of residually finite commutative principal ideal domains. The analysis is done through the use of the maximal subsemigroups and groups corresponding to an idempotent of the corresponding semigroup and in the case of residually finite PID’s employing the available analysis of the Euler-Fermat Theorem as given in [11]. In particular the case when this set of solutions is a union of groups is handled. As a simple application we show a not yet noticed group structure of the set of solutions to x^n = x connected with the message space of RSA cryptosystems and Fermat pseudoprimes.
Keywords: equation X^s = X^m; finite commutative ring with identity element; idempotent; maximal group corresponding to an idempotent; maximal semigroup corresponding to an idempotent; residually finite commutative principal ideal domains; set of solutions
Project no.: GA17-02804S (CEP)
Funding provider: GA ČR

Institution: Institute of Computer Science AS ČR (web)
Document availability information: Fulltext is available on demand via the digital repository of the Academy of Sciences.
Original record: http://hdl.handle.net/11104/0273446

Permalink: http://www.nusl.cz/ntk/nusl-361692


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