Coprime solutions to axb (mod n)

Otokar Grošek 1  and Štefan Porubský 2
  • 1 Institute of Computer Science and Mathematics, Slovak University of Technology, FEI STU, Ilkovičova 3, 812 19 Bratislava 1, Slovak Republic
  • 2 Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věží 2, 182 07 Prague 8, Czech Republic


It is well known that a congruence axb (mod n) has a solution if and only if , and, if the condition is satisfied, the number of incongruent solutions equals . In 2010, Alomair, Clark and Poovendran proved that the congruence axb (mod n) has a solution coprime to n if and only if , as an auxiliary result playing a key role in a problem related to an electronic signature. In this paper we provide a concise proof of this result, together with a closed formula for the number of incongruent solutions coprime to n as well. Moreover, a bound is presented for the probability that, for randomly chosen , this congruence possesses at least one solution coprime to n.

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JMC is a forum for original research articles in the area of mathematical cryptology. Works in the theory of cryptology and articles linking mathematics with cryptology are welcome. Submissions from all areas of mathematics significant for cryptology are published, including but not limited to, algebra, algebraic geometry, coding theory, combinatorics, number theory, probability and stochastic processes.