Invertible matrices over some quotient rings: identification, generation, and analysis
-
Viktoriya V. Vysotskaya
and
Lev I. Vysotsky
Abstract
We study matrices over quotient rings modulo univariate polynomials over a two-element field. Lower bounds for the fraction of the invertible matrices among all such matrices of a given size are obtained. An efficient algorithm for calculating the determinant of matrices over these quotient rings and an algorithm for generating random invertible matrices (with uniform distribution on the set of all invertible matrices) are proposed and analyzed. An effective version of the latter algorithm for quotient rings modulo polynomials of form xr − 1 is considered and analyzed. These methods may find practical applications for generating keys of cryptographic schemes based on quasi-cyclic codes such as LEDAcrypt.
Originally published in Diskretnaya Matematika (2021) 33,№2, 46–65 (in Russian).
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Articles in the same Issue
- Contents
- Group service system with three queues and load balancing
- Formulas for the numbers of sequences containing a given pattern given number of times
- On a generalization of class of negative binomial distributions
- Invertible matrices over some quotient rings: identification, generation, and analysis
- On synthesis of reversible circuits consisting of NOT, CNOT, 2-CNOT gates with small number of additional inputs
- Computation of distributions of statistics by means of Markov chains
Articles in the same Issue
- Contents
- Group service system with three queues and load balancing
- Formulas for the numbers of sequences containing a given pattern given number of times
- On a generalization of class of negative binomial distributions
- Invertible matrices over some quotient rings: identification, generation, and analysis
- On synthesis of reversible circuits consisting of NOT, CNOT, 2-CNOT gates with small number of additional inputs
- Computation of distributions of statistics by means of Markov chains
