To compute i n mod m where i is an integer, it is undesirable to use this "obvious" syntax because the powering will be performed first over the integers (possibly resulting in a very large integer) before reduction modulo m. The following functions for polynomial and matrix arithmetic over finite rings and fields are known to mod. The extension α is a RootOf a monic univariate irreducible polynomial of degree n over the integers mod m. Elements of finite fields of characteristic m with q = m n elements are represented as polynomials in α where α is a simple algebraic extension over the integers mod m. If the modulus m is a prime integer, then all coefficient arithmetic is done in the finite field of integers modulo m. all rational coefficients will be reduced to integers in the range − iquo m − 1, 2, iquo m, 2. When assigned the value mods, the symmetric representation is used i.e. all rational coefficients will be reduced to integers in the range 0, m − 1. When assigned the value modp (the default), the positive representation for integers modulo m is used i.e. The environment variable `mod` may be assigned either the modp function or the mods function. The operator syntax e mod m is equivalent to the function call `mod`(e, m). It incorporates facilities for doing finite field arithmetic and polynomial and matrix arithmetic over finite fields, including factorization. The mod operator evaluates the expression e over the integers modulo m. Computation with polynomials over the integers modulo mĬomputation over the integers modulo m using positive representationĬomputation over the integers modulo m using symmetric representation
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