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A modular multiplicative inverse of an integer a with respect to the modulus m is a solution of the linear congruence a x ≡ 1 ( mod m ) . {\displaystyle ax\equiv 1{\pmod {m}}.} The previous result says that a solution exists if and only if gcd( a , m ) = 1 , that is, a and m must be relatively prime (i.e. coprime).
The modular inverse of aR mod N is REDC((aR mod N) −1 (R 3 mod N)). Modular exponentiation can be done using exponentiation by squaring by initializing the initial product to the Montgomery representation of 1, that is, to R mod N, and by replacing the multiply and square steps by Montgomery multiplies.
The modular multiplicative inverse is defined by the following rules: Existence: There exists an integer denoted a −1 such that aa −1 ≡ 1 (mod m) if and only if a is coprime with m. This integer a −1 is called a modular multiplicative inverse of a modulo m.
In mathematics, set inversion is the problem of characterizing the preimage X of a set Y by a function f, i.e., X = f −1 (Y ) = {x ∈ R n | f(x) ∈ Y }. It can also be viewed as the problem of describing the solution set of the quantified constraint "Y(f (x))", where Y( y) is a constraint, e.g. an inequality, describing the set Y.
A set of nested modules, of which the modular decomposition is an example, can be used to guide the recursive solution of many combinatorial problems on graphs, such as recognizing and transitively orienting comparability graphs, recognizing and finding permutation representations of permutation graphs, recognizing whether a graph is a cograph ...
In modular arithmetic, the modular multiplicative inverse of a is also defined: it is the number x such that ax ≡ 1 (mod n). This multiplicative inverse exists if and only if a and n are coprime. For example, the inverse of 3 modulo 11 is 4 because 4 ⋅ 3 ≡ 1 (mod 11). The extended Euclidean algorithm may be used to compute it.
The modular product of graphs. In graph theory, the modular product of graphs G and H is a graph formed by combining G and H that has applications to subgraph isomorphism.It is one of several different kinds of graph products that have been studied, generally using the same vertex set (the Cartesian product of the sets of vertices of the two graphs G and H) but with different rules for ...
Suppose is a directed graph without 2-dicycles, is the set of all dipaths in , and is the 0-1 incidence matrix of () versus . Then A {\displaystyle A} is totally unimodular if and only if every simple arbitrarily-oriented cycle in G {\displaystyle G} consists of alternating forwards and backwards arcs.