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  2. Modular arithmetic - Wikipedia

    en.wikipedia.org/wiki/Modular_arithmetic

    Given the Euler's totient function φ(m), any set of φ(m) integers that are relatively prime to m and mutually incongruent under modulus m is called a reduced residue system modulo m. [5] The set {5, 15} from above, for example, is an instance of a reduced residue system modulo 4.

  3. Modulo - Wikipedia

    en.wikipedia.org/wiki/Modulo

    In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation. Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor. [1]

  4. Modular equation - Wikipedia

    en.wikipedia.org/wiki/Modular_equation

    In that sense a modular equation becomes the equation of a modular curve. Such equations first arose in the theory of multiplication of elliptic functions (geometrically, the n 2 -fold covering map from a 2- torus to itself given by the mapping x → n · x on the underlying group) expressed in terms of complex analysis .

  5. Primitive root modulo n - Wikipedia

    en.wikipedia.org/wiki/Primitive_root_modulo_n

    In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n.

  6. Satisfiability modulo theories - Wikipedia

    en.wikipedia.org/wiki/Satisfiability_modulo_theories

    In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable.It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involving real numbers, integers, and/or various data structures such as lists, arrays, bit vectors, and strings.

  7. Modulo (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Modulo_(mathematics)

    Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. [3] Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n.

  8. Montgomery modular multiplication - Wikipedia

    en.wikipedia.org/wiki/Montgomery_modular...

    A straightforward algorithm to multiply numbers in Montgomery form is therefore to multiply aR mod N, bR mod N, and R′ as integers and reduce modulo N. For example, to multiply 7 and 15 modulo 17 in Montgomery form, again with R = 100, compute the product of 3 and 4 to get 12 as above.

  9. Modular multiplicative inverse - Wikipedia

    en.wikipedia.org/wiki/Modular_multiplicative_inverse

    The congruence relation, modulo m, partitions the set of integers into m congruence classes. Operations of addition and multiplication can be defined on these m objects in the following way: To either add or multiply two congruence classes, first pick a representative (in any way) from each class, then perform the usual operation for integers on the two representatives and finally take the ...

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