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  2. Modulo - Wikipedia

    en.wikipedia.org/wiki/Modulo

    Modulo operations might be implemented such that a division with a remainder is calculated each time. For special cases, on some hardware, faster alternatives exist. For example, the modulo of powers of 2 can alternatively be expressed as a bitwise AND operation (assuming x is a positive integer, or using a non-truncating definition):

  3. 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.

  4. 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 ...

  5. Multiplicative group of integers modulo n - Wikipedia

    en.wikipedia.org/wiki/Multiplicative_group_of...

    Hence another name is the group of primitive residue classes modulo n. In the theory of rings , a branch of abstract algebra , it is described as the group of units of the ring of integers modulo n .

  6. 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.

  7. Barrett reduction - Wikipedia

    en.wikipedia.org/wiki/Barrett_reduction

    In modular arithmetic, Barrett reduction is an algorithm designed to optimize the calculation of [1] without needing a fast division algorithm.It replaces divisions with multiplications, and can be used when is constant and <.

  8. Primitive root modulo n - Wikipedia

    en.wikipedia.org/wiki/Primitive_root_modulo_n

    This derives from the fact that a sequence (g k modulo n) always repeats after some value of k, since modulo n produces a finite number of values. If g is a primitive root modulo n and n is prime, then the period of repetition is n − 1. Permutations created in this way (and their circular shifts) have been shown to be Costas arrays.

  9. Reduced residue system - Wikipedia

    en.wikipedia.org/wiki/Reduced_residue_system

    Every number in a reduced residue system modulo n is a generator for the additive group of integers modulo n. A reduced residue system modulo n is a group under multiplication modulo n . If { r 1 , r 2 , ... , r φ( n ) } is a reduced residue system modulo n with n > 2, then ∑ r i ≡ 0 mod n {\displaystyle \sum r_{i}\equiv 0\!\!\!\!\mod n} .