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Montgomery modular multiplication relies on a special representation of numbers called Montgomery form. The algorithm uses the Montgomery forms of a and b to efficiently compute the Montgomery form of ab mod N. The efficiency comes from avoiding expensive division operations. Classical modular multiplication reduces the double-width product ab ...
The n th roots of unity form under multiplication a cyclic group of order n, and in fact these groups comprise all of the finite subgroups of the multiplicative group of the complex number field. A generator for this cyclic group is a primitive n th root of unity. The n th roots of unity form an irreducible representation of any cyclic group of ...
Hypercomplex number. In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.
In modular arithmetic, the integers coprime (relatively prime) to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n .
Field extension. In mathematics, particularly in algebra, a field extension (denoted ) is a pair of fields , such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. [1] [2] [3] For example, under the usual notions of addition and multiplication, the complex numbers are ...
The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Zn; it has φ ( n) elements, φ being Euler's totient function, and is denoted as U ( n ...
A finite field of order q exists if and only if q is a prime power p k (where p is a prime number and k is a positive integer). In a field of order p k, adding p copies of any element always results in zero; that is, the characteristic of the field is p. If q = p k, all fields of order q are isomorphic (see § Existence and uniqueness below). [1]
The Schönhage–Strassen algorithm is based on the fast Fourier transform (FFT) method of integer multiplication. This figure demonstrates multiplying 1234 × 5678 = 7006652 using the simple FFT method. Base 10 is used in place of base 2 w for illustrative purposes. The Schönhage–Strassen algorithm is an asymptotically fast multiplication ...