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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 Montgomery ladder is an -coordinate only algorithm for elliptic curve point multiplication and is based on the double and add rules over a specific set of curves known as Montgomery curve. The algorithm has a conditional branching such that the condition depends on a secret bit.
A Montgomery curve of equation = + +. A Montgomery curve over a field K is defined by the equation,: = + + for certain A, B ∈ K and with B(A 2 − 4) ≠ 0.. Generally this curve is considered over a finite field K (for example, over a finite field of q elements, K = F q) with characteristic different from 2 and with A ≠ ±2 and B ≠ 0, but they are also considered over the rationals with ...
In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as square-and-multiply algorithms or binary exponentiation.
Peter Lawrence Montgomery (September 25, 1947 – February 18, 2020) was an American mathematician who worked at the System Development Corporation and Microsoft Research.He is best known for his contributions to computational number theory and mathematical aspects of cryptography, including the Montgomery multiplication method for arithmetic in finite fields, the use of Montgomery curves in ...
Modular exponentiation can be performed with a negative exponent e by finding the modular multiplicative inverse d of b modulo m using the extended Euclidean algorithm. That is: c = be mod m = d−e mod m, where e < 0 and b ⋅ d ≡ 1 (mod m). Modular exponentiation is efficient to compute, even for very large integers.
Barrett reduction. In modular arithmetic, Barrett reduction is a reduction algorithm introduced in 1986 by P.D. Barrett. [1] A naive way of computing. would be to use a fast division algorithm. Barrett reduction is an algorithm designed to optimize this operation assuming is constant, and , replacing divisions by multiplications.
Modular multiplicative inverse. In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. [1] In the standard notation of modular arithmetic this congruence is written as.
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