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Modular exponentiation is efficient to compute, even for very large integers. On the other hand, computing the modular discrete logarithm – that is, finding the exponent e when given b, c, and m – is believed to be difficult. This one-way function behavior makes modular exponentiation a candidate for use in cryptographic algorithms.
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.
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.
PARI/GP online calculator - https://pari.math.u-bordeaux.fr/gp.html (PARI/GP is a widely used computer algebra system designed for fast computations in number theory (factorizations, algebraic number theory, elliptic curves, modular forms, L functions...), but also contains a large number of other useful functions to compute with mathematical ...
The runtime bottleneck of Shor's algorithm is quantum modular exponentiation, which is by far slower than the quantum Fourier transform and classical pre-/post-processing. There are several approaches to constructing and optimizing circuits for modular exponentiation.
The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers, published by Arnold Schönhage and Volker Strassen in 1971. [1] It works by recursively applying fast Fourier transform (FFT) over the integers modulo 2 n + 1 {\displaystyle 2^{n}+1} .
The Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. [ 1 ] [ 2 ] [ 3 ] It is a divide-and-conquer algorithm that reduces the multiplication of two n -digit numbers to three multiplications of n /2-digit numbers and, by repeating this reduction, to at most n log 2 3 ...
The simplest method is the double-and-add method, [2] similar to square-and-multiply in modular exponentiation. The algorithm works as follows: The algorithm works as follows: To compute sP , start with the binary representation for s : s = s 0 + 2 s 1 + 2 2 s 2 + ⋯ + 2 n − 1 s n − 1 {\displaystyle s=s_{0}+2s_{1}+2^{2}s_{2}+\cdots +2 ...