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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 = b e 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.
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.
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.
A modular multiplicative inverse of a modulo m can be found by using the extended Euclidean algorithm. The Euclidean algorithm determines the greatest common divisor (gcd) of two integers, say a and m. If a has a multiplicative inverse modulo m, this gcd must be 1. The last of several equations produced by the algorithm may be solved for this gcd.
This is widely used in modular arithmetic, because this allows reducing modular exponentiation with large exponents to exponents smaller than n. Euler's theorem is used with n not prime in public-key cryptography , specifically in the RSA cryptosystem , typically in the following way: [ 10 ] if y = x e ( mod n ) , {\displaystyle y=x^{e}{\pmod ...
Exponentiation with Montgomery reduction O ( M ( n ) k ) {\displaystyle O(M(n)\,k)} On stronger computational models, specifically a pointer machine and consequently also a unit-cost random-access machine it is possible to multiply two n -bit numbers in time O ( n ).
When the numbers involved are large, it is more efficient to reduce modulo p multiple times during the computation. Regardless of the specific algorithm used, this operation is called modular exponentiation. For example, consider Z 17 ×. To compute 3 4 in this group, compute 3 4 = 81, and then divide 81 by 17, obtaining a remainder of 13.
Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is O(k log 2 n log log n) = Õ(k log 2 n), where k is the number of times we test a random a, and n is the value we want to test for primality; see Miller–Rabin primality test for details.