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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 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 modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n.
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. Thus 3 4 = 13 in the group Z 17 ×. The discrete logarithm is just the inverse operation.
The properties involving multiplication, division, and exponentiation generally require that a and n are integers. Identity: (a mod n) mod n = a mod n. n x mod n = 0 for all positive integer values of x. If p is a prime number which is not a divisor of b, then ab p−1 mod p = a mod p, due to Fermat's little theorem. Inverse: [(−a mod n) + (a ...
This can be accomplished via modular exponentiation, which is the slowest part of the algorithm. The gate thus defined satisfies U r = I {\displaystyle U^{r}=I} , which immediately implies that its eigenvalues are the r {\displaystyle r} -th roots of unity ω r k = e 2 π i k / r {\displaystyle \omega _{r}^{k}=e^{2\pi ik/r}} .
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In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae , published in 1801.