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Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = b e mod m. From the definition of division, it follows that 0 ≤ c < m. For example, given b = 5, e = 3 and m = 13, dividing 5 3 = 125 by 13 leaves a remainder of c = 8.
For example, when computing x 2 k −1, the binary method requires k−1 multiplications and k−1 squarings. However, one could perform k squarings to get x 2 k and then multiply by x −1 to obtain x 2 k −1. To this end we define the signed-digit representation of an integer n in radix b as
In mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation by a positive integer power that requires a minimal number of multiplications. Using the form of the shortest addition chain , with multiplication instead of addition, computes the desired exponent (instead of multiple) of the base .
t 1 = 3 is the modular multiplicative inverse of 7 × 11 (mod 5), t 2 = 6 is the modular multiplicative inverse of 5 × 11 (mod 7) and t 3 = 6 is the modular multiplicative inverse of 5 × 7 (mod 11). Thus, X = 3 × (7 × 11) × 4 + 6 × (5 × 11) × 4 + 6 × (5 × 7) × 6 = 3504. and in its unique reduced form X ≡ 3504 ≡ 39 (mod 385) since ...
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation.. Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor.
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.
As another example, consider the product 7 ⋅ 15 mod 17 but with R = 10. Using the extended Euclidean algorithm, compute −5 ⋅ 10 + 3 ⋅ 17 = 1, so N′ will be −3 mod 10 = 7. The Montgomery forms of 7 and 15 are 70 mod 17 = 2 and 150 mod 17 = 14, respectively.
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}} .