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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.
For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution).
where a −1 is the modular multiplicative inverse of a modulo m. I.e., it satisfies the equation = The multiplicative inverse of a only exists if a and m are coprime. Hence without the restriction on a, decryption might not be possible. It can be shown as follows that decryption function is the inverse of the encryption function,
First, the input is mapped to its multiplicative inverse in GF(2 8) = GF(2) [x]/(x 8 + x 4 + x 3 + x + 1), Rijndael's finite field. Zero, as the identity, is mapped to itself. This transformation is known as the Nyberg S-box after its inventor Kaisa Nyberg. [2] The multiplicative inverse is then transformed using the following affine ...
The multiplicative inverse x ≡ a −1 (mod m) may be efficiently computed by solving Bézout's equation a x + m y = 1 for x, y, by using the Extended Euclidean algorithm. In particular, if p is a prime number, then a is coprime with p for every a such that 0 < a < p; thus a multiplicative inverse exists for all a that is not congruent to zero ...
For example, given b = 5, e = 3 and m = 13, dividing 5 3 = 125 by 13 leaves a remainder of c = 8. 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).
Any involution is a bijection.. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x ↦ −x), reciprocation (x ↦ 1/x), and complex conjugation (z ↦ z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the ...
The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function. If the inequality is strict ( a < b , a > b ) and the function is strictly monotonic, then the inequality remains strict.