Modular multiplicative inverse: Information from Answers.com

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The modular multiplicative inverse of an integer a modulo m is an integer x such that

$a^{-1} \equiv x \pmod{m}.$

That is, it is the multiplicative inverse in the ring of integers modulo m. This is equivalent to

$ax \equiv 1 \pmod{m}.$

The multiplicative inverse of a modulo m exists iff a and m are coprime (i.e., if gcd(a, m) = 1). If the modular multiplicative inverse of a modulo m exists, the operation of division by a modulo m can be defined as multiplying by the inverse, which is in essence the same concept as division in the field of reals.

1 Explanation
2 Computation
- 2.1 Extended Euclidean Algorithm
- 2.2 Direct Modular Exponentiation
3 Implementation in Python
4 Example
5 See also

Explanation

When the inverse exists, it is always unique in Z_m where m is the modulus. Therefore, the x that is selected as the modular multiplicative inverse is generally a member of Z_m for most applications.

For example,

$3^{-1} \equiv x \pmod{11}$

yields

$3x \equiv 1 \pmod{11}.$

The smallest x that solves this congruence is 4; therefore, the modular multiplicative inverse of 3 (mod 11) is 4. However, another x that solves the congruence is 15 (easily found by adding m, which is 11, to the found inverse).

Computation

Extended Euclidean Algorithm

The modular multiplicative inverse of a modulo m can be found with the extended Euclidean algorithm. The algorithm finds solutions to Bézout's identity

$ax + by = \gcd(a, b)\,$

where a, b are given and x, y, and gcd(a, b) are the integers that the algorithm discovers. So, since the modular multiplicative inverse is the solution to

$ax \equiv 1 \pmod{m},$

by the definition of congruence, m | ax - 1, which means that m is a divisor of ax - 1. This, in turn, means that

$ax - 1 = qm.\,$

Rearranging produces

$ax - qm = 1,\,$

with a and m given, x the inverse, and q an integer multiple that will be discarded. This is the exact form of equation that the extended Euclidean algorithm solves—the only difference being that gcd(a, m)=1 is predetermined instead of discovered. Thus, a needs to be coprime to the modulus, or the inverse won't exist. The inverse is x, and q is discarded.

This algorithm runs in time O(log(m)²), assuming |a|<m, and is generally more efficient than exponentiation.

Direct Modular Exponentiation

The direct modular exponentiation method, as an alternative to the extended Euclidean algorithm, is as follows:

According to Euler's theorem, if a is coprime to m, that is, gcd(a, m) = 1, then

$a^{\varphi(m)} \equiv 1 \pmod{m}$

where φ(m) is Euler's totient function. This follows from the fact that a belongs to the multiplicative group (Z/mZ)^* iff a is coprime to m. Therefore the modular multiplicative inverse can be found directly:

$a^{\varphi(m)-1} \equiv a^{-1} \pmod{m}$

This method is generally slower than the extended Euclidean algorithm, but is sometimes used when an implementation for modular exponentiation is already available. Some disadvantages of this method include:

the required knowledge of φ(m), whose most efficient computation requires m's factorization. Factorization is widely believed to be a mathematically hard problem.
exponentiation. Though it can be implemented more efficiently using modular exponentiation, which is a form of binary exponentiation, it is still a taxing operation with computational complexity O(log φ(m)).

Implementation in Python

def extended_gcd(a, b):
    x, last_x = 0, 1
    y, last_y = 1, 0
 
    while b:
        quotient = a // b
        a, b = b, a % b
        x, last_x = last_x - quotient*x, x
        y, last_y = last_y - quotient*y, y
 
    return (last_x, last_y, a)
 
def inverse_mod(a, m):
    x, q, gcd = extended_gcd(a, m)
 
    if gcd == 1:
        # x is the inverse, but we want to be sure a positive number is returned.
        return (x + m) % m
    else:
        # if gcd != 1 then a and m are not coprime and the inverse does not exist.
        return None

Example

>>> a = 1234
>>> m = 9997
>>> inverse_mod(a, m)
3119
>>> (a * inverse_mod(a, m)) % m
1

	Inversive congruential generator
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Modular multiplicative inverse

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