Modular multiplicative inverse
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The multiplicative inverse of a modulo m exists if and only if 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.
Suppose we wish to find modular multiplicative inverse x of 3 modulo 11.
This is the same as finding x such that
Working in we find one value of x that satisfies this congruence is 4 because
and there are no other values of x in that satisfy this congruence. Therefore, the modular multiplicative inverse of 3 modulo 11 is 4.
Once we have found the inverse of 3 in , we can find other values of x in that also satisfy the congruence. They may be found by adding multiples of m = 11 to the found inverse. Generalizing, all possible x for this example can be formed from
Extended Euclidean algorithm 
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where a and 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
by the definition of congruence, m | ax − 1, which means that m is a divisor of ax − 1. This, in turn, means that
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.
This algorithm runs in time O(log(m)2), assuming |a| < m, and is generally more efficient than exponentiation.
Using Euler's theorem 
As an alternative to the extended Euclidean algorithm, Euler's theorem may be used to compute modular inverse:
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:
In the special case when m is a prime, the modular inverse is given by the above equation as:
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. However, calculating φ(m) is trivial in some common cases such as when m is known to be prime or a power of a prime.
- the relative cost of exponentiation. Though it can be implemented more efficiently using modular exponentiation, when large values of m are involved this is most efficiently computed with the Montgomery reduction method. This algorithm itself requires a modular inverse mod m, which is what we wanted to calculate in the first place. Without the Montgomery method, we're left with standard binary exponentiation which requires division mod m at every step, a slow operation when m is large. Furthermore, any kind of modular exponentiation is a taxing operation with computational complexity O(log φ(m)) = O(log m).
The modular multiplicative inverse has many applications in algorithms, particularly those related to number theory, since many such algorithms rely heavily on the theory of modular arithmetic. As a simple example, consider the exact division problem where you have a list of odd word-sized numbers each divisible by k and you wish to divide them all by k. One solution is as follows:
- Use the extended Euclidean algorithm to compute k−1, the modular multiplicative inverse of k mod 2w, where w is the number of bits in a word. This inverse will exist since the numbers are odd and the modulus has no odd factors.
- For each number in the list, multiply it by k−1 and take the least significant word of the result.
On many machines, particularly those without hardware support for division, division is a slower operation than multiplication, so this approach can yield a considerable speedup. The first step is relatively slow but only needs to be done once.
See also 
- Inversive congruential generator
- Modular arithmetic
- Number theory
- Public-key cryptography
- Rational reconstruction (mathematics)