Tool to compute modular power. Modular Exponentiation (or power modulo) is the result of the calculus a^b mod n. It is often used in informatics and cryptography.

Modular Exponentiation - dCode

Tag(s) : Arithmetics

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Solver limited to integer solutions < 10000

It consists in an exponentiation followed by a modulus, but it exists optimized algorithms with big numbers to return a fast result without having to actually perform the calculation (called fast, thanks to mathematical simplifications).

__Example:__ $$ 12^{34} \equiv 16 \mod 56 $$

The word *power* indicates the name of the operation, and *exponent* to indicate the operand.

Calculating the last $ x $ digits of $ a ^ b $ is equivalent to calculating $ a ^ b \mod n $ with $ n = 10 ^ x $ (the number $ 1 $ followed by $ x $ zeros)

__Example:__ $ 3 ^ 9 = 19683 $ and $ 3 ^ 9 \mod 100 = 83 $ (the last 2 digits)

There are several algorithms, here is the shortest one in pseudocode:`// pseudocode`

function powmod(base b, exponent e, modulus m)

if m = 1 then return 0

var c := 1

for var a from 1 to e

c := (c * b) mod m

end for

return c

This calculation is known as the discrete logarithm problem. Some solutions can be found by brute force but there is no trivial general solution.

Calculus uses exponent and modulos that are generally defined over the natural number domain set N. It is possible to use rational numbers but it is not handled here.

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