Power Mod Calculator

Enter Calculation

Base

Exponent

Modulus

Results

Modular Exponentiation Result

9

7^256 mod 13

Base	7
Exponent	256
Modulus	13

What Is the Power Mod Calculator?

The Power Mod Calculator computes (base^exponent) mod m — the remainder left after raising a base to a power and dividing by a modulus. This single operation, called modular exponentiation, is everywhere in number theory and cryptography: it powers RSA encryption, Diffie-Hellman key exchange, primality tests, and hash functions. Computing it directly (calculating the giant power first, then taking the remainder) is impossible for large exponents, so we use a fast algorithm instead.

How to Use It

Enter three whole numbers: the base, the exponent (zero or positive), and the modulus m (a positive integer greater than 1). Press calculate and the tool returns the result in the range 0 to m−1. Negative bases are reduced into the proper non-negative residue automatically.

The Formula Explained

The calculator evaluates:

$$\text{result} = \text{Base}^{\text{Exponent}} \bmod \text{Modulus}$$

Rather than building the enormous number base^exponent, the calculator uses square-and-multiply (also called binary exponentiation). It reads the exponent in binary. Starting with a result of 1, it repeatedly squares the base modulo m; whenever the current binary digit of the exponent is 1, it multiplies that squared value into the running result, again reducing modulo m. Because every intermediate number stays smaller than \(m^2\), the computation is fast even for exponents with hundreds of digits.

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Square-and-multiply: scan exponent bits, squaring each step and multiplying when a bit is 1, reducing mod m throughout.

Worked Example

Compute \(7^{256} \bmod 13\). The order of 7 modulo 13 divides 12, and \(7^{12} \equiv 1\). Since \(256 = 12 \times 21 + 4\), we get

$$7^{256} \equiv 7^{4} = 2401 \equiv 9 \pmod{13}$$

So the answer is 9 — found instantly here without ever forming the 217-digit number \(7^{256}\).

Each step squares the running value and reduces mod m, multiplying in the base where exponent bits are set.

FAQ

What if the modulus is 1? Every integer is congruent to 0 modulo 1, so the result is 0.

Can the exponent be 0? Yes. Any base to the power 0 equals 1, so the result is 1 mod m (which is 1 when m > 1).

Why not just compute the power directly? For large exponents the intermediate number would have astronomically many digits and overflow. Modular reduction at every step keeps values small and the method efficient.