Modular exponentiation finds the remainder of a power under a modulus. Repeated squaring avoids huge intermediate values by reducing each squared power modulo m before continuing.
⚠Use a nonnegative exponent and modulus at least 2. Extremely large inputs may exceed normal browser integer precision.
How Modular Exponentiation Works
Modular exponentiation rewrites a large power as a product of squared powers. Each step is reduced modulo m, which makes the calculation fast and suitable for cryptography and number theory.
Base Reduction
Start by replacing a with a mod m.
Repeated Squaring
Square powers while reducing after each multiplication.
Exponent Bits
Multiply only the powers needed by the exponent.
Verification
The final answer is the same remainder as direct exponentiation would give.
Example: For 7^128 mod 13, repeated squaring keeps reducing powers modulo 13 and returns the final remainder without expanding 7^128.
Applications of Modular Exponentiation
RSA EncryptionDiffie-HellmanPrimality TestsModular ArithmeticContest Math
Frequently Asked Questions
What is a modular exponentiation calculator?▼
It computes a^b modulo m without expanding the huge power directly. The result is the remainder after division by m.
What formula does modular exponentiation use?▼
It uses repeated squaring: reduce the base modulo m, square repeatedly, and multiply selected powers according to the exponent.
How do I use this modular power calculator?▼
Enter the base, exponent, and modulus, then calculate. The tool returns a^b mod m with a short derivation.
Why not calculate a^b directly first?▼
Large powers grow extremely fast. Reducing after each multiplication keeps the numbers small and exact for modular arithmetic.
Where is modular exponentiation used?▼
It is used in RSA encryption, Diffie-Hellman key exchange, primality tests, modular arithmetic, and contest math.
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