Enter a base, non-negative exponent, and positive modulus to compute modular powers with repeated-squaring steps.
| Bit step | Exponent state | Current power | Multiply? | Result |
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This is exact integer modular arithmetic. It is useful for math/programming exercises, but this page is not a cryptographic protocol implementation or key-management system.
Compute a^b mod n and verify it with binary exponentiation, exponent-bit counts and a copyable square-and-multiply trace.
Enter a base, non-negative exponent, and positive modulus. The calculator uses exact BigInt repeated squaring to compute a^b mod n without constructing the full power.
The step table explains modular exponentiation for math and programming work. It is not a cryptographic protocol implementation, key generator, or security audit tool.
The goal is a^b mod n, not to construct a^b as one enormous integer first. Square-and-multiply repeatedly reduces intermediate values modulo n, keeping the computation practical even when the exponent is large.
A modulus of zero has no valid remainder operation, and cryptographic or number-theory conventions normally use a positive modulus. If the base is negative, the canonical nonnegative residue should still lie from zero through n minus one.
For a manageable exponent, compute the power independently and divide by the modulus to check the remainder. For larger exponents, use the displayed square-and-multiply rows to confirm each squaring or multiplication is followed by reduction.
The arithmetic is used inside RSA, Diffie-Hellman, primality tests, and many algorithms, but a correct residue does not validate a key, prove a number prime, or make an arbitrary protocol secure. Those tasks require additional rules.
These calculations are integer operations. Avoid spreadsheet or programming environments that silently round very large integers to floating point. Copy the exact base, exponent, modulus, and residue when reproducing the result elsewhere.
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