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Exact BigInt modular arithmetic for values up to 10¹⁸: (a ± b) mod m, (a × b) mod m, a⁻¹ mod m via extended Euclid, and aᵇ mod m via fast exponentiation.
Execution
Fill in the form, run the tool, and review the result in one place.
Samples
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Tool usage guide
The Modular Arithmetic Calculator performs exact BigInt modular calculations for integers up to 10¹⁸, including addition, subtraction, multiplication, modular inverse via the Extended Euclidean algorithm, and modular exponentiation using fast square-and-multiply.
Background
Studying public-key cryptography and verifying square-and-multiply steps for small modular exponentiation examples.
Problem
Need to compute 17⁵ mod 13 with intermediate binary exponentiation steps.
How to use
Select 'Power' operation, input a = 17, b = 5, and m = 13, then compute.
operation: power, a: 17, b: 5, m: 13Outcome
Displays 5 = 101₂, tracks powers 17¹ ≡ 4 and 17⁴ ≡ 9, and outputs final result 10 mod 13.
Background
Testing key generation algorithms that require finding modular inverses via the Extended Euclidean algorithm.
Problem
Determine if 5 has an inverse modulo 18 and calculate the Bézout identity.
How to use
Select 'Inverse' operation, enter a = 5 and m = 18, leaving b blank.
All results are reduced to the standard non-negative residue range [0, m−1].
A modular inverse a⁻¹ mod m exists if and only if gcd(a, m) = 1 (a and m are coprime).
It uses the binary square-and-multiply algorithm to compute aᵇ mod m in logarithmic time O(log b).
The calculator uses exact BigInt arithmetic supporting values up to 10¹⁸ without precision loss.
Modular inversion only requires base a and modulus m to solve the Bézout identity a·x + m·y = 1.
operation: inverse, a: 5, m: 18Outcome
Confirms gcd(5, 18) = 1, outputs identity 5 × (-7) + 18 × 2 = 1, and returns canonical inverse 11.