# Modular Arithmetic Calculator (Add / Subtract / Multiply / Inverse / Power)

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.

> Canonical page: https://elysiatools.com/en/tools/modulo-arithmetic-converter

- **Category:** Math & Numbers

- **Keywords:** modular arithmetic, modulo, mod, congruence, modular inverse, fast power, square and multiply, extended euclid, number theory, remainder

## Overview

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.

## Inputs

- **Operation** (select)
- **Value a** (text): First operand (the base in power mode, the value being inverted in inverse mode).
- **Value b** (text): Second operand — the exponent in power mode; unused in inverse mode.
- **Modulus m** (text): Modulus, m ≥ 2. Results are reduced to the canonical range \[0, m−1\].

## When to use

- Computing modular inverses for cryptography homework or RSA key generation exercises.
- Evaluating large power congruences like aᵇ mod m without floating-point overflow.
- Solving discrete mathematics and number theory congruences in the canonical range \[0, m−1\].

## How it works

- Select the arithmetic operation: addition, subtraction, multiplication, inverse, or power.
- Enter operand a, modulus m (where m ≥ 2), and second operand or exponent b if required.
- The tool executes exact BigInt operations, applying Euclidean reduction or binary exponentiation step by step.
- Review the resulting congruence alongside intermediate reduction steps, Bézout coefficients, or bitwise decomposition.

## Use cases

- Verifying manual steps in number theory problem sets involving modular division and inverses.
- Demonstrating RSA cryptographic steps such as modular exponentiation and decryption key verification.
- Simplifying modular reductions for competitive programming challenges with large 64-bit moduli.

## Frequently asked questions

### What is the canonical form returned by the calculator?

All results are reduced to the standard non-negative residue range [0, m−1].

### When does a modular inverse exist?

A modular inverse a⁻¹ mod m exists if and only if gcd(a, m) = 1 (a and m are coprime).

### How does the tool handle large exponents?

It uses the binary square-and-multiply algorithm to compute aᵇ mod m in logarithmic time O(log b).

### Is there a limit on input number size?

The calculator uses exact BigInt arithmetic supporting values up to 10¹⁸ without precision loss.

### Why is operand b disabled or ignored in inverse mode?

Modular inversion only requires base a and modulus m to solve the Bézout identity a·x + m·y = 1.

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