Modular Inverse Calculator
Modular multiplicative inverse a⁻¹ mod m
Client-side computation for analysis, verification, and learning. Do not paste production secrets into untrusted sites; prefer local tooling for key material.
Educational / engineering use. RSA mode uses tiny primes for illustration only—not for real key generation.
a
modulus m
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What is Modular Inverse Calculator?
Finds the integer x such that (a × x) ≡ 1 (mod m). Essential in cryptography, particularly in RSA key generation and solving linear congruences.
Modular Inverse Formula
Exists only when gcd(a,m)=1.
For prime m (Fermat): x = a^(m−2) mod m
Step-by-Step
- Enter a and modulus m.
- Check gcd(a, m).
- Apply Extended Euclidean Algorithm.
- Extract inverse coefficient x.
- Normalize x to [0, m-1].
Worked Example
Use Extended Euclidean:
Where Modular Inverse Calculator Is Applied
- RSA private key calculation (d=e−1 mod φ(n))
- Solving linear congruences in number theory
- Elliptic Curve Cryptography (ECC) point operations
- Secret sharing schemes (Shamir's Secret Sharing)
- Error correction codes (Reed-Solomon)
- Cryptographic protocol implementations
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