RSA Mathematics Calculator

RSA math is built from modular exponentiation, Euler totients, and modular inverses.

Use the result as a learning aid. For classwork, still show the formula and intermediate reasoning so the final answer is not just a black-box number.

RSA is a public-key cryptosystem built from modular arithmetic. In the textbook construction, choose two distinct primes p and q, form n=pq, and compute φ(n)=(p−1)(q−1). Choose a public exponent e satisfying gcd(e,φ(n))=1, then find the private exponent d such that ed≡1 (mod φ(n)).

A small integer message m is encrypted as c=mᵉ mod n and decrypted as m=cᵈ mod n. The modular inverse relation between e and d is what makes the round trip work under the assumptions of textbook RSA. Practical systems use much larger primes and standardized padding; raw textbook RSA should not be used directly to protect real data.

For educational inputs, verify that p and q really are prime, that e is coprime to φ(n), and that the message is represented as an integer in the allowed range, typically 0≤m<n. Reusing tiny primes is insecure; the page is designed only to illustrate the number theory.

How to verify a small RSA exercise. First check that p and q are distinct primes. Then compute n and φ(n), confirm gcd(e,φ)=1, and verify that e·d leaves remainder 1 when divided by φ. Finally, encrypt the message and decrypt the ciphertext; the recovered residue should match the original message modulo n.

For hand calculations, repeated squaring is far safer than constructing mᵉ or cᵈ directly. The same fast modular-exponentiation method used elsewhere on CalcNovaHub keeps intermediate numbers small. Remember that this is textbook number theory: real applications additionally need secure key sizes, protected private-key operations, and standardized encoding and padding to resist attacks.