How does a toy RSA system use different keys to encrypt and decrypt?

Public-key encryption separates two roles. A recipient publishes an encryption key while protecting a paired decryption key. A sender uses the recipient’s public key to produce ciphertext, a transformed representation intended to hide the message. The recipient uses the private key to recover the encoded message. Publishing the encryption key is part of the design, not an accidental disclosure.

The arithmetic example is intentionally insecure. Its modulus is n = 33 and its public exponent is e = 3. A pre-encoded message number m is transformed into c = m³ mod 33. The notation mod means the remainder after division. For the initial message two, eight is the ciphertext.

The paired private exponent is d = 7. Applying c⁷ mod 33 recovers two. The factors of 33 are three and eleven. Among the integers from one through thirty-two, twenty share no factor with 33 except one; that count is written φ(33) = 20. The exponents are paired because 3 × 7 = 21 leaves remainder one when divided by twenty. This relationship supports the reversible arithmetic here, but the tiny factors and message space make it unsuitable for secrecy.

Real RSA encryption needs a carefully specified encoding scheme with appropriate randomized padding, suitable key sizes and secure implementation. A sender also needs a trustworthy association between a public key and the intended recipient. A mathematically valid round trip alone does not establish those properties.

Encryption and digital signatures serve different purposes. Encryption aims at confidentiality for a recipient; signatures support verification associated with a signing key. Simply reversing the labels of raw exponent operations does not produce interchangeable secure schemes. This lesson shows public-to-private encryption roles, not proof of a sender’s identity or factual truth.