RSA Primes, Key Pairs and Limits

Choose distinct primes p=5 and q=11. Their product n=55 is the modulus, and (p−1)(q−1)=40. Choose public exponent e=3, coprime with 40, and d=27 because 3×27=81 is congruent to 1 modulo 40. The public arithmetic pair is (n,e); the private capability depends on secret information including d in this model.

These values are intentionally tiny so every step can be checked by hand. They can be factored immediately and must never be used to protect real data. The lesson explains why prime structure helps define inverse operations; it does not claim that publishing the two prime factors is compatible with a secure private key.

For this raw teaching operation, an encoded message is an integer m with 0≤m<n. Compute ciphertext c=mᵉ mod n, then recover m=cᵈ mod n. For m=7, 7³=343 and 343 mod 55=13. The inverse modular exponent gives 7 again. A full byte or arbitrary text string is not automatically a valid single message under modulus 55.

Python’s three-argument pow computes modular exponentiation directly. It avoids first constructing the unrestricted huge power in this demonstration. Independent fixtures check all 55 permitted messages, including zero and messages sharing a factor with n, rather than testing only the convenient value seven. A round trip is evidence for the arithmetic example, not evidence that a cryptographic design is secure.

p, q = 5, 11
n = p * q
phi = (p - 1) * (q - 1)
e, d = 3, 27
assert (e * d) % phi == 1
message = 7
ciphertext = pow(message, e, n)
recovered = pow(ciphertext, d, n)
print(n, ciphertext, recovered)
print(all(pow(pow(value, e, n), d, n) == value for value in range(n)))

Practical RSA encryption and signature schemes require appropriate parameters, encoding/padding, randomness where specified and careful implementation. Raw textbook exponentiation is deterministic and has structural weaknesses. Encryption and signing are different complete schemes, not interchangeable instructions to “encrypt with either key” for any purpose.

RSA-based transport of a small symmetric secret can appear in a suitably defined complete protocol, but secure recipient authentication is still necessary. The TLS 1.3 teaching flow elsewhere uses ephemeral agreement instead of historical RSA key transport. Key compromise, weak random generation, small moduli and implementation mistakes can invalidate a design. In an examination evaluation, link the vulnerability to the relevant mechanism rather than describing public-key cryptography as mathematically unbreakable.

Attempt the task before opening its worked response. Record each change of state, explain the rule used, and check the result against the original requirements. Code examples labelled Python 3 use that language’s conventions; pseudocode states its own assumptions.