Abstract
RSA key pairs are normally generated from two large primes p and q. We consider what happens if they are generated from two integers s and r, where r is prime, but unbeknownst to the user, s is not. Under most circumstances, the correctness of encryption and decryption depends on the choice of the public and private exponents e and d. In some cases, specific ( s , r ) pairs can be found for which encryption and decryption will be correct for any ( e , d ) exponent pair. Certain s exist, however, for which encryption and decryption are correct for any odd prime r ∤ s . We give necessary and sufficient conditions for s with this property.
Highlights
RSA key pairs are normally generated from two large primes p and q
We consider the case of RSA encryption and decryption where at least one of ( p, q) is a composite number s
This situation might arise in the presence of a flawed primality tester or in the classroom when a teacher wishes to demonstrate in RSA what happens if one of ( p, q) is not a true prime
Summary
Consider the RSA public-key cryptosystem and its operations of encryption and decryption [1]. For a composite number s (that the user incorrectly believes is prime) and a true prime r used to generate keys with standard two-prime RSA, encryption and decryption exponents would be chosen using the (incorrect) pseudo-totient φ0 (n = sr ) = (s − 1)(r − 1), choosing (e, d) such that ed ≡ 1. Let λ(n) denote the Carmichael function, the maximum order of any element in Un. By Lagrange’s theorem, and the fact that for integers a and b, a | b ↔ all divisors of a | b, we see that those tuples (s, r, e, d) with the property λ(n) | (ed − 1) are exactly those that are witness-free.
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