Abstract
We explore the class of positive integers n that admit idempotent factorizations n = p ¯ q ¯ such that λ ( n ) ∣ ( p ¯ − 1 ) ( q ¯ − 1 ) , where λ is the Carmichael lambda function. Idempotent factorizations with p ¯ and q ¯ prime have received the most attention due to their cryptographic advantages, but there are an infinite number of n with idempotent factorizations containing composite p ¯ and/or q ¯ . Idempotent factorizations are exactly those p ¯ and q ¯ that generate correctly functioning keys in the Rivest–Shamir–Adleman (RSA) 2-prime protocol with n as the modulus. While the resulting p ¯ and q ¯ have no cryptographic utility and therefore should never be employed in that capacity, idempotent factorizations warrant study in their own right as they live at the intersection of multiple hard problems in computer science and number theory. We present some analytical results here. We also demonstrate the existence of maximally idempotent integers, those n for which all bipartite factorizations are idempotent. We show how to construct them, and present preliminary results on their distribution.
Highlights
Certain square-free positive integers n can be factored into two numbers ( p, q) such that λ(n) ∣( p − 1)(q − 1), where λ is the Carmichael lambda function
For any square-free p, a composite non-Carmichael qcan be found such that n = pqis an idempotent factorization
Rather than view idempotency as an all-or-nothing property of a bipartite factorization, it may be viewed as a ratio between 0 and 1
Summary
Certain square-free positive integers n can be factored into two numbers ( p, q) such that λ(n) ∣. These, are not the only idempotent factorizations While they do not use the term themselves, Huthnance and Warndof [3] describe idempotent factorizations n = pq, where pand qare either primes or Carmichael numbers, noting that such integers generate correct RSA keys. These values are a subset of idempotent factorizations as we define them here, as there are an infinite number of idempotent tuples (n, p, q) with composite pand/or qwhere neither pnor qare Carmichael numbers.
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