Abstract
Previous work established the set of square-free integers n with at least one factorization n=p¯q¯ for which p¯ and q¯ are valid RSA keys, whether they are prime or composite. These integers are exactly those with the property λ(n)∣(p¯−1)(q¯−1), where λ is the Carmichael totient function. We refer to these integers as idempotent, because ∀a∈Zn,ak(p¯−1)(q¯−1)+1≡na for any positive integer k. This set was initially known to contain only the semiprimes, and later expanded to include some of the Carmichael numbers. Recent work by the author gave the explicit formulation for the set, showing that the set includes numbers that are neither semiprimes nor Carmichael numbers. Numbers in this last category had not been previously analyzed in the literature. While only the semiprimes have useful cryptographic properties, idempotent integers are deserving of study in their own right as they lie at the border of hard problems in number theory and computer science. Some idempotent integers, the maximally idempotent integers, have the property that all their factorizations are idempotent. We discuss their structure here, heuristics to assist in finding them, and algorithms from graph theory that can be used to construct examples of arbitrary size.
Highlights
Previous work established the set of square-free integers n with at least one factorization n = pqfor which pand qare valid RSA keys, whether they are prime or composite
While only the semiprimes have useful cryptographic properties, idempotent integers are deserving of study in their own right as they lie at the border of hard problems in number theory and computer science
In [3], we introduced the notion of idempotent integers, the set of square-free integers n that can be factored into two positive integers pand qsuch that λ(n) ∣ ( p − 1)(q − 1), where λ is the Carmichael totient function
Summary
We will call ai the predecessor of pi and pi the successor of ai It is a known property of the function λ that λ(n) = lcm(a1 , a2 , . Let p, q be prime, consider a semiprime n = pq. It is a known property of λ that λ(n) ∣ φ(n). Each factorj≠i k≠i,j ization corresponds to a single equation in n, pand qthat represents a possible idempotent factorization. The first eight square-free n with three or more factors and fully composite idempotent factorizations are shown in Table 1 [3]. The smallest integer with two fully composite idempotent factorizations is 2730, when factored into 10*273 and 21*130. The complete list of all n < 2 with fully composite idempotent factorizations is available at [5]
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