Idempotent Factorizations of Square-Free Integers

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

Search Heuristics and Constructive Algorithms for Maximally Idempotent Integers

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

Composite Numbers That Give Valid RSA Key Pairs for Any Coprime p

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

Using Antifuse-Based FPGAs in Performance Critical Digital Designs

We present experimental results on the use of antifuse-based FPGA's in a performance critical digital design, performed at the recently completed Thayer Rapid Prototyping Facility. Our case study focuses on the design of a special purpose ALU for gene sequence analysis. Our work indicates the existence of highly nonlinear relationships between design changes and critical path lengths, due to the overwhelming influence of routing on performance. This suggests that the standard paradigms for digital design are not appropriate for FPGA's. We compare our results to previous work using SRAMbased technology, and discuss the implications of our results for digital design and rapid prototyping. 1.0 Introduction We have previously reported in [1] the results of experimental investigations concerning the use of SRAM-based FPGA's in high performance digital designs. This paper describes similar experiments using antifuse-based devices. Both efforts were performed at the Thayer Rapid Prototyping F..

The Digital World: Teaching Technological Literacy to a Multidisciplinary Audience

We report our experience with the development and execution of a course entitled "The Digital World", designed to increase the fluency and comfort level of non-science students with digital technology. The course relies heavily on computer-aided instruction, including the extensive use of electronic lectures and multimedia. We describe our successes and failures, and present analyses of student performance by gender, class, and field of study. 1.0 Introduction In the fall of 1991, we began the development of a course entitled "The Digital World". Like existing courses elsewhere, this one would teach the basics of digital technology, but it with two important differences: 1) it would be accessible to non-science majors, and 2) it would have no prerequisites. Our inquiries at Dartmouth and elsewhere indicated a growing consensus on the importance of technological literacy to a liberal arts education (see for example [NSF86], [NSF89], and [NSF90]). We felt existing efforts were inadequat..