A one line factoring algorithm

We describe a variant of Fermat’s factoring algorithm which is competitive with SQUFOF in practice but has heuristic run time complexity O(n1/3) as a general factoring algorithm. We also describe a sparse class of integers for which the algorithm is particularly effective. We provide speed comparisons between an optimised implementation of the algorithm described and the tuned assortment of factoring algorithms in the Pari/GP computer algebra package

Fooling primality tests on smartcards

We analyse whether the smartcards of the JavaCard platform correctly validate primality of domain parameters. The work is inspired by the paper Prime and prejudice: primality testing under adversarial conditions, where the authors analysed many open-source libraries and constructed pseudoprimes fooling the primality testing functions. However, in the case of smartcards, often there is no way to invoke the primality test directly, so we trigger it by replacing (EC)DSA and (EC)DH prime domain parameters by adversarial composites. Such a replacement results in vulnerability to Pohlig-Hellman style attacks, leading to private key recovery. Out of nine smartcards (produced by five major manufacturers) we tested, all but one have no primality test in parameter validation. As the JavaCard platform provides no public primality testing API, the problem cannot be fixed by an extra parameter check, %an additional check before the parameters are passed to existing (EC)DSA and (EC)DH functions, making it difficult to mitigate in already deployed smartcards

Complete 2017 Program

Program and schedule of events for the 28th Annual John Wesley Powell Student Research Conference

Giuga\u27s Primality Conjecture for Number Fields

Giuseppe Giuga conjectured in 1950 that a natural number n is prime if and only if it satisfies the congruence 1n-1+2n-1+ ... + (n-1)n-1 = -1 mod n. Progress in validating or disproving the conjecture has been minimal, with the most significant advance being the knowledge that a counter-example would need at least 19,907 digits. To gain new insights into Giuga\u27s conjecture, we explore it in the broader context of number fields. We present a generalized version of the conjecture and prove generalizations of many of the major results related to the conjecture. We introduce the concept of a Giuga ideal and perform computational searches for partial counter-examples to the generalized conjecture. We investigate the relationship between the existence of a counter-example in one number field with the existence of counter-examples in others, with a particular focus on quadratic extensions. This paper lays the preliminary foundation for answering the question: When does the existence of a counter-example in a number field imply the existence of a counter-example in the integers