Kevin S. McCurley

A key distribution system equivalent to factoring

We propose a variation of the Diffie and Hellman key distribution scheme for which we can prove that decryption of a single key requires the ability to factor a number that is the product of two large primes. The practical advantage of such a scheme is that it will still be secure if the cryptanalyst knows a very fast algorithm for either factoring or computing discrete logarithms, but not for both. Using these keys in the ElGamal public-key cryptosystem provides a scheme for which the decryption of a message requires the ability to factor the modulus and break the original Diffie and Hellman scheme.

Open Problems in Number Theoretic Complexity

Publisher Summary In the past decade, there has been a resurgence of interest in computational problems of a number theoretic nature. This period has been characterized by a growing awareness of the practical aspects of number theoretic computations and at the same time by an increased understanding of the relevance of deep theory to the problems that arise. This chapter presents a collection of 36 open problems in number theoretic complexity. Questions about the integers have natural generalizations to rings of integers in an algebraic number field, and questions about elliptic curves may generalize to arbitrary abelian varieties. The chapter presents the problems that arose from many different places and times.

Breaking the Ong-Schnorr-Shamir signature scheme for quadratic number fields

Recently Ong, Schnorr, and Shamir [OSS1, OSS2] have presented new public key signature schemes based on quadratic equations. We will refer to these as the OSS schemes. The security of the schemes rest in part on the difficulty of finding solutions to

On the distribution of running times of certain integer factoring algorithms

X^2 - KY^2 \equiv M(mod{\mathbf{ }}n),

A rigorous subexponential algorithm for computation of class groups

(1) where n is the product of two large rational primes. In the original OSS scheme [OSS1], K, M, X, and Y were to be rational integers. However, when this version succumbed to an attack by Pollard [PS,S1], a new version was introduced [OSS2], where M, X, and Y were to be quadratic integers, i. e. elements of the ring \( Z[\sqrt d ] \). In this paper we will show that the OSS system in \( Z[\sqrt d ] \) is also breakable The method by which we do this is to reduce the problem of solving the congruence over the ring \( Z[\sqrt d ] \) to the problem of solving the congruence over the integers, for which we can use Pollard’s algorithm.

Solving bivariate quadratic congruences in random polynomial time

Let Q be a set of primes having relative density δ among the primes, with 0<δ<1, and let ψ(x,y,Q) be the number of positive integers ≤x that have no prime factors from Q exceeding y. We prove that if y→∞, then ψ(x,y,Q)∼xpδ(u), where u = (logx)(logy), and ϱδ is the continuous solution of the differential delay equation up′δ(u) = −δϱδ(u−1), ϱδ(u) = 1, 0≤u≤1. This generalizes work by Dickman, de Bruijn, and Hildebrand, who considered the case where Q consists of all primes (and δ = 1).

Sieving the positive integers by small primes

Abstract There are several algorithms for computing the prime decomposition of integers whose running times essentially depend on the size of the second largest prime factor of the input. For several such algorithms, we give uniform estimates for the number of inputs n with 1 ≤ n ≤ x for which the algorithm will halt in at most t steps. As a consequence we derive the best known lower bound for the number of integers n ≤ x that can be completely factored in random polynomial time.

A key distribution scheme based on factoring

The popularization of the Internet has brought fundamental changes to the world, because it allows a universal method of communication between computers. This carries enormous benefits with it, but also raises many security considerations. Cryptography is a fundamental technology used to provide security of computer networks, and there is currently a widespread engineering effort to incorporate cryptography into various aspects of the Internet. The system-level engineering required to provide security services for the Internet carries some important lessons for researchers whose study is focused on narrowly defined problems. It also offers challenges to the cryptographic research community by raising new questions not adequately addressed by the existing body of knowledge. This paper attempts to summarize some of these lessons and challenges for the cryptographic research community.