Accelerating the CM method

Given a prime q and a negative discriminant D, the CM method constructs an elliptic curve E/\Fq by obtaining a root of the Hilbert class polynomial H_D(X) modulo q. We consider an approach based on a decomposition of the ring class field defined by H_D, which we adapt to a CRT setting. This yields two algorithms, each of which obtains a root of H_D mod q without necessarily computing any of its coefficients. Heuristically, our approach uses asymptotically less time and space than the standard CM method for almost all D. Under the GRH, and reasonable assumptions about the size of log q relative to |D|, we achieve a space complexity of O((m+n)log q) bits, where mn=h(D), which may be as small as O(|D|^(1/4)log q). The practical efficiency of the algorithms is demonstrated using |D| > 10^16 and q ~ 2^256, and also |D| > 10^15 and q ~ 2^33220. These examples are both an order of magnitude larger than the best previous results obtained with the CM method.Comment: 36 pages, minor edits, to appear in the LMS Journal of Computation and Mathematic

Efficient Algorithms for gcd and Cubic Residuosity in the Ring of Eisenstein Integers

We present simple and efficient algorithms for computing gcd and cubic residuosity in the ring of Eisenstein integers, Z[zeta] , i.e. the integers extended with zeta , a complex primitive third root of unity. The algorithms are similar and may be seen as generalisations of the binary integer gcd and derived Jacobi symbol algorithms. Our algorithms take time O(n^2) for n bit input. This is an improvement from the known results based on the Euclidian algorithm, and taking time O(n· M(n)), where M(n) denotes the complexity of multiplying n bit integers. The new algorithms have applications in practical primality tests and the implementation of cryptographic protocols. The technique underlying our algorithms can be used to obtain equally fast algorithms for gcd and quartic residuosity in the ring of Gaussian integers, Z[i]

Grained integers and applications to cryptography

To meet the requirements of the modern communication society, cryptographic techniques are of central importance. In modern cryptography, we try to build cryptographic primitives, whose security can be reduced to solving a particular number theoretic problem for which no fast algorithmic method is known by now. Thus, any advance in the understanding of the nature of such problems indirectly gives insight in the analysis of some of the most practical cryptographic techniques. In this work we analyze exactly this aspect much more deeply: How can we use some of the purely theoretical results in number theory to answer very practical questions on the security of widely used cryptographic algorithms and how can we use such results in concrete implementations? While trying to answer these kinds of security-related questions, we always think two-fold: From a cryptographic, security-ensuring perspective and from a cryptanalytic one. After we outlined -- with a special focus on the historical development of these results -- the necessary analytic and algorithmic foundations of number theory, we first delve into the question how point addition on certain elliptic curves can be done efficiently. The resulting formulas have their application in the cryptanalysis of crypto systems that are insecure if factoring integers can be done efficiently. The rest of the thesis is devoted to the study of integers, all of whose prime factors are neither too small nor too large. We show with the help of two applications how one can use the properties of such kinds of integers to answer very practical questions in the design and the analysis of cryptographic primitives: The optimization of a hardware-realization of the cofactorization step of the General Number Field Sieve and the analysis of different standardized key-generation algorithms

Computing Hilbert class polynomials with the Chinese Remainder Theorem

We present a space-efficient algorithm to compute the Hilbert class polynomial H_D(X) modulo a positive integer P, based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(|D|^(1/2+o(1))log P) space and has an expected running time of O(|D|^(1+o(1)). We describe practical optimizations that allow us to handle larger discriminants than other methods, with |D| as large as 10^13 and h(D) up to 10^6. We apply these results to construct pairing-friendly elliptic curves of prime order, using the CM method.Comment: 37 pages, corrected a typo that misstated the heuristic complexit