Principal Ideal Problem Research Articles

An algorithm for solving the principal ideal problem with subfields

An algorithm for solving the principal ideal problem with subfields

An Improved Multilinear Map and its Applications

Cryptographic multilinear maps have extensive applications. However, current constructions of multilinear maps suffer from the zeroizing attacks. For a candidate construction of multilinear maps described by Garg, Gentry, and Halevi (GGH13), Hu & Jia recently presented an efficient attack, which broke the GGH13-based applications of multipartite key exchange (MPKE) and witness encryption (WE) based on the hardness of 3-exact cover problem. By introducing random matrix, the author presents an improvement of the GGH13 map, which supports the applications for public tools of encoding in the GGH13 map, such as MPKE and WE. The security of the construction depends upon new hardness assumption. Moreover, the author's improvement destroys the structure of the ring element in the principal ideal lattice problem, and avoids potential attacks using algorithm of solving short principal ideal lattice generator.

Quantum algorithms for one-dimensional infrastructures

Infrastructures are group-like objects that make their appearance in arithmetic geometry in the study of computational problems related to number fields and function fields over finite fields. The most prominent computational tasks of infrastructures are the computation of the circumference of the infrastructure and the generalized discrete logarithms. Both these problems are not known to have efficient classical algorithms for an arbitrary infrastructure. Our main contributions are polynomial time quantum algorithms for one-dimensional infrastructures that satisfy certain conditions. For instance, these conditions are always fulfilled for infrastructures obtained from number fields and function fields, both of unit rank one. Since quadratic number fields give rise to such infrastructures, this algorithm can be used to solve Pell's equation and the principal ideal problem. In this sense we generalize Hallgren's quantum algorithms for quadratic number fields, while also providing a polynomial speedup over them. Our more general approach shows that these quantum algorithms can also be applied to infrastructures obtained from complex cubic and totally complex quartic number fields. Our improved way of analyzing the performance makes it possible to show that these algorithms succeed with constant probability independent of the problem size. In contrast, the lower bound on the success probability due to Hallgren decreases as the fourth power of the logarithm of the circumference. Our analysis also shows that fewer qubits are required. We also contribute to the study of infrastructures, and show how to compute efficiently within infrastructures.

An algorithm for the principal ideal problem in indefinite quaternion algebras

AbstractDeciding whether an ideal of a number field is principal and finding a generator is a fundamental problem with many applications in computational number theory. For indefinite quaternion algebras, the decision problem reduces to that in the underlying number field. Finding a generator is hard, and we present a heuristically subexponential algorithm.

Computing the unit group, class group, and compact representations in algebraic function fields

Number fields and global function fields have many similar properties. Both have many applications to cryptography and coding theory, and the main computational problems for number fields, such as computing the ring of integers and computing the class group and the unit group, have analogues over function fields. The complexity of the number field problems has been studied extensively and these problems have been the source of some exponential speedups by quantum computation. In this paper we study the analogous problems in function fields. We show that there are efficient quantum algorithms for computing the unit group, the class group and for solving the principal ideal problem in function fields of arbitrary degree. We show that compact representations exist, which allows us to show that the principal ideal problem is in NP. Unlike the number field case, we are also able to show that these compact representations can be computed efficiently.

Fully Homomorphic Encryption, Approximate Lattice Problem and LWE

In this paper, we first introduce a new concept of approximate lattice problem (ALP), which is an extension of learning with errors (LWE). Next, we propose two ALP-based public key encryption schemes. Then, we construct two new fully homomorphic encryption scheme (FHE) based on respectively approximate principal ideal lattice problem with related modulus (APIP-RM) and approximate lattice problem with related modulus (ALP-RM). Moreover, we also extend our ALP-RM-based FHE to the ALP problem with unrelated modulus (ALP-UM). Our work is different from previous works in three aspects: (1) We extend the LWE problem to the ALP problem. This ALP problem is similar to the closest vector problem in lattice. We believe that this problem is independent of interest. (2) We construct a new FHE by using a re-randomizing method, which is different from the squashing decryption in previous works. (3) The expansion rate is merely O( k ) with k a security parameter in Our FHE, which can be improved to O(log k ) by using dimension reduction [BV11], whereas all previous schemes are at least O( k* log k ) [BV11, Gen11, LNV11]. Our method can also decrease a factor k of the expansion rate in their schemes.

Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem

We give polynomial-time quantum algorithms for three problems from computational algebraic number theory. The first is Pell's equation. Given a positive nonsquare integer d , Pell's equation is x 2 − dy 2 = 1 and the goal is to find its integer solutions. Factoring integers reduces to finding integer solutions of Pell's equation, but a reduction in the other direction is not known and appears more difficult. The second problem we solve is the principal ideal problem in real quadratic number fields. This problem, which is at least as hard as solving Pell's equation, is the one-way function underlying the Buchmann--Williams key exchange system, which is therefore broken by our quantum algorithm. Finally, assuming the generalized Riemann hypothesis, this algorithm can be used to compute the class group of a real quadratic number field.

Cryptographic protocols on real hyperelliptic curves

We present public-key cryptographic protocols for key exchange, digital signatures, and encryption whose security is based on the presumed intractability of solving the principal ideal problem, or equivalently, the distance problem, in the real model of a hyperelliptic curve. Our protocols represent a significant improvement over existing protocols using real hyperelliptic curves. Theoretical analysis and numerical experiments indicate that they are comparable to the imaginary model in terms of efficiency, and hold much more promise for practical applications than previously believed.

Subexponential Algorithms for Class Group and Unit Computations

We describe in detail the implementation of an algorithm which computes the class group and the unit group of a general number field, and solves the principal ideal problem. The basic ideas of this algorithm are due to J. Buchmann. New ideas are the use of LLL-reduction of an ideal in a given direction which replaces the notion of neighbour, and the use ofcomplexlogarithmic embeddings of elements which plays a crucial role. Heuristically the algorithm performs in sub-exponential time with respect to the discriminant for fixed degree, and performs well in practice.

Nonfinitely generated closed ideals in group algebras

The main result of this paper is that for every locally compact abelian group G, which is not compact, the convolution algebra L 1( G) contains a closed ideal which is not finitely generated. This gives a negative solution to the principal ideal problem in group algebras, which has been open for some time [5, p. 352; 6, p. 38] and [8, p. 154].