Discrete logarithms in free groups

Structure computation and discrete logarithms in finite abelian p-groups

We present a generic algorithm for computing discrete logarithms in a finite abelian p-group H, improving the Pohlig-Hellman algorithm and its generalization to noncyclic groups by Teske. We then give a direct method to compute a basis for H without using a relation matrix. The problem of computing a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian group G is addressed, yielding a Monte Carlo algorithm to compute the structure of G using O(|G|^0.5) group operations. These results also improve generic algorithms for extracting pth roots in G.

离散对数问题的困难实例及其密码学应用

Hard Instances of Discrete Logarithm Problem and Cryptographic Applications

本文研究了离散对数问题在有限生成群中的困难性，并提出了具有NP难离散对数问题的密码协议方案。

The Discrete Logarithm Problem in Bergman's non-representable ring

Bergman's Ring $E_p$, parameterized by a prime number $p$, is a ring with $p^5$ elements that cannot be embedded in a ring of matrices over any commutative ring. This ring was discovered in 1974. In 2011, Climent, Navarro and Tortosa described an efficient implementation of $E_p$ using simple modular arithmetic, and suggested that this ring may be a useful source for intractable cryptographic problems. We present a deterministic polynomial time reduction of the Discrete Logarithm Problem in $E_p$ to the classical Discrete Logarithm Problem in $\Zp$, the $p$-element field. In particular, the Di…

The ElGamal cryptosystem over circulant matrices

In this paper we study extensively the discrete logarithm problem in the group of non-singular circulant matrices. The emphasis of this study was to find the exact parameters for the group of circulant matrices for a secure implementation. We tabulate these parameters. We also compare the discrete logarithm problem in the group of circulant matrices with the discrete logarithm problem in finite fields and with the discrete logarithm problem in the group of rational points of an elliptic curve.

A Deterministic Algorithm for the Discrete Logarithm Problem in a Semigroup

The discrete logarithm problem in a finite group is the basis for many protocols in cryptography. The best general algorithms which solve this problem have time complexity of $\mathcal{O}(\sqrt{N}\log N)$, and a space complexity of $\mathcal{O}(\sqrt{N})$ where $N$ is the order of the group. (If $N$ is unknown, a simple modification would achieve a time complexity of $\mathcal{O}(\sqrt{N}(\log N)^2)$.) These algorithms require the inversion of some group elements or rely on finding collisions and the existence of inverses, and thus do not adapt to work in the general semigroup setting. For sem…

半直积离散对数问题的亚指数量子算法

A Subexponential Quantum Algorithm for the Semidirect Discrete Logarithm Problem

本文首次对半直积离散对数问题（SDLP）进行专门安全性分析，通过建立其与群作用的联系，构造出求解 SDLP 的亚指数量子算法，从而厘清了该问题的复杂性及其与已知计算问题的关系。