Cryptanalysis of an MOR cryptosystem based on a finite associative algebra

Abstract

The Shor algorithm is effective for public-key cryptosystems based on an abelian group. At CRYPTO 2001, Paeng (2001) presented a MOR cryptosystem using a non-abelian group, which can be considered as a candidate scheme for post-quantum attack. This paper analyses the security of a MOR cryptosystem based on a finite associative algebra using a quantum algorithm. Specifically, let L be a finite associative algebra over a finite field F. Consider a homomorphism φ: Aut(L) → Aut(H)×Aut(I), where I is an ideal of L and H ≅ L/I. We compute dim Im(φ) and dim Ker(φ), and combine them by dim Aut(L) = dim Im(φ)+dim Ker(φ). We prove that Im(φ) = Stab_Comp(H,I)(μ + B ²(H, I)) and Ker(φ) ≅ Z ¹(H, I). Thus, we can obtain dim Im(φ), since the algorithm for the stabilizer is a standard algorithm among abelian hidden subgroup algorithms. In addition, Z ¹(H, I) is equivalent to the solution space of the linear equation group over the Galois fields GF(p), and it is possible to obtain dim Ker(φ) by the enumeration theorem. Furthermore, we can obtain the dimension of the automorphism group Aut(L). When the map ϕ ∈ Aut(L), it is possible to effectively compute the cyclic group 〈ϕ〉 and recover the private key a. Therefore, the MOR scheme is insecure when based on a finite associative algebra in quantum computation.

摘要

创新点

1997年Shor量子算法的出现对基于交换群的传统公钥密码构成了威胁。目前, 量子算法对基于非交换群的问题没有有效算法。2001年Paeng等人基于非交换群提出了MOR方案。这可看做ELGamal的模拟。之后, 基于各种具体的非交换群, 对该方案进行了安全分析并得到一些结果。在本文中, 我们基于结合代数分析了该方案的安全性并得到如下结果。设L是有限结合代数, 当映射φ∈Aut(L), 存在有效的量子算法求解循环群〈φ〉并能恢复密钥。这说明基于结合代数的MOR方案在量子攻击下是不安全的。