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Cryptanalysis of an MOR cryptosystem based on a finite associative algebra

基于有限结合代数的 MOR 公钥密码安全性分析

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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 HL/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) = StabComp(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。

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References

  1. 1

    Deutsch D, Jozsa R. Rapid solution of problems by quantum computation. Proc Roy Soc A-Math Phys Eng, 1992, 439: 553–558

  2. 2

    Simon D R. On the power of quantum computation. SIAM J Comput, 1997, 26: 1474–1483

  3. 3

    Shor P W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev, 1999, 41: 303–332

  4. 4

    Grover L K. Quantum mechanics helps in searching for a needle in a haystack. Phys Rev Lett, 1997, 79: 325–328

  5. 5

    Mosca M, Ekert A. The hidden subgroup problem and eigenvalue estimation on a quantum computer. Quantum Comput Quantum Commun, 1999: 174–188

  6. 6

    Ko K H, Lee S J, Cheon J H, et al. New public-key cryptosystem using braid groups. In: Proceedings of 20th Annual International Cryptology Conference, Santa Barbara, 2000. 166–183

  7. 7

    Paeng S H, Ha K C, Kim J H, et al. New public key cryptosystem using finite non Abelian groups. In: Proceedings of 21st Annual International Cryptology Conference, Santa Barbara, 2001. 470–485

  8. 8

    Lempken W, van Tran T, Magliveras S S, et al. A public key cryptosystem based on non-abelian finite groups. J Cryptol, 2009, 22: 62–74

  9. 9

    Mahalanobis A. A simple generalization of the ElGamal cryptosystem to non-abelian groups II. Commun Algebra, 2012, 40: 3583–3596

  10. 10

    Paeng S H. On the security of cryptosystem using automorphism groups. Inf Process Lett, 2003, 88: 293–298

  11. 11

    Tobias C. Security analysis of the MOR cryptosystem. In: Proceedings of 6th International Workshop on Practice and Theory in Public Key Cryptography, Miami, 2002. 175–186

  12. 12

    Lee I S, Kim W H, Kwon D, et al. On the security of MOR public key cryptosystem. In: Proceedings of 10th International Conference on the Theory and Application of Cryptology and Information Security, Jeju Island, 2004. 387–400

  13. 13

    Korsten A. Cryptanalysis of MOR and discrete logarithms in inner automorphism groups. In: Proceedings of 2nd Western European Workshop on Research in Cryptology, Bochum, 2008. 78–89

  14. 14

    Mahalanobis A. A simple generalization of ElGamal cryptosystem to non-abelian groups. Commun Algebra, 2006, 40: 3583–3596

  15. 15

    Babai L, Beals R, Seress A. Polynomial-time theory of matrix groups. In: Proceedings of 41st Annual ACM Symposium on Theory of Computing. New York: ACM, 2009. 55–64

  16. 16

    Friedl K, Ivanyos G, Magniez F, et al. Hidden translation and orbit coset in quantum computing. In: Proceedings of 35th Annual ACM Symposium on Theory of Computing. New York: ACM, 2003. 1–9

  17. 17

    Hallgren S, Russell A, Ta-Shma A. The hidden subgroup problem and quantum computation using group representations. SIAM J Comput, 2003, 32: 916–934

  18. 18

    Childs A M, van Dam W. Quantum algorithms for algebraic problems. Rev Mod Phys, 2010, 82: 1–52

  19. 19

    Wei H Z, Wang Y X. Enumeration theorems of solutions of some matrix equations over finite field (in Chinese). J Hebei Normal Univ, 1993, 17: 1–13

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Correspondence to Haiqing Han.

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Wu, W., Zhang, H., Wang, H. et al. Cryptanalysis of an MOR cryptosystem based on a finite associative algebra. Sci. China Inf. Sci. 59, 32111 (2016). https://doi.org/10.1007/s11432-015-5447-y

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Keywords

  • MOR cryptosystem
  • cryptanalysis
  • quantum algorithm
  • finite associative algebra
  • hidden subgroup problem
  • stabilizer

关键词

  • MOR 公钥密码
  • 密码分析
  • 量子算法
  • 有限结合代数
  • 隐藏子群问题
  • 稳定化子