# 암호해독을 위한 양자 계산 알고리즘의 최근 연구동향

• Bae, Eunok (Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University) ;
• Kim, Jeong San (Department of Applied Mathematics and Institute of Natural Sciences, Kyung Hee University) ;
• Lee, Soojoon (Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University)
• 배은옥 (경희대학교 수학과, 기초과학 연구소) ;
• 김정산 (경희대학교 응용수학과, 자연과학종합연구원) ;
• 이수준 (경희대학교 수학과, 기초과학 연구소)
• Accepted : 2018.03.12
• Published : 2018.04.25

#### Abstract

In this paper, we mainly introduce some quantum computational algorithms that have exponential speedups over the best known classical algorithms, and summarize recent research achievements in quantum algorithms that can affect existing cryptosystems. Finally, we suggest a research direction that can improve these results more progressively.

#### Acknowledgement

Supported by : 한국연구재단

#### References

1. J. Kim, Y. Lim, E. Bae, and D. Kim, "A research on the technique of cryptosystem security analysis using quantum computational algorithms" (in Korean), National Security Research Institute Report (Grant No. 2017-013, 2017).
2. P. W. Shor, "Algorithms for quantum computation: discrete logarithms and factoring," in Proc. 35th Annual IEEE Symposium on the Foundations of Computer Science (IEEE Computer Society Press, Piscataway, NJ, USA, 1994), SIAM J. Comput. 26, 1484-1509 (1997).
3. L. K. Grover, "A fast quantum mechanical algorithm for database search" in Proc. 28th Annual ACM Symposium on Theory of Computing (ACM, NY, USA, 1996), Phys. Rev. Lett. 79, 325-328 (1997).
4. D. Boneh and R. Lipton, "Quantum cryptanalysis of hidden linear functions," in Proc. Crypto'95, LNCS 963, 427-437 (1995).
5. A. Y. Kitaev, "Quantum measurements and the abelian stabilizer problem," arXiv:quant-ph/9511026v1 (1995).
6. M. Ettinger and P. Hoyer, "A quantum observable for the graph isomorphism problem," arXiv:quant-ph/9901029v1 (1999).
7. S. Hallgren, "The hidden subgroup problem and quantum computing using group representations," SIAM J. Comput. 32, 916-934 (2003). https://doi.org/10.1137/S009753970139450X
8. M. Grigni, L. Schulman, M. Vazirani, and U. Vazirani, "Quantum mechanical algorithms for the non-abelian hidden subgroup problem," in Proc. 33rd Annual ACM Symposium on Theory of Computing (2001), Combinatorica 24, 137-154 (2004).
9. K. Friedl, G. Ivanyos, F. Magniez, M. Santha, and P. Sen, "Hidden translation and translating coset in quantum computing," in Proc. 35th Annual ACM Symposium on Theory of Computing (2003), SIAM J. Comput. 43, 1-24 (2014).
10. G. Kuperberg, "A subexponential-time quantum algorithm for the dihedral hidden subgroup problem," SIAM J. Comput. 35, 170-188 (2005). https://doi.org/10.1137/S0097539703436345
11. M. Ettinger, P. Hoyer, and E. Knill, "The quantum query complexity of the hidden subgroup problem is polynomial," Inf. Process. Lett. 91, 43-48 (2004). https://doi.org/10.1016/j.ipl.2004.01.024
12. D. Gavinsky, "Quantum solution to the hidden subgroup problem for poly-near-hamiltonian groups," Quantum Inf. Comput. 4, 229-235 (2004).
13. Y. Inui and F. Le Gall, "Efficient quantum algorithm for the hidden subgroup problem over a class of semi-direct product groups," Quantum Inf. Comput. 7, 559-570 (2007).
14. C. Moore, D. N. Rockmore, A. Russell, and L. J. Schulman, "The power of strong Fourier sampling: Quantum algorithms for affine groups and hidden shifts," in Proc. 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SIAM, Philadelphia, USA, 2004), SIAM J. Comput. 37, 938-958 (2007).
15. O. Regev, "A subexponential-time algorithm for the dihedral hidden subgroup problem with polynomial space," arXiv: quant-ph/0406151v1 (2004).
16. D. Bacon, A. Childs, and W. van Dam, "From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups," in Proc. 46th Annual IEEE Symposium on the Foundations of Computer Science, 469-478 (2005).
17. O. Regev, "Quantum computation and lattice problems," in Proc. 43rd Annual IEEE Symposium on the Foundations of Computer Science, 520-529 (2002).
18. S. Hallgren, C. Moore, M. Rotteler, A. Russell, and P. Sen, "Limitations of quantum coset states for graph isomorphism," in Proc. 38th Annual ACM Symposium on Theory of Computing, 604-617 (2006).
19. W. van Dam, S. Hallgren, and L. Ip, "Quantum algorithms for some hidden shift problems," SIAM J. Comput. 36, 763-778 (2006). https://doi.org/10.1137/S009753970343141X
20. I. B. Damgard, "On the randomness of Legendre and Jacobi sequences," in Proc. Advances in Cryptology-CRYPTO 1988, 403, 163-172 (1990).
21. M. Ozols, M. Roetteler, and J. Roland, "Quantum rejection sampling," in Proc. 3rd Innovations in Theoretical Computer Science Conference, 290-308 (2012).
22. O. Regev, "Quantum computation and lattice problems," SIAM J. Comput. 33, 738-760 (2004). https://doi.org/10.1137/S0097539703440678
23. S. Hallgren, "Polynomial-time quantum algorithm for Pell's equation and the principal ideal problem," in Proc. 34th Annual ACM Symposium on Theory of Computing (2002), J. ACM 54, 1-19 (2007).
24. S. Hallgren, "Fast quantum algorithms for computing the unit group and class group of a number field," in Proc. 37th Annual ACM Symposium on Theory of Computing, 468-474 (2005).
25. A. Schmidt and U. Vollmer, "Polynomial-time quantum algorithm for the computation of the unit group of a number field," in Proc. 37th Annual ACM Symposium on Theory of Computing, 475-480 (2005).
26. K. Eisentrager, S. Hallgren, A. Kitaev, and F. Song, "A quanum algorithm for computing the unit group of an arbitrary degree number field," in Proc. 46th Annual ACM Symposium on Theory of Computing, 293-302 (2014).
27. J. F. Biasse and F. Song, "Efficient quantum algorithms for computing class groups and solving the principal ideal problem in arbitrary degree number fields," in Proc. 27th Annual ACM-SIAM Symposium on Discrete Algorithms, (2016).
28. E. Bae and S. Lee, "Quantum algorithm for continuous hidden shift problems" in preparation.
29. C. Gentry and S. Halevi, "Implementing gentry's fullyhomomorphic encryption scheme," in Proc. Eurocrypt 2011, 132-150 (2011).
30. V. Lyubashevsky, C. Peikert, and O. Regev, "On ideal lattices and learning with errors over rings," in Proc. Advances in cryptology-CRYPTO 2010, 6110, 1-23 (2010).
31. Z. Brakerski and V. Vaikuntanathan, "Fully homomorphic encryption from ring-LWE and security for key dependent messages," in Proc. Advances in cryptology-Eurocrypt 2011, 6841, 505-524 (2011).