Abstract
Recently there has been a great deal of interest in the power of “Quantum Computers” [4, 15, 18]. The driving force is the recent beautiful result of Shor that shows that discrete log and factoring are solvable in random quantum polynomial time [15]. We use a method similar to Shor’s to obtain a general theorem about quantum polynomial time. We show that any cryptosystem based on what we refer to as a ‘hidden linear form’ can be broken in quantum polynomial time. Our results imply that the discrete log problem is doable in quantum polynomial time over any group including Galois fields and elliptic curves. Finally, we introduce the notion of ‘junk bits’ which are helpful when performing classical computations that are not injective.
Supported in part by NSF CCR-9304718.
Chapter PDF
Similar content being viewed by others
References
R. Beals, Computing Fourier Transform over S n in QP, unpublished manuscript.
C. Bennett, Logical reversibility of computation, IBM J. Res. Develop. vol. 17, 1973, pp. 525–532.
C. Bennett, E. Bernstein, G. Brassard, U. Vazirani, Strengths and Weaknesses of Quantum Computing, to appear.
E. Bernstein and U. Vazirani, Quantum Complexity Theory, Proc. 25th ACM Symp. on Theory of Computation, 1993.
D. Coppersmith, An Approximate Fourier Transform Useful in Quantum Factoring, IBM Research Report 19642, 1994.
W. Diffie and M. Hellman, New Directions in Cryptography, IEEE transactions on Information Theory, vol. 22, no. 6, pp. 644–654, 1976.
N. Koblitz, Elliptic Curve Cryptosystems, Mathematics of Computations 48, 1987, pp. 203–209.
S. Lang, Algebra.
U. Maurer and Y. Yacobi, Non-interactive public-key cryptography, EuroCrypt91, pp.498–507, 1991.
K. McCurley, A Key Distribution System Equivalent to Factoring, Journal of Cryptology, vol. 1, no. 2, pp. 95–105.
V. Miller, Uses of Elliptic Curves in Cryptography, In Proceedings of Crypto 1985, pp. 417–426.
B. Preneel, R. Govaerts, J. Vandewalle, Hash Functions Based on Block Ciphers: A Synthetic Approach, in Proc. of Advances in Cryptology, CRYPTO’ 93.
J. P. Serre, Linear Representations of Finite Groups, Springer-Verlag, 1977.
D. Simon, On the Power of Quantum Computation, Proc. FOCS, 1994, pp. 116–123.
P. Shor, Algorithms for Quantum Computation, Proc. FOCS, 1994, pp. 124–134.
L. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, 1982.
J. Buchmann and H. Williams, A Key Exchange System Based on Imaginary Quadratic Fields, Journal of Cryptology, vol. 1, no. 2, pp. 107–118, 1988.
A. Yao, Quantum Circuit Complexity, Proc. 34th IEEE Symp. on Foundations of Computer Science, 1993, pp. 352–360.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Boneh, D., Lipton, R.J. (1995). Quantum Cryptanalysis of Hidden Linear Functions. In: Coppersmith, D. (eds) Advances in Cryptology — CRYPT0’ 95. CRYPTO 1995. Lecture Notes in Computer Science, vol 963. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44750-4_34
Download citation
DOI: https://doi.org/10.1007/3-540-44750-4_34
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60221-7
Online ISBN: 978-3-540-44750-4
eBook Packages: Springer Book Archive