Abstract
The noisy polynomial interpolation problem is a new intractability assumption introduced last year in oblivious polynomial evaluation. It also appeared independently in password identification schemes, due to its connection with secret sharing schemes based on Lagrange’s polynomial interpolation. This paper presents new algorithms to solve the noisy polynomial interpolation problem. In particular, we prove a reduction from noisy polynomial interpolation to the lattice shortest vector problem, when the parameters satisfy a certain condition that we make explicit. Standard lattice reduction techniques appear to solve many instances of the problem. It follows that noisy polynomial interpolation is much easier than expected. We therefore suggest simple modifications to several cryptographic schemes recently proposed, in order to change the intractability assumption. We also discuss analogous methods for the related noisy Chinese remaindering problem arising from the well-known analogy between polynomials and integers.
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References
M. Ajtai. Generating hard instances of lattice problems. In Proc. of 28th STOC, pages 99–108. ACM, 1996. Available at [11] as TR96-007 report.
M. Ajtai. The shortest vector problem in L 2 is NP-hard for randomized reductions. In Proc. 30th ACM STOC, 1998. Available at [11] as TR97-047.
S. Ar, R. Lipton, R. Rubinfeld, and M. Sudan. Reconstructing algebraic functions from mixed data. SIAM J. of Computing, 28(2):488–511, 1999.
A. O. L. Atkin. The number of points on an elliptic curve modulo a prime. Email on the Number Theory mailing list, 1988.
A. O. L. Atkin. The number of points on an elliptic curve modulo a prime. Email on the Number Theory mailing list, 1991.
D. Boneh. Finding smooth integers in short intervals using CRT decoding. In Proc. of 32nd STOC. ACM, 2000.
J.H. Conway and N.J.A. Sloane. Sphere Packings, Lattices and Groups. Springer-Verlag, 1998. Third edition.
D. Coppersmith. Small solutions to polynomial equations, and low exponent RSA vulnerabilities. J. of Cryptology, 10(4):233–260, 1997.
M.J. Coster, A. Joux, B.A. LaMacchia, A.M. Odlyzko, C.-P. Schnorr, and J. Stern. Improved low-density subset sum algorithms. Comput. Complexity, 2:111–128, 1992.
N. Courtois, A. Klimov, J. Patarin, and A. Shamir. Efficient algorithms for solving overdefined systems of multivariate polynomial equations. In Proc. of Eurocrypt 2000, volume 1807 of LNCS, pages 392–407. Springer-Verlag, 2000.
ECCC. http://www.eccc.uni-trier.de/eccc/. The Electronic Colloquium on Computational Complexity.
N. D. Elkies. Explicit isogenies. Draft, 1991.
C. Ellison, C. Hall, R. Milbert, and B. Schneier. Protecting secret keys with personal entropy. Future Generation Computer Systems, 16(4):311–318, 2000. Available at http://www.counterpane.com/.
N. Gilboa. Two party RSA key generation. In Proc. of Crypto’ 99, volume 1666 of LNCS, pages 116–129. Springer-Verlag, 1999.
O. Goldreich, D. Ron, and M. Sudan. Chinese remaindering with errors. In Proc. of 31st STOC. ACM, 1999. Also available at [11]. The Electronic Colloquium on Computational Complexity.
M. Gruber and C. G. Lekkerkerker. Geometry of Numbers. North-Holland, 1987.
V. Guruswami and M. Sudan. Improved decoding of Reed-Solomon and algebraicgeometric codes. IEEE Trans. on Information Theory, 45(6):1757–1767, 1999. An extended abstract appeared in the Proc. of IEEE FOCS’ 98.
A. Joux and J. Stern. Lattice reduction: A toolbox for the cryptanalyst. J. of Cryptology, 11:161–185, 1998.
A. Kipnis and A. Shamir. Cryptanalysis of the HFE public key cryptosystem by relinearization. In Proc. of Crypto’ 99, volume 1666 of LNCS, pages 19–30. Springer-Verlag, 1999.
J. C. Lagarias and A. M. Odlyzko. Solving low-density subset sum problems. Journal of the Association for Computing Machinery, January 1985.
A. K. Lenstra, H. W. Lenstra, and L. Lovász. Factoring polynomials with rational coefficients. Math. Ann., 261:515–534, 1982.
R. Lercier and F. Morain. Counting the number of points on elliptic curves over finite fields: strategies and performances. In Proc. of Eurocrypt’ 95, volume 921 of LNCS, pages 79–94. Springer-Verlag, 1995.
J. E. Mazo and A. M. Odlyzko. Lattice points in high-dimensional spheres. Monatsh. Math., 110:47–61, 1990.
D. Micciancio. The shortest vector problem is NP-hard to approximate within some constant. In Proc. 39th IEEE FOCS, 1998. Available at [11] as TR98-016.
F. Monrose, M. Reiter, and S. Wetzel. Password hardening based on keystroke dynamics. In Proc. of 6th Conf. on Computer and Communications Security. ACM, 1999.
M. Naor and B. Pinkas. Oblivious transfer and polynomial evaluation. In Proc. of 31st STOC, pages 245–254. ACM, 1999.
P. Nguyen and J. Stern. Merkle-Hellman revisited: a cryptanalysis of the Qu-Vanstone cryptosystem based on group factorizations. In Proc. of Crypto’ 97, volume 1294 of LNCS, pages 198–212. Springer-Verlag, 1997.
R. Peralta and E. Okamoto. Faster factoring of integers of a special form. IEICE Trans. Fund. of Electronics, Communications, and Computer Sciences, 79(4), 1996.
B. Pinkas, 1999. Private communication.
C.-P. Schnorr. A hierarchy of polynomial lattice basis reduction algorithms. Theoretical Computer Science, 53:201–224, 1987.
C. P. Schnorr and M. Euchner. Lattice basis reduction: improved practical algorithms and solving subset sum problems. Math. Programming, 66:181–199, 1994.
C.P. Schnorr and H.H. Hörner. Attacking the Chor-Rivest cryptosystem by improved lattice reduction. In Proc. of Eurocrypt’95, volume 921 of LNCS, pages 1–12. Springer-Verlag, 1995.
R. Schoof. Counting points on elliptic curves over finite fields. J. Théor. Nombres Bordeaux, 7:219–254, 1995.
V. Shoup. Number Theory Library (NTL) version 3.7a. Can be obtained at http://www.shoup.net/ntl/.
C. L. Siegel. Lectures on the Geometry of Numbers. Springer-Verlag, 1989.
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Bleichenbacher, D., Nguyen, P.Q. (2000). Noisy Polynomial Interpolation and Noisy Chinese Remaindering. In: Preneel, B. (eds) Advances in Cryptology — EUROCRYPT 2000. EUROCRYPT 2000. Lecture Notes in Computer Science, vol 1807. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45539-6_4
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