Abstract
We present a technique to recover f ∈ ℚ(ζ p ) where ζ p is a primitive p th root of unity for a prime p, given its norm \(g = f * \bar{f}\) in the totally real field \(\mathbb{Q}(\zeta_{p}+\zeta_{p}^{-1})\). The classical method of solving this problem involves finding generators of principal ideals by enumerating the whole class group associated with ℚ(ζ p ), but this approach quickly becomes infeasible as p increases. The apparent hardness of this problem has led several authors to suggest the problem as one suitable for cryptography. We describe a technique which avoids enumerating the class group, and instead recovers f by factoring N f , the absolute norm of f, (for example with a subexponential sieve algorithm), and then running the Gentry-Szydlo polynomial time algorithm for a number of candidates. The algorithm has been tested with an implementation in PARI.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Borevich, Z.I., Shafarevich, I.R.: Number Theory. Academic Press, London (1966)
Buchmann, J., Maurer, M., Möller, B.: Cryptography based on number fields with large regulator. Journal de Théorie des Nombres de Bordeaux, 293–307 (2000)
Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, vol. 138. Springer, Heidelberg (1993)
Cohen, H., Lenstra, H.: Heuristics on class groups of number fields. In: Number Theory. Lecture Notes in Mathematics, vol. 1068, pp. 33–62. Springer, Heidelberg (1983)
Elser, V.: Private Communication
Elser, V.: Bit retrieval: intractability and application to digital watermarking, http://arxiv.org/abs/math.NT/0309387
Elser, V.: Phase retrieval challenges, http://www.cecm.sfu.ca/~veit/
Gentry, C.: Private Communication
Gentry, C., Szydlo, M.: Cryptanalysis of the Revised NTRU signature scheme. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 299–320. Springer, Heidelberg (2002)
Hoffstein, J.: Private Communication
Hoffstein, J., Lieman, D., Silverman, J.H.: Polynomial Rings and Efficient Public Key Authentication. In: Blum, M., Lee, C.H. (eds.) Proc. International Workshop on Cryptographic Techniques and E-Commerce (CrypTEC 1999), City University of Hong Kong Press, Hong Kong (1999)
Hoffstein, J., Silverman, J.H.: Polynomial Rings and Efficient Public Key Authentication II. In: Proceedings of a Conference on Cryptography and Number Theory (CCNT 1999), Birkhauser, Basel (1999)
Lenstra, A.K., Lenstra Jr., H.W., Lovász, L.: Factoring Polynomials with Rational Coefficients. Mathematische Ann 261, 513–534 (1982)
Micciancio, D.: The Shortest Vector in a Lattice is Hard to Approximate to within Some Constant. In: Proc. 39th Symposium on Foundations of Computer Science, pp. 92–98 (1998)
Nguyen, P., Stern, J.: Lattice Reduction in Cryptology: An Update. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 85–112. Springer, Heidelberg (2000)
Schnorr, C.-P.: A Hierarchy of Polynomial Time Lattice Basis Reduction Algorithms. Theoretical Computer Science 53, 201–224 (1987)
Shoup, V.: NTL: A Library for Doing Number Theory, Available at http://www.shoup.net/ntl/
Szydlo, Michael: Hypercubic Lattice Reduction. Eurocrypt 2003 (2003)
Washington, L.: Introduction to Cyclotomic Fields. Graduate Texts in Mathematics 83 (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Howgrave-Graham, N., Szydlo, M. (2004). A Method to Solve Cyclotomic Norm Equations \(f * \bar{f}\) . In: Buell, D. (eds) Algorithmic Number Theory. ANTS 2004. Lecture Notes in Computer Science, vol 3076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24847-7_20
Download citation
DOI: https://doi.org/10.1007/978-3-540-24847-7_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22156-2
Online ISBN: 978-3-540-24847-7
eBook Packages: Springer Book Archive