Abstract
Inversion in Galois Fields is a famous primitive permutation for designing cryptographic algorithms e.g. for Rijndael because it has suitable differential and linear properties. Inputs and outputs are usually transformed by addition (e.g. XOR) to key bits. We call this construction the APA (Add-Permute-Add) scheme. In this paper we study its pseudorandomness in terms of k-wise independence.
We show that the pairwise independence of the APA construction is related to the impossible differentials properties. We notice that inversion has many impossible differentials, so x -> 1/(x+a)+b is not pairwise independent.
In 1998, Vaudenay proposed the random harmonic permutation h:x -> a/(x-b)+c. Although it is not perfectly 3-wise independent (despite what was originally claimed), we demonstrate in this paper that it is almost 3-wise independent. In particular we show that any distinguisher limited to three queries between this permutation and a perfect one has an advantage limited to 3/q where q is the field order. This holds even if the distinguisher has access to h − 1.
Finally, we investigate 4-wise independence and we suggest the cross-ratio as a new tool for cryptanalysis of designs involving inversion.
Chapter PDF
Similar content being viewed by others
References
Advanced Encryption Standard (AES). Federal Information Processing Standards Publication #197. U.S. Department of Commerce, National Institute of Standards and Technology (2001)
Biham, E., Biryukov, A., Shamir, A.: Cryptanalysis of Skipjack Reduced to 31 Rounds using Impossible Differentials. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 12–23. Springer, Heidelberg (1999)
Carter, J.L., Wegman, M.N.: Universal Classes of Hash Functions. Journal of Computer and System Sciences 18, 143–154 (1979)
Courtois, N.T., Pieprzyk, J.: Cryptanalysis of Block Ciphers with Overdefined Systems of Equations
Daemen, J., Rijmen, V.: The Design of Rijndael. In: Daemen, J., Rijmen, V. (eds.) Information Security and Cryptography, Springer, Heidelberg (2002)
Even, S., Mansour, Y.: A Construction of a Cipher from a Single Pseudorandom Permutation. In: Matsumoto, T., Imai, H., Rivest, R.L. (eds.) ASIACRYPT 1991. LNCS, vol. 739, pp. 210–224. Springer, Heidelberg (1993); Also in Journal of Cryptology 10, 151–161 (1997)
Jakobsen, T., Knudsen, L.R.: The Interpolation Attack on Block Ciphers. In: Biham, E. (ed.) FSE 1997. LNCS, vol. 1267, pp. 28–40. Springer, Heidelberg (1997)
Kilian, J., Rogaway, P.: How to Protect DES Against Exhaustive Key Search. Journal of Cryptology 14, 17–35 (2001)
Murphy, S., Robshaw, M.J.B.: Essential Algebraic Structure within the AES. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 1–16. Springer, Heidelberg (2002)
Nyberg, K.: Differentially Uniform Mappings for Cryptography. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 55–64. Springer, Heidelberg (1994)
Rees, E.G.: Notes on Geometry. Springer, Heidelberg (1983)
Rijmen, V., Daemen, J., Preneel, B., Bosselaers, A., De Win, E.: The Cipher SHARK. In: Gollmann, D. (ed.) FSE 1996. LNCS, vol. 1039, pp. 99–112. Springer, Heidelberg (1996)
Russel, A., Wang, H.: How to Fool an Unbounded Adversary with a Short Key. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 133–148. Springer, Heidelberg (2002)
Shannon, C.E.: Communication Theory of Secrecy Systems. Bell system technical journal 28, 656–715 (1949)
Stinson, D.R.: Universal Hashing and Authentication Codes. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 74–85. Springer, Heidelberg (1992)
Vaudenay, S.: Provable Security for Block Ciphers by Decorrelation. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 249–275. Springer, Heidelberg (1998)
Vaudenay, S.: Adaptive-Attack Norm for Decorrelation and Super-Pseudorandomness. In: Heys, H.M., Adams, C.M. (eds.) SAC 1999. LNCS, vol. 1758, pp. 49–61. Springer, Heidelberg (2000)
Vaudenay, S.: Decorrelation: a Theory for Block Cipher Security. To appear in the Journal of Cryptology
Wegman, M.N., Carter, J.L.: New Hash Functions and their Use in Authentication and Set Equality. Journal of Computer and System Sciences 22, 265–279 (1981)
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
Aoki, K., Vaudenay, S. (2004). On the Use of GF-Inversion as a Cryptographic Primitive. In: Matsui, M., Zuccherato, R.J. (eds) Selected Areas in Cryptography. SAC 2003. Lecture Notes in Computer Science, vol 3006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24654-1_17
Download citation
DOI: https://doi.org/10.1007/978-3-540-24654-1_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21370-3
Online ISBN: 978-3-540-24654-1
eBook Packages: Springer Book Archive