Abstract
This paper addresses the question how often the square code of an arbitrary l-dimensional subcode of the code GRS k (a, b) is exactly the code GRS2k-1(a, b * b). To answer this question we first introduce the notion of gaps of a code which allows us to characterize such subcodes easily. This property was first used and stated by Wieschebrink where he applied the Sidelnikov–Shestakov attack to break the Berger–Loidreau cryptosystem.
Similar content being viewed by others
References
Berger T., Loidreau P.: How to mask the structure of codes for a cryptographic use. Des. Codes Cryptogr. 35, 63–79 (2005)
Bernstein D.J.: Grover vs. McEliece. PQCrypto 2010(36), 73–80 (2010)
Diffie W., Hellman M.: New directions in cryptography. IEEE Trans. Inf. Theory. IT-22, 644–654 (1976)
Duursma I., Kirov R., Park S.: Distance bounds for algebraic geometric codes. J. Pure Appl. Algebra. 215, 1863–1878 (2011)
Geil O., Munuera C., Ruano D., Torres F.: On the order bounds for one-point AG codes. Adv. Math. Commun. 5, 489–504 (2011)
Høholdt T., van Lint J.H., Pellikaan R.: Algebraic geometry codes. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory. vol. 1, pp. 871–961. Elsevier, Amsterdam (1998).
Kasami T., Lin S., Peterson W.W.: New generalizations of the Reed–Muller codes part I: Primitive codes. IEEE Trans. Inf. Theory IT-14, 189–199 (1968)
Li Y.X., Deng R.H., Wang X.M.: On the equivalence of McEliece’s and Niederreiter’s public-key cryptosystems. IEEE Trans. Inf. Theory IT- 40, 27–273 (1994)
McEliece R.J.: A public-key cryptosystem based on algebraic coding theory. DSN Prog. Rep. 42(−44), 114–116 (1978)
Niederreiter H.: Knapsack-type crypto systems and algebraic coding theory. Prob. Control Inf. Theory 15(2), 159–166 (1986)
Overbeck R., Sendrier N.: Code-based cryptography. Post Quantum Cryptogr. 6, 95–145 (2009)
Sidelnikov V.M., Shestakov S.O.: On the insecurity of cryptosystems based on generalized Reed–Solomon codes. Discret. Math. Appl. 2, 439–444 (1992)
Schmidt W.M.: A lower bound for the number of solutions of equations over finite fields. J. Number Theory 6, 448–480 (1974)
Stichtenoth H.: Algebraic Function Fields and Codes. Springer, Universitext (1993)
Wieschebrink C.: An attack on the modified Niederreiter encryption scheme. In: PKC 2006. Lecture Notes in Computer Science, vol. 3958, pp. 14–26, Berlin, Springer (2006).
Wieschebrink C.: Cryptoanalysis of the Niederreiter public key scheme based on GRS subcodes. In: Post-Quantum Cryptography. Lecture Notes in Computer Science, vol. 6061, pp. 6–72. Berlin, Springer (2010).
Author information
Authors and Affiliations
Corresponding author
Additional information
This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.
Rights and permissions
About this article
Cite this article
Márquez-Corbella, I., Martínez-Moro, E. & Pellikaan, R. The non-gap sequence of a subcode of a generalized Reed–Solomon code. Des. Codes Cryptogr. 66, 317–333 (2013). https://doi.org/10.1007/s10623-012-9694-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-012-9694-2