On Computing Nearest Neighbors with Applications to Decoding of Binary Linear Codes

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9056)


We propose a new decoding algorithm for random binary linear codes. The so-called information set decoding algorithm of Prange (1962) achieves worst-case complexity \(2^{0.121n}\). In the late 80s, Stern proposed a sort-and-match version for Prange’s algorithm, on which all variants of the currently best known decoding algorithms are build. The fastest algorithm of Becker, Joux, May and Meurer (2012) achieves running time \(2^{0.102n}\) in the full distance decoding setting and \(2^{0.0494n}\) with half (bounded) distance decoding.

In this work we point out that the sort-and-match routine in Stern’s algorithm is carried out in a non-optimal way, since the matching is done in a two step manner to realize an approximate matching up to a small number of error coordinates. Our observation is that such an approximate matching can be done by a variant of the so-called High Dimensional Nearest Neighbor Problem. Namely, out of two lists with entries from \({\mathbb F}_2^m\) we have to find a pair with closest Hamming distance. We develop a new algorithm for this problem with sub-quadratic complexity which might be of independent interest in other contexts.

Using our algorithm for full distance decoding improves Stern’s complexity from \(2^{0.117n}\) to \(2^{0.114n}\). Since the techniques of Becker et al apply for our algorithm as well, we eventually obtain the fastest decoding algorithm for binary linear codes with complexity \(2^{0.097n}\). In the half distance decoding scenario, we obtain a complexity of \(2^{0.0473n}\).


Linear codes Nearest neighbor problem Approximate matching Meet-in-the-middle 


  1. 1.
    Agarwal, P.K., Edelsbrunner, H., Schwarzkopf, O.: Euclidean Minimum Spanning Trees and Bichromatic Closest Pairs. Discrete & Computational Geometry 6, 407–422 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Alekhnovich, M.: More on Average Case vs Approximation Complexity. In: 44th Symposium on Foundations of Computer Science (FOCS), pp. 298–307 (2003)Google Scholar
  3. 3.
    Becker, A., Coron, J.-S., Joux, A.: Improved Generic Algorithms for Hard Knapsacks. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 364–385. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  4. 4.
    Becker, A., Joux, A., May, A., Meurer, A.: Decoding random binary linear codes in \(2^{n/20}\): How \(1 + 1 = 0\) improves information set decoding. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 520–536. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  5. 5.
    Brakerski, Z., Vaikuntanathan, V.: Efficient Fully Homomorphic Encryption from (Standard) LWE. In: FOCS, pp. 97–106 (2011)Google Scholar
  6. 6.
    Dubiner, M., Bucketing coding and information theory for the statistical high-dimensional nearest-neighbor problem. IEEE Transactions on Information Theory 56(8), 4166–4179 (2010)Google Scholar
  7. 7.
    Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: STOC, pp. 197–206 (2008)Google Scholar
  8. 8.
    Har-Peled, S., Indyk, P.: Rajeev Motwani Approximate Nearest Neighbor: Towards Removing the Curse of Dimensionality. Theory of Computing 8(1), 321–350 (2012)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Hopper, N.J., Blum, M.: Secure Human Identification Protocols. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 52–66. Springer, Heidelberg (2001) CrossRefGoogle Scholar
  10. 10.
    Horowitz, E., Sahni, S.: Computing partitions with applications to the knapsack problem. J. ACM 21(2), 277–292 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Howgrave-Graham, N., Joux, A.: New Generic Algorithms for Hard Knapsacks. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 235–256. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  12. 12.
    Kiltz, E., Pietrzak, K., Cash, D., Jain, A., Venturi, D.: Efficient Authentication from Hard Learning Problems. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 7–26. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  13. 13.
    Lee, P.J., Brickell, E.F.: An Observation on the Security of McEliece’s Public-Key Cryptosystem. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 275–280. Springer, Heidelberg (1988) CrossRefGoogle Scholar
  14. 14.
    May, A., Meurer, A., Thomae, E.: Decoding Random Linear Codes in \(\tilde{\mathcal{O}}(2^{0.054n})\). In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 107–124. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  15. 15.
    McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. Jet Propulsion Laboratory DSN Progress Report 42–44, 114–116 (1978)Google Scholar
  16. 16.
    Peikert, C.: Brent Waters Lossy trapdoor functions and their applications. In: STOC, pp. 187–196 (2008)Google Scholar
  17. 17.
    Peters, C.: Information-Set Decoding for Linear Codes over \(\mathbb{F}_q\). In: Sendrier, N. (ed.) PQCrypto 2010. LNCS, vol. 6061, pp. 81–94. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  18. 18.
    Prange, E.: The Use of Information Sets in Decoding Cyclic Codes. IRE Transaction on Information Theory 8(5), 5–9 (1962)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: STOC, pp. 84–93 (2005)Google Scholar
  20. 20.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM 56(6) (2009)Google Scholar
  21. 21.
    Stern, J.: A method for finding codewords of small weight. In: Proceedings of the 3rd International Colloquium on Coding Theory and Applications, London, UK, pp. 106–113. Springer (1989)Google Scholar
  22. 22.
    Sudan, M.: Algorithmic Introduction to Coding Theory. Lecture Notes (available online) (2001)Google Scholar
  23. 23.
    Andoni, A., Indyk, P., Nguyen, H.L., Razenshteyn, I.: Beyond Locality-Sensitive Hashing. In: SODA, pp. 1018–1028 (2014)Google Scholar
  24. 24.
    Valiant, G.: Finding Correlations in Subquadratic Time, with Applications to Learning Parities and Juntas. In: FOCS, pp. 11–20 (2012)Google Scholar

Copyright information

© International Association for Cryptologic Research 2015

Authors and Affiliations

  1. 1.Faculty of Mathematics, Horst Görtz Institute for IT-SecurityRuhr-University BochumBochumGermany

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