Chinese remaindering with errors

IEEE Transactions on Information Theory

Volumn 46, Issue 4, 2000, Pages 1330-1338

  Goldreich, Oded ^a   Ron, Dana ^b   Sudan, Madhu ^c  

^a WEIZMANN INSTITUTE OF SCIENCE   (Israel)

^b TEL AVIV UNIVERSITY   (Israel)

^c USA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; DECODING; DIGITAL ARITHMETIC; ENCODING (SYMBOLS); ERROR CORRECTION; NUMBER THEORY;

CHINESE REMAINDER THEOREM; LATTICE REDUCTION; LIST DECODING; REDUNDANT RESIDUE NUMBER SYSTEM CODES;

CODES (SYMBOLS);

EID: 0034228438     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/18.850672     Document Type: Article

References (42)

1. S. Ar, R. Lipton, R. Rubinfeld, and M. Sudan, Reconstructing algebraic functions from mixed data, SIAM J. Comput, vol. 28, no. 2, pp. 488-511, 1999.
2. C. Asmuth and J. Bloom, A modular approach to key safeguarding, IEEE Trans. Inform. Theory, vol. IT-29, pp. 208-210, Mar. 1983.
3. F. Barsie and P. Maestrini, Error detection and correction by product codes in residue number systems, IEEE Trans. Inform. Theory, vol. IT-24, pp. 640-643, Sept. 1978.
4. E. R. Berlekamp, Algebraic Coding Theory. New York: McGraw-Hill, 1968.
5. _, Bounded distance +1 soft-decision reed-solomon decoding, IEEE Trans. Inform.Theory, vol. 42, pp. 704-720, May 1996.
6. A. Borodin and I. Munro, The Computational Complexity of Algebraic and Numeric Problems. New York: American Elsevier, 1975.
7. D. E. R. Denning, Cryptography and Data Security. Reading, MA: Addison-Wesley, 1983.
8. I. M. Duursma, Decoding codes from curves and cyclic codes, Ph.D. dissertation, Eindhoven Inst. Technol., Eindhoven, The Netherlands, 1993.
9. P. Elias, List Decoding for Noisy Channels, MIT, Res. Lab. Electron., Tech. Rep. 335, 1957.
10. _, Error-correcting codes for list decoding, IEEE Trans. Inform. Theory, vol. 37, pp. 5-12, Jan. 1991.
11. O. Goldreich, D. Ron, and M. Sudan. (1998) Chinese Remaindering with Errors, TR 98-082. [Online]. Available: http://www.eccc.uni-trier. de/eccc/
12. O. Goldreich, R. Rubinfeld, and M. Sudan, Learning polynomials with queries: The highly noisy case, in Proc. 36th Annu. Symp. Foundations of Computer Science, Milwaukee, WI, Oct. 23-25, 1998, pp. 294-303. Revised version available from ECCC.
13. V. Guruswami and M. Sudan, Improved decoding for Reed-Solomon and algebraic-geometric codes, IEEE Trans. Inform. Theory, vol. 45, pp. 1757-1767, Sept. 1998. Preliminary version in FOCS.
14. J. Håstad and M. Naslund, The security of all RSA and discrete log bits, Manuscript, Feb. 1999.
15. E. Kaltofen, Polynomial factorization 1987-1991, in LATIN'92 (Lecture Notes in Computer Science), I. Simon, Ed. New York: Springer-Verlag, 1992, vol. 583, pp. 294-313.
16. R. M. Karp and M. O. Rabin, Efficient Randomized Pattern-Matching Algorithms, Aiken Computation Lab., Harvard Univ., Tech. Rep. TR-31-81, 1981.
17. D. E. Knuth, The analysis of algorithms, Actes du Congres International des Mathematiciens, vol. 3, pp. 269-274, 1970.
18. R. Kotter, A unified description of an error locating procedure for linear codes, in Proc. Algebraic and Combinatorial Coding Theory Conf., Voneshta Voda, Bulgaria, 1992.
19. H. Krishna, B. Krishna, K.-Y Lin, and J.-D. Sun, Computational Number Theory and Digital Signal Processing: Fast Algorithms and Error Control Techniques. Boca Raton, EL: CRC 1994.
20. E. Kushilevitz and N. Nisan, Communication Complexity. Cambridge, U.K.: Cambridge Univ. Press, 1997.
21. A. K. Lenstra, H. W. Lenstra, and L. Lovasz, Factoring polynomials with rational coefficients, Mathematische Annalen, vol. 261, pp. 515-534, 1982.
22. H. W. Lenstra, Integer programming with a fixed number of variables, Math. Operations Res., vol. 8, pp. 538-548, 1983.
23. L. Lovasz, An Algorithmic Theory of Numbers, Graphs and Convexity. Philadephia, PA: SIAM, 1986.
24. F. J. Mac Williams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1981.
25. D. M. Mandelbaum, On a class of arithmetic codes and a decoding algorithm, IEEE Trans. Inform. Theory, vol. IT-22, pp. 85-88, Jan. 1976.
26. _, Further results on decoding arithmetic residue codes, IEEE Trans. Inform. Theory, vol. IT-24, pp. 643-644, Sept. 1978.
27. J. L. Massey, Shift register synthesis and BCH decoding, IEEE Trans. Inform, Theory, vol. IT-15, pp. 122-127, Jan. 1969.
28. B. Mishra, Algorithmic Algebra. New York: Springer-Verlag, 1993.
29. R. Pellikaan, On Decoding Linear Codes by Error Correcting Pairs, preprint, 1988. Eindhoven Univ. Technol.
30. W. W. Peterson, Encoding and error-correction procedures for BoseChaudhuri codes, IRE Trans. Inform. Theory, vol. IT-6, pp. 459-470, Sept. 1960.
31. A. Schonhage and V Strassen, Schnelle multiplikation grosser zahlen, Computing, vol. 7, pp. 281-292, 1971.
32. A. Shamir, How to share a secret, Commun. ACM, vol. 22, pp. 612-613, Nov. 1979.
33. M. A. Shokrollahi and H. Wasserman, Decoding algebraic-geometric codes beyond the error-correction bound, in Proc. 30th Annu. ACM Symp. Theory of Computing, Dallas, TX, May 23-26, 1998, pp. 241-248.
34. M. Sipser and D. A. Spielman, Expander codes, IEEE Trans. Inform. Theory, vol. 42, pp. 1710-1722, Nov. 1996.
35. M. A. Soderstrand, W. K. Jenkins, G. A. Jullien, and F. J. Taylor, Residue Number System Arithmetic: Modern Applications in Digital Signal Processing. New York: IEEE Press, 1986.
36. D. A. Spielman, Linear-time encodable and decodable error-correcting codes IEEETrans. Inform. Theory, vol. 42, pp. 1723-1732, Nov. 1996.
37. M. Sudan, Decoding of Reed-Solomon codes beyond the error-correction bound, J. Complexity, vol. 13, no. 1, pp. 180-193, 1997.
38. J. H. van Lint, Introduction to Coding Theory. New York: Springer-Verlag, 1982.
39. L. G. Valiant, The complexity of computing the permanent, Theor. Comput. Sci., vol. 8, no. 2, pp. 189-201, Apr. 1979.
40. A. Vardy, Algorithmic complexity in coding theory and the minimum distance problem, in Proc. 29th Annu. ACM Symp. Theory of Computing, El Paso, TX, May 4-6, 1997, pp. 92-109.
41. R. W. Watson and C. W. Hastings, Self-checked computation using residue arithmetic, Proc. IEEE, vol. 54, pp. 1920-1931, Dec. 1966.
42. L. Welch and E. R. Berlekamp, Error Correction of Algebraic Block Codes, U.S. Patent 4 633 470, issued Dec. 1986.