# American Institute of Mathematical Sciences

August  2015, 9(3): 311-339. doi: 10.3934/amc.2015.9.311

## Ideal forms of Coppersmith's theorem and Guruswami-Sudan list decoding

 1 Microsoft Research New England, One Memorial Drive, Cambridge, MA 02142, United States 2 Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA 19104, United States

Received  February 2014 Published  July 2015

We develop a framework for solving polynomial equations with size constraints on solutions. We obtain our results by showing how to apply a technique of Coppersmith for finding small solutions of polynomial equations modulo integers to analogous problems over polynomial rings, number fields, and function fields. This gives us a unified view of several problems arising naturally in cryptography, coding theory, and the study of lattices. We give (1) a polynomial-time algorithm for finding small solutions of polynomial equations modulo ideals over algebraic number fields, (2) a faster variant of the Guruswami-Sudan algorithm for list decoding of Reed-Solomon codes, and (3) an algorithm for list decoding of algebraic-geometric codes that handles both single-point and multi-point codes. Coppersmith's algorithm uses lattice basis reduction to find a short vector in a carefully constructed lattice; powerful analogies from algebraic number theory allow us to identify the appropriate analogue of a lattice in each application and provide efficient algorithms to find a suitably short vector, thus allowing us to give completely parallel proofs of the above theorems.
Citation: Henry Cohn, Nadia Heninger. Ideal forms of Coppersmith's theorem and Guruswami-Sudan list decoding. Advances in Mathematics of Communications, 2015, 9 (3) : 311-339. doi: 10.3934/amc.2015.9.311
##### References:
 [1] M. Ajtai, The shortest vector problem in $L_2$ is NP-hard for randomized reductions,, in Proc. 30th Ann. ACM Symp. Theory Comput., (1998), 10. doi: 10.1145/276698.276705. Google Scholar [2] M. Ajtai, R. Kumar and D. Sivakumar, A sieve algorithm for the shortest lattice vector problem,, in Proc. 33rd Ann. ACM Symp. Theory Comput., (2001), 601. doi: 10.1145/380752.380857. Google Scholar [3] M. Alekhnovich, Linear Diophantine equations over polynomials and soft decoding of Reed-Solomon codes,, IEEE Trans. Inf. Theory, 51 (2005), 2257. doi: 10.1109/TIT.2005.850097. Google Scholar [4] P. Beelen and K. Brander, Efficient list decoding of a class of algebraic-geometry codes,, Adv. Math. Commun., 4 (2010), 485. doi: 10.3934/amc.2010.4.485. Google Scholar [5] P. Beelen and K. Brander, Key equations for list decoding of Reed-Solomon codes and how to solve them,, J. Symb. Comput., 45 (2010), 773. doi: 10.1016/j.jsc.2010.03.010. Google Scholar [6] D. J. Bernstein, Reducing lattice bases to find small-height values of univariate polynomials,, in Algorithmic Number Theory: Lattices, (2008), 421. Google Scholar [7] D. J. Bernstein, List decoding for binary Goppa codes,, in Coding and Cryptology, (2011), 62. doi: 10.1007/978-3-642-20901-7_4. Google Scholar [8] J.-F. Biasse and G. Quintin, An algorithm for list decoding number field codes,, in 2012 IEEE Int. Symp. Inf. Theory Proc., (2012), 91. doi: 10.1109/ISIT.2012.6284696. Google Scholar [9] D. Bleichenbacher and P. Q. Nguyen, Noisy polynomial interpolation and noisy Chinese remaindering,, in Adv. Crypt. - EUROCRYPT 2000, (2000), 53. doi: 10.1007/3-540-45539-6_4. Google Scholar [10] J. Blömer and A. May, New partial key exposure attacks on RSA,, in Adv. Crypt. - CRYPTO 2003, (2003), 27. doi: 10.1007/978-3-540-45146-4_2. Google Scholar [11] D. Boneh, Finding smooth integers in short intervals using CRT decoding,, in Proc. 32nd Ann. ACM Symp. Theory Comput., (2000), 265. doi: 10.1145/335305.335337. Google Scholar [12] D. Boneh, G. Durfee and Y. Frankel, An attack on RSA given a small fraction of the private key bits,, in Adv. Crypt. - ASIACRYPT '98, (1998), 25. doi: 10.1007/3-540-49649-1_3. Google Scholar [13] J. Buchmann, T. Takagi and U. Vollmer, Number field cryptography,, in High Primes and Misdemeanours, (2004), 111. Google Scholar [14] M. F. I. Chowdhury, C.-P. Jeannerod, V. Neiger, É. Schost and G. Villard, Faster algorithms for multivariate interpolation with multiplicities and simultaneous polynomial approxima-tions,, IEEE Trans. Inf. Theory, 61 (2015), 2370. doi: 10.1109/TIT.2015.2416068. Google Scholar [15] H. Cohen, A Course in Computational Algebraic Number Theory,, Springer-Verlag, (1993). doi: 10.1007/978-3-662-02945-9. Google Scholar [16] H. Cohn and N. Heninger, Approximate common divisors via lattices,, in Proc. 10th Algor. Number Theory Symp., (2013), 271. doi: 10.2140/obs.2013.1.271. Google Scholar [17] D. Coppersmith, Small solutions to polynomial equations, and low exponent RSA vulnerabilities,, J. Cryptology, 10 (1997), 233. doi: 10.1007/s001459900030. Google Scholar [18] D. Coppersmith, Finding small solutions to small degree polynomials,, in Cryptography and Lattices, (2001), 20. doi: 10.1007/3-540-44670-2_3. Google Scholar [19] D. Coppersmith, N. Howgrave-Graham and S. V. Nagaraj, Divisors in residue classes, constructively,, Math. Comp., 77 (2008), 531. doi: 10.1090/S0025-5718-07-02007-8. Google Scholar [20] N. Coxon, List decoding of number field codes,, Des. Codes Cryptogr., 72 (2014), 687. doi: 10.1007/s10623-013-9803-x. Google Scholar [21] C. Fieker and M. E. Pohst, On lattices over number fields,, in Algorithmic Number Theory, (1996), 133. doi: 10.1007/3-540-61581-4_48. Google Scholar [22] C. Fieker and D. Stehlé, Short bases of lattices over number fields,, in Algorithmic Number Theory, (2010), 157. doi: 10.1007/978-3-642-14518-6_15. Google Scholar [23] J. von zur Gathen, Hensel and Newton methods in valuation rings,, Math. Comp., 42 (1984), 637. doi: 10.2307/2007608. Google Scholar [24] J. von zur Gathen and J. Gerhard, Modern Computer Algebra, 2nd edition,, Cambridge Univ. Press, (2003). Google Scholar [25] J. von zur Gathen and E. Kaltofen, Factorization of multivariate polynomials over finite fields,, Math. Comp., 45 (1985), 251. doi: 10.2307/2008063. Google Scholar [26] P. Giorgi, C.-P. Jeannerod and G. Villard, On the complexity of polynomial matrix computations,, in Proc. 2003 Int. Symp. Symb. Algebr. Comput., (2003), 135. doi: 10.1145/860854.860889. Google Scholar [27] E. Grigorescu and C. Peikert, List decoding Barnes-Wall lattices,, in Proc. 27th IEEE Conf. Comput. Compl., (2012), 316. doi: 10.1109/CCC.2012.33. Google Scholar [28] V. Guruswami and A. Rudra, Explicit codes achieving list decoding capacity: error correction with optimal redundancy,, IEEE Trans. Inf. Theory, 54 (2008), 135. doi: 10.1109/TIT.2007.911222. Google Scholar [29] V. Guruswami, A. Sahai and M. Sudan, "Soft-decision'' decoding of Chinese remainder codes,, in Proc. 41st Ann. Symp. Found. Comp. Sci., (2000), 159. doi: 10.1109/SFCS.2000.892076. Google Scholar [30] V. Guruswami and M. Sudan, Improved decoding of Reed-Solomon and algebraic-geometry codes,, IEEE Trans. Inf. Theory, 45 (1999), 1757. doi: 10.1109/18.782097. Google Scholar [31] N. Howgrave-Graham, Approximate integer common divisors,, in Cryptography and Lattices, (2001), 51. doi: 10.1007/3-540-44670-2_6. Google Scholar [32] M.-D. Huang and D. Ierardi, Efficient algorithms for the Riemann-Roch problem and for addition in the Jacobian of a curve,, J. Symb. Comput., 18 (1994), 519. doi: 10.1006/jsco.1994.1063. Google Scholar [33] T. Kailath, Linear Systems,, Prentice-Hall, (1980). Google Scholar [34] S. V. Konyagin and T. Steger, On polynomial congruences,, Math. Notes, 55 (1994), 596. doi: 10.1007/BF02110354. Google Scholar [35] A. K. Lenstra, Factoring polynomials over algebraic number fields,, in Computer Algebra, (1983), 245. doi: 10.1007/3-540-12868-9_108. Google Scholar [36] A. K. Lenstra, Factoring multivariate polynomials over algebraic number fields,, SIAM J. Comput., 16 (1987), 591. doi: 10.1137/0216040. Google Scholar [37] H. W. Lenstra, Algorithms in algebraic number theory,, Bull. Amer. Math. Soc., 26 (1992), 211. doi: 10.1090/S0273-0979-1992-00284-7. Google Scholar [38] A. K. Lenstra, H. W. Lenstra and L. Lovász, Factoring polynomials with rational coefficients,, Math. Ann., 261 (1982), 515. doi: 10.1007/BF01457454. Google Scholar [39] D. Lorenzini, An Invitation to Arithmetic Geometry,, Amer. Math. Soc., (1996). doi: 10.1090/gsm/009. Google Scholar [40] V. Lyubashevsky, C. Peikert and O. Regev, On ideal lattices and learning with errors over rings,, in Adv. Crypt. - EUROCRYPT 2010, (2010), 1. doi: 10.1007/978-3-642-13190-5_1. Google Scholar [41] K. Manders and L. Adleman, NP-complete decision problems for quadratic polynomials,, in Proc. 8th Ann. ACM Symp. Theory Comp., (1976), 23. doi: 10.1145/800113.803627. Google Scholar [42] R. C. Mason, Diophantine Equations over Function Fields,, Cambridge Univ. Press, (1984). doi: 10.1017/CBO9780511752490. Google Scholar [43] A. May, New RSA Vulnerabilities Using Lattice Reduction Methods,, Ph.D. thesis, (2003). Google Scholar [44] A. May, Using LLL-reduction for solving RSA and factorization problems,, in The LLL Algorithm, (2010), 315. doi: 10.1007/978-3-642-02295-1_10. Google Scholar [45] M. Naor and B. Pinkas, Oblivious transfer and polynomial evaluation,, in Proc. 31st Ann. ACM Symp. Theory Comp., (1999), 245. doi: 10.1145/301250.301312. Google Scholar [46] F. Parvaresh and A. Vardy, Correcting errors beyond the Guruswami-Sudan radius in polynomial time,, in Proc. 46th IEEE Symp. Found. Comp. Sci., (2005), 285. doi: 10.1109/SFCS.2005.29. Google Scholar [47] C. Peikert and A. Rosen, Lattices that admit logarithmic worst-case to average-case connection factors,, in Proc. 39th Ann. ACM Symp. Theory Comp., (2007), 478. doi: 10.1145/1250790.1250860. Google Scholar [48] M. Rosen, Number Theory in Function Fields,, Springer-Verlag, (2002). doi: 10.1007/978-1-4757-6046-0. Google Scholar [49] M. A. Shokrollahi and H. Wasserman, List decoding of algebraic-geometric codes,, IEEE Trans. Inf. Theory, 45 (1999), 432. doi: 10.1109/18.748993. Google Scholar [50] V. Shoup, OAEP reconsidered,, in Adv. Crypt. - CRYPTO 2001, (2001), 239. doi: 10.1007/3-540-44647-8_15. Google Scholar [51] H. Stichtenoth, Algebraic Function Fields and Codes,, 2nd edition, (2010). doi: 10.1007/978-3-540-76878-4. Google Scholar [52] M. Sudan, Ideal error-correcting codes: Unifying algebraic and number-theoretic algorithms, in Applied Algebra, (2001), 36. doi: 10.1007/3-540-45624-4_4. Google Scholar [53] V. Vassilevska Williams, Multiplying matrices faster than Coppersmith-Winograd,, in Proc. 44th ACM Symp. Theory Comp., (2012), 887. doi: 10.1145/2213977.2214056. Google Scholar

show all references

##### References:
 [1] M. Ajtai, The shortest vector problem in $L_2$ is NP-hard for randomized reductions,, in Proc. 30th Ann. ACM Symp. Theory Comput., (1998), 10. doi: 10.1145/276698.276705. Google Scholar [2] M. Ajtai, R. Kumar and D. Sivakumar, A sieve algorithm for the shortest lattice vector problem,, in Proc. 33rd Ann. ACM Symp. Theory Comput., (2001), 601. doi: 10.1145/380752.380857. Google Scholar [3] M. Alekhnovich, Linear Diophantine equations over polynomials and soft decoding of Reed-Solomon codes,, IEEE Trans. Inf. Theory, 51 (2005), 2257. doi: 10.1109/TIT.2005.850097. Google Scholar [4] P. Beelen and K. Brander, Efficient list decoding of a class of algebraic-geometry codes,, Adv. Math. Commun., 4 (2010), 485. doi: 10.3934/amc.2010.4.485. Google Scholar [5] P. Beelen and K. Brander, Key equations for list decoding of Reed-Solomon codes and how to solve them,, J. Symb. Comput., 45 (2010), 773. doi: 10.1016/j.jsc.2010.03.010. Google Scholar [6] D. J. Bernstein, Reducing lattice bases to find small-height values of univariate polynomials,, in Algorithmic Number Theory: Lattices, (2008), 421. Google Scholar [7] D. J. Bernstein, List decoding for binary Goppa codes,, in Coding and Cryptology, (2011), 62. doi: 10.1007/978-3-642-20901-7_4. Google Scholar [8] J.-F. Biasse and G. Quintin, An algorithm for list decoding number field codes,, in 2012 IEEE Int. Symp. Inf. Theory Proc., (2012), 91. doi: 10.1109/ISIT.2012.6284696. Google Scholar [9] D. Bleichenbacher and P. Q. Nguyen, Noisy polynomial interpolation and noisy Chinese remaindering,, in Adv. Crypt. - EUROCRYPT 2000, (2000), 53. doi: 10.1007/3-540-45539-6_4. Google Scholar [10] J. Blömer and A. May, New partial key exposure attacks on RSA,, in Adv. Crypt. - CRYPTO 2003, (2003), 27. doi: 10.1007/978-3-540-45146-4_2. Google Scholar [11] D. Boneh, Finding smooth integers in short intervals using CRT decoding,, in Proc. 32nd Ann. ACM Symp. Theory Comput., (2000), 265. doi: 10.1145/335305.335337. Google Scholar [12] D. Boneh, G. Durfee and Y. Frankel, An attack on RSA given a small fraction of the private key bits,, in Adv. Crypt. - ASIACRYPT '98, (1998), 25. doi: 10.1007/3-540-49649-1_3. Google Scholar [13] J. Buchmann, T. Takagi and U. Vollmer, Number field cryptography,, in High Primes and Misdemeanours, (2004), 111. Google Scholar [14] M. F. I. Chowdhury, C.-P. Jeannerod, V. Neiger, É. Schost and G. Villard, Faster algorithms for multivariate interpolation with multiplicities and simultaneous polynomial approxima-tions,, IEEE Trans. Inf. Theory, 61 (2015), 2370. doi: 10.1109/TIT.2015.2416068. Google Scholar [15] H. Cohen, A Course in Computational Algebraic Number Theory,, Springer-Verlag, (1993). doi: 10.1007/978-3-662-02945-9. Google Scholar [16] H. Cohn and N. Heninger, Approximate common divisors via lattices,, in Proc. 10th Algor. Number Theory Symp., (2013), 271. doi: 10.2140/obs.2013.1.271. Google Scholar [17] D. Coppersmith, Small solutions to polynomial equations, and low exponent RSA vulnerabilities,, J. Cryptology, 10 (1997), 233. doi: 10.1007/s001459900030. Google Scholar [18] D. Coppersmith, Finding small solutions to small degree polynomials,, in Cryptography and Lattices, (2001), 20. doi: 10.1007/3-540-44670-2_3. Google Scholar [19] D. Coppersmith, N. Howgrave-Graham and S. V. Nagaraj, Divisors in residue classes, constructively,, Math. Comp., 77 (2008), 531. doi: 10.1090/S0025-5718-07-02007-8. Google Scholar [20] N. Coxon, List decoding of number field codes,, Des. Codes Cryptogr., 72 (2014), 687. doi: 10.1007/s10623-013-9803-x. Google Scholar [21] C. Fieker and M. E. Pohst, On lattices over number fields,, in Algorithmic Number Theory, (1996), 133. doi: 10.1007/3-540-61581-4_48. Google Scholar [22] C. Fieker and D. Stehlé, Short bases of lattices over number fields,, in Algorithmic Number Theory, (2010), 157. doi: 10.1007/978-3-642-14518-6_15. Google Scholar [23] J. von zur Gathen, Hensel and Newton methods in valuation rings,, Math. Comp., 42 (1984), 637. doi: 10.2307/2007608. Google Scholar [24] J. von zur Gathen and J. Gerhard, Modern Computer Algebra, 2nd edition,, Cambridge Univ. Press, (2003). Google Scholar [25] J. von zur Gathen and E. Kaltofen, Factorization of multivariate polynomials over finite fields,, Math. Comp., 45 (1985), 251. doi: 10.2307/2008063. Google Scholar [26] P. Giorgi, C.-P. Jeannerod and G. Villard, On the complexity of polynomial matrix computations,, in Proc. 2003 Int. Symp. Symb. Algebr. Comput., (2003), 135. doi: 10.1145/860854.860889. Google Scholar [27] E. Grigorescu and C. Peikert, List decoding Barnes-Wall lattices,, in Proc. 27th IEEE Conf. Comput. Compl., (2012), 316. doi: 10.1109/CCC.2012.33. Google Scholar [28] V. Guruswami and A. Rudra, Explicit codes achieving list decoding capacity: error correction with optimal redundancy,, IEEE Trans. Inf. Theory, 54 (2008), 135. doi: 10.1109/TIT.2007.911222. Google Scholar [29] V. Guruswami, A. Sahai and M. Sudan, "Soft-decision'' decoding of Chinese remainder codes,, in Proc. 41st Ann. Symp. Found. Comp. Sci., (2000), 159. doi: 10.1109/SFCS.2000.892076. Google Scholar [30] V. Guruswami and M. Sudan, Improved decoding of Reed-Solomon and algebraic-geometry codes,, IEEE Trans. Inf. Theory, 45 (1999), 1757. doi: 10.1109/18.782097. Google Scholar [31] N. Howgrave-Graham, Approximate integer common divisors,, in Cryptography and Lattices, (2001), 51. doi: 10.1007/3-540-44670-2_6. Google Scholar [32] M.-D. Huang and D. Ierardi, Efficient algorithms for the Riemann-Roch problem and for addition in the Jacobian of a curve,, J. Symb. Comput., 18 (1994), 519. doi: 10.1006/jsco.1994.1063. Google Scholar [33] T. Kailath, Linear Systems,, Prentice-Hall, (1980). Google Scholar [34] S. V. Konyagin and T. Steger, On polynomial congruences,, Math. Notes, 55 (1994), 596. doi: 10.1007/BF02110354. Google Scholar [35] A. K. Lenstra, Factoring polynomials over algebraic number fields,, in Computer Algebra, (1983), 245. doi: 10.1007/3-540-12868-9_108. Google Scholar [36] A. K. Lenstra, Factoring multivariate polynomials over algebraic number fields,, SIAM J. Comput., 16 (1987), 591. doi: 10.1137/0216040. Google Scholar [37] H. W. Lenstra, Algorithms in algebraic number theory,, Bull. Amer. Math. Soc., 26 (1992), 211. doi: 10.1090/S0273-0979-1992-00284-7. Google Scholar [38] A. K. Lenstra, H. W. Lenstra and L. Lovász, Factoring polynomials with rational coefficients,, Math. Ann., 261 (1982), 515. doi: 10.1007/BF01457454. Google Scholar [39] D. Lorenzini, An Invitation to Arithmetic Geometry,, Amer. Math. Soc., (1996). doi: 10.1090/gsm/009. Google Scholar [40] V. Lyubashevsky, C. Peikert and O. Regev, On ideal lattices and learning with errors over rings,, in Adv. Crypt. - EUROCRYPT 2010, (2010), 1. doi: 10.1007/978-3-642-13190-5_1. Google Scholar [41] K. Manders and L. Adleman, NP-complete decision problems for quadratic polynomials,, in Proc. 8th Ann. ACM Symp. Theory Comp., (1976), 23. doi: 10.1145/800113.803627. Google Scholar [42] R. C. Mason, Diophantine Equations over Function Fields,, Cambridge Univ. Press, (1984). doi: 10.1017/CBO9780511752490. Google Scholar [43] A. May, New RSA Vulnerabilities Using Lattice Reduction Methods,, Ph.D. thesis, (2003). Google Scholar [44] A. May, Using LLL-reduction for solving RSA and factorization problems,, in The LLL Algorithm, (2010), 315. doi: 10.1007/978-3-642-02295-1_10. Google Scholar [45] M. Naor and B. Pinkas, Oblivious transfer and polynomial evaluation,, in Proc. 31st Ann. ACM Symp. Theory Comp., (1999), 245. doi: 10.1145/301250.301312. Google Scholar [46] F. Parvaresh and A. Vardy, Correcting errors beyond the Guruswami-Sudan radius in polynomial time,, in Proc. 46th IEEE Symp. Found. Comp. Sci., (2005), 285. doi: 10.1109/SFCS.2005.29. Google Scholar [47] C. Peikert and A. Rosen, Lattices that admit logarithmic worst-case to average-case connection factors,, in Proc. 39th Ann. ACM Symp. Theory Comp., (2007), 478. doi: 10.1145/1250790.1250860. Google Scholar [48] M. Rosen, Number Theory in Function Fields,, Springer-Verlag, (2002). doi: 10.1007/978-1-4757-6046-0. Google Scholar [49] M. A. Shokrollahi and H. Wasserman, List decoding of algebraic-geometric codes,, IEEE Trans. Inf. Theory, 45 (1999), 432. doi: 10.1109/18.748993. Google Scholar [50] V. Shoup, OAEP reconsidered,, in Adv. Crypt. - CRYPTO 2001, (2001), 239. doi: 10.1007/3-540-44647-8_15. Google Scholar [51] H. Stichtenoth, Algebraic Function Fields and Codes,, 2nd edition, (2010). doi: 10.1007/978-3-540-76878-4. Google Scholar [52] M. Sudan, Ideal error-correcting codes: Unifying algebraic and number-theoretic algorithms, in Applied Algebra, (2001), 36. doi: 10.1007/3-540-45624-4_4. Google Scholar [53] V. Vassilevska Williams, Multiplying matrices faster than Coppersmith-Winograd,, in Proc. 44th ACM Symp. Theory Comp., (2012), 887. doi: 10.1145/2213977.2214056. Google Scholar
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