O. Goldreich, D. Ron and M. Sudan. Chinese Remaindering with Errors. TR98-062, available from ECCC, at http://www.eccc.uni-trier.de/eccc/, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of such possible schemes is briefly outlined in [57] Even though this scheme is broken by the algorithm of [57] this idea, using a hidden algebraic number field, deserves more attention and further study. 11 Chinese Remaindering HNP CR HNP The following problem is motivated by the papers [6, 17, 23]. Let A 0 be a su#ciently large integer. CR HNP: Recover a number # [0, A] such that for k prime numbers p 1 , p k , chosen independently and uniformly at random from the interval [2 n 1 , 2 ] we are given k pairs (p i , MSB #,p i (#) i = 1, k, for some # 0. ....

O. Goldreich, D. Ron and M. Sudan, `Chinese remaindering with errors', IEEE Trans. Inform. Theory , 46 (2000), 1330--1338. 16

....cases, random vectors are generated uniformly in a Voronoi region of a lattice using closest point search. The closely related shortest vector problem has been used in assessing the quality of noncryptographic random number generators [50, pp. 89 113] and in decoding of Chinese remainder codes [38], 40] It also has important applications in cryptography [5] 7] Another related problem of paramount importance in cryptography [13] 70] is that of lattice basis reduction. These search problems will be discussed in Section VI. The choice of method for solving the closest point problem ....

O. Goldreich, D. Ron, and M. Sudan, "Chinese remaindering with errors, " IEEE Trans. Inform. Theory, vol. 46, pp. 1330--1338, July 2000.

....algorithms to decode even in the presence of a large number of errors [18, 5] Motivation behind our work. Another family of algebraic codes that have received some attention recently are number theoretic redundant residue codes called the Chinese Remainder codes (henceforth, CRT codes) [12, 3]. Here the messages are identi ed with integers with absolute value at most K (for some parameter K that governs the rate) and a message m is encoded by its residues modulo n primes p 1 p 2 p n . If K = p 1 p 2 p k and n k, this gives a redundant encoding of m and the ....

....di erent alphabets) has distance n k 1. The distance claim is a simple consequence of the Chinese Remainder Theorem, hence the name for these codes. In light of the progress in decoding algorithms for Reed Solomon and algebraic geometric codes, there has also been progress on decoding CRT codes [3, 1, 6] in the presence of very high noise, and the performance of the best known algorithm matches the number of errors correctable for Reed Solomon codes. Since Reed Solomon codes are a speci c example of the more general family of AG codes, it is natural to ask if CRT codes can also be realized as ....

[Article contains additional citation context not shown here]

O. Goldreich, D. Ron and M. Sudan. Chinese Remaindering with errors. IEEE Trans. on Information Theory, to appear. Preliminary version appeared in Proc. of the 31st Annual ACM Symposium on Theory of Computing, Atlanta, Georgia, May 1999, pp. 225-234.

....decoding algorithms to decode even in the presence of a large number of errors [18, 5] Motivation behind our work. Another family of algebraic codes that have received some study are number theoretic redundant residue codes called the Chinese Remainder codes (henceforth called CRT codes) [12, 3]. Here the messages are identified with integers with absolute value at most K (for some parameter K that governs the rate) and a message m is encoded by its residues modulo n primes p 1 p 2 p n . If K = p 1 p 2 p k and n k, this gives a redundant encoding of m and the ....

.... code (which is different from usual codes in that symbols in different codeword positions are over different alphabets) has distance n k 1. In light of the progress in decoding algorithms for Reed Solomon and algebraic geometric codes, there has also been progress on decoding CRT codes [3, 1, 6] in the presence of very high noise, and the performance of the best known algorithm matches the number of errors correctable for Reed Solomon codes [5] There is quite a bit of similarity between Reed Solomon and CRT codes: both are redundant residue codes that are MDS (see [10] Since ....

[Article contains additional citation context not shown here]

O. Goldreich, D. Ron and M. Sudan. Chinese Remaindering with errors. IEEE Trans. on Information Theory, to appear. Preliminary version appeared in Proc. of the 31st Annual ACM Symposium on Theory of Computing, Atlanta, Georgia, May 1999, pp. 225-234.

....In both cases, random vectors are generated uniformly inside a Voronoi region of a lattice using closest point search. The closely related shortest vector problem has been used in assessing the quality of random number generators [30, pp. 89 113] and in the decoding of Chinese remainder codes [23]. It also has important applications in cryptography [4,6] Another related problem of paramount importance in cryptography [11, 43] is that of lattice basis reduction. These search problems will be discussed in Section VI. The choice of method for solving the closest point problem depends on the ....

O. Goldreich, D. Ron, and M. Sudan, "Chinese remaindering with errors," IEEE Trans. Inform. Theory, vol. 46, pp. 1330-- 1338, July 2000.

....that dist(C(a) C(b) is greater than or equal to d. Given an [n; k; d] q error correcting code C and a word c 2 Sigma n , consider the problem of finding all codewords in C that are close to c. This problem is a generalization of the standard decoding problem, and is denoted as list decoding ([14, 8, 9]) We say that an [n; k; d] q error correcting code C is list decodeable, if there is an efficient procedure that given a word c 2 Sigma n produces a short list containing all words a 2 Sigma k for which C(a) is close to c. Definition 2 (List decodeable ECC) Let Sigma be an alphabet of ....

O. Goldreich, D. Ron and M. Sudan. Chinese Remaindering with Errors. Proc. 31st Symp. Theory of Computing, pp. 225-234, 1999.

.... corresponds to the integer size; Lagrange s interpolation corresponds to Chinese remainders; and polynomial evaluation corresponds to the modulo operation (in fact, a polynomial P evaluated at x 0 can also be viewed as the remainder of P (x) modulo the linear polynomial x Gamma x 0 ) We refer to [15] for some examples. The noisy polynomial interpolation and polynomial reconstruction problems then become the following ones: Problem 9 (Noisy Chinese remaindering) Let 0 N B, and p 1 ; pn be coprime integers. Given n sets S 1 ; Sn where each S i = fr i;j g 1jm contains m ....

....noisy polynomial interpolation. 13 Problem 10 (Chinese remaindering with errors) Given as input integers t, B and n points (r 1 ; p 1 ) r n ; pn ) 2 N 2 where the p i s are coprime, output all numbers 0 N B such that N j r i (mod p i ) for at least t values of i. We refer to [15] for a history of the latter problem, which is beyond the scope of this article. We will only mention that the best decoding algorithm known for the problem is the recent lattice based work of Boneh [6] which improves previous work of Goldreich et al. 15] The algorithm works in polynomial time ....

[Article contains additional citation context not shown here]

O. Goldreich, D. Ron, and M. Sudan. Chinese remaindering with errors. In Proc. of 31st STOC. ACM, 1999. Also available at [11]. 3 holding for more or all choices of the parameters. 17

....a concatenated code. However, such results will not obtain the tight result described above. The tight result above is also due to Guruswami and Sudan [18] 6. A list decoder correcting n Gamma q 2kn log pn log p 1 errors for the Chinese remainder codes was given by Goldreich, Ron, and Sudan [14]. Boneh [7] recently improved this bound to correct n Gamma q kn log pn log p 1 errors. Even more recently Guruswami, Sahai, and Sudan [16] improve this to correct n Gamma p nk errors. 4.2 List decoding results: Implicit version For the implicit list decoding problem, some fairly strong ....

Oded Goldreich, Dana Ron, and Madhu Sudan. Chinese remaindering with errors. Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pages 225--234, Atlanta, Georgia, 1-4 May 1999.

....value of the permanent. By taking one entry from such a list at random, we obtain a probabilistic polynomial time algorithm with an inverse polynomial success ratio. In a related development (after this paper was submitted to STACS) Goldreich, Ron, and Sudan published a technical report in ECCC [GRS98] showing that if there is a polynomial time algorithm B that is able to guess the permanent of a random n Theta n matrix on 2n bit integers modulo a random n bit prime with inverse polynomial success probability, then P #P = BPP. To prove this result, they develop algorithmic tools to decode an ....

O. Goldreich and D. Ron and M. Sudan. Chinese remaindering with errors. ECCC Technical Report TR 98-062, October 29, 1998. Available at www.eccc.uni-trier.de.

No context found.

O. Goldreich, D. Ron and M. Sudan. Chinese Remaindering with Errors. TR98-062, available from ECCC, at http://www.eccc.uni-trier.de/eccc/, 1998.

No context found.

O. Goldreich, D. Ron and M. Sudan. Chinese Remaindering with Errors. TR98-062, available from ECCC, at http://www.eccc.uni-trier.de/eccc/, 1998.

....h # we may not be able to uniquely determine a from y, but we may still be able to find a small list containing all possible candidates for a in that is, a list of all a #am y# h this is called a List Decoding Problem. Recently, there has been a series of interesting results (see [3, 6, 9, 27]) on e#cient algorithms for list decoding of the Chinese Remainder Code and other codes as well. However, these results assume a noise vector e bounded in the Hamming weight norm, that is, e is assumed to have zero coordinates except for less than h non zero ones, which are allowed to be ....

....conditions under which the algorithm successfully recovers an interval I of width at most 2h consisting of all solutions to our decoding problem. We remark that our algorithm for the Lee norm uses lattice reduction techniques more directly than the list decoding algorithm for the Hamming norm [6], which first transforms the problem to an algebraic interpolation problem and then applies lattice techniques to the algebraic problem. Moreover, in these list decoders, the number of solutions (size of the list) is small, that is, bounded by a polynomial in the length of the input, so the ....

O. Goldreich, D. Ron, and M. Sudan, `Chinese remaindering with errors', IEEE Transactions on Information Theory , 46 (July 2000), 1330--1338.

.... paradigms for correcting algebraic codes: The unique decoding algorithm abstracts from a large collection of (unique) error correcting algorithms for algebraic codes [30, 4, 27, 42, 5] In fact, an elegant unification of these results (see [29, 18, 8] or the technical report version of this paper [11]) provides the inspiration for our algorithm. The list decoding algorithm abstracts from the recent works on list decoding algorithms for algebraic codes [1, 37, 33, 13] We stress however, that the translation of the above mentioned algorithms to our case is not immediate. In particular, the ....

....Remainder code are of use. In Section 5, we describe the applicability of the Chinese Remainder code and the decoding algorithm in secret sharing as an alternate to Shamir s scheme [32] The latter used the properties of the Reed Solomon code) In the technical report version of this paper [11], an application to the average case complexity of computing the permanent of random matrices is also given. One more case where the list decoding algorithm for the CRT code has been utilized is in a result of Hastad and Naslund [14] who utilize the list decoding algorithm for the CRT code (as ....

[Article contains additional citation context not shown here]

O. Goldreich, D. Ron and M. Sudan. Chinese Remaindering with Errors. TR98-062, available from ECCC, at http://www.eccc.uni-trier.de/eccc/, 1998.

....of a concatenated code. However, such results will not obtain the tight result described above. The tight result above is due to Guruswami and Sudan [18] 5. A list decoder correcting n Gamma q 2kn log pn log p1 errors for the Chinese remainder codes was given by Goldreich, Ron, and Sudan [14]. Boneh [7] recently improved this bound to correct n Gamma q kn log pn log p1 errors. Even more recently Guruswami, Sahai, and Sudan [16] improve this to correct n Gamma p nk errors. 4.2 List decoding results: Implicit version For the implicit list decoding problem, some fairly strong ....

Oded Goldreich, Dana Ron, and Madhu Sudan. Chinese remaindering with errors. Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pages 225--234, Atlanta, Georgia, 1-4 May 1999.

....underlying the error correcting algorithms and unveiling an ideal theoretic view of decoding. When all weights are equal to 1, we obtain the more commonly studied list decoding problem. List decoding algorithms for the Chinese Remainder Code were given recently by Goldreich, Ron, and Sudan [5], and improved by Boneh [1] Their algorithms work for t p 2knlog pn =log p 1 and t p knlog pn =log p 1 , respectively. We improve upon the algorithms above by using our soft decision decoding algorithm with a non trivial choice of weights, and solve the list decoding problem provided t p ....

....in the literature in coding theory (see [15, 9] and the references there in) and its redundancy property has been exploited often in theoretical computer science as well. Mandelbaum gave a decoding algorithm for this code, correcting n Gammak 2 errors. 1 Recently, Goldreich, Ron, and Sudan [5] gave a list decoding algorithm for this code. Formally, the list decoding problem has as its inputs a vector hp 1 ; pn i, an integer k (specifying the CRT code) a vector hr 1 ; r n i and an agreement parameter t. The goal is to find a list of all messages m 2 M such that r i ....

[Article contains additional citation context not shown here]

O. Goldreich, D. Ron and M. Sudan. Chinese Remaindering with errors. IEEE Trans. on Information Theory, to appear. Preliminary version appeared in Proc. of 31st STOC, 1999, pp. 225-234.

No context found.

O. Goldreich, D. Ron and M. Sudan. Chinese Remaindering with Errors. Available from ECCC, 1998.

....underlying the error correcting algorithms and unveiling an ideal theoretic view of decoding. When all weights are equal to 1, we obtain the more commonly studied list decoding problem. List decoding algorithms for the Chinese Remainder Code were given recently by Goldreich, Ron, and Sudan [5], and improved by Boneh [1] Their algorithms work for t p 2knlog pn =log p 1 and t p knlog pn =log p 1 , respectively. We improve upon the algorithms above by using our soft decision decoding algorithm with a non trivial choice of weights, and solve the list decoding problem provided t ....

....now in the literature in coding theory (see [15, 9] and the references there in) and its redundancy property has been exploited often in theoretical computer science as well. Mandelbaum gave a decoding algorithm for this code, correcting n k 2 errors. 1 Recently, Goldreich, Ron, and Sudan [5] gave a list decoding algorithm for this code. Formally, the list decoding problem has as its inputs a vector hp 1 ; pn i, an integer k (specifying the CRT code) a vector hr 1 ; r n i and an agreement parameter t. The goal is to find a list of all messages m 2 M such that r i ....

[Article contains additional citation context not shown here]

O. Goldreich, D. Ron and M. Sudan. Chinese Remaindering with errors. IEEE Trans. on Information Theory, to appear. Preliminary version appeared in Proc. of 31st STOC, 1999, pp. 225-234.

....underlying the errorcorrecting algorithms and unveiling an ideal theoretic view of decoding. When all weights are equal to 1, we obtain the more commonly studied list decoding problem. List decoding algorithms for the Chinese Remainder Code were given recently by Goldreich, Ron, and Sudan [5], and improved by Boneh [1] Their algorithms work for t q 2kn log pn log p1 and t q kn log pn log p1 , respectively. We improve upon the algorithms above, by using our soft decision decoding algorithm with a non trivial choice of weights, and solve the list decoding problem provided ....

....in the literature in coding theory (see [14, 9] and the references there in) and its redundancy property has been exploited often in theoretical computer science as well. Mandelbaum gave a decoding algorithm for this code, correcting n Gammak 2 errors. 1 Recently, Goldreich, Ron, and Sudan [5] gave a list decoding algorithm for this code. Formally, the list decoding problem has as its inputs a vector hp 1 ; p n i, an integer k (specifying the CRT code) a vector hr 1 ; r n i and an agreement parameter t. The goal is to find a list of all messages m 2 M such that r i = ....

[Article contains additional citation context not shown here]

O. Goldreich, D. Ron and M. Sudan. Chinese Remaindering with errors. IEEE Trans. on Information Theory, to appear. Preliminary version appeared in Proc. of 31st STOC, 1999, pp. 225-234.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC