Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials with queries: The highly noisy case. SIAM Journal on Discrete Mathematics, 13(4):535--570, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the algorithm of [53] requires . We also mention that the Poly HNP is related to several other polynomial interpolation and approximation problems. In particular, it has links with noisy polynomial interpolation [5, 24, 52, 61] learning and reconstructing polynomials with noisy queries [2, 3, 18, 30] and sparse polynomial interpolation [13, 22, 33, 62, 63] 11 10 HNP over Unknown Algebraic Number Fields ANF HNP A HNP of a di#erent spirit has recently been introduced and studied in [57] To describe this problem we need to recall some basic facts of the theory of algebraic number ....

O. Goldreich, R. Rubinfeld and M. Sudan, `Learning polynomials with queries: the highly noisy case', Electronic Colloq. on Comp. Compl., Univ. of Trier, TR

....(in a more general form) by Boneh [3] This algorithm explicitly lists all solutions to our decoding problem in polynomialtime whenever 2h n k, where k = log K log p min , and p min = min 1#i#n p i . We also remark that the result of [25] is a Lee norm analogue of Hamming norm results of [2, 7, 10, 11, 16, 19, 22, 24, 25, 26, 27] on noisy polynomial reconstruction problem and algebraic geometry codes list decoding. Finally, several possible cryptographic applications of noisy polynomial reconstruction have been outlined in [14, 15] It would be interesting to study possible cryptographic applications of noisy Chinese ....

O. Goldreich, R. Rubinfeld and M. Sudan, `Learning polynomials with queries: the highly noisy case', Electronic Colloq. on Comp. Compl., Univ. of Trier, TR

....half the distance of the code there may be more than one possible decoding. The problem of nding all possible decodings in such cases is called the list decoding problem. For Reed Solomon codes the number of possible decodings is constant as long as the number of errors is less than N NK (see [GRS95]) Using techniques introduced by Sudan [Su97] and subsequently improved in [RR00] and [GS99] one can produce this list with a randomized polynomial time algorithm. In its most ecient form [GS99] the algorithm can handle up to dN NK 1e errors. We could use this list decoding algorithm for our ....

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: The highly noisy case. In Proc. 36th IEEE Symp. on Foundations of Comp. Science, pages 294-303. IEEE, 1995.

.... with ecient algorithms (the list decoding algorithm for Reed Solomon codes due to Guruswami and Sudan [10] is a case in point) The Johnson bound was originally proved by Johnson [15] in the context of constant weight codes, and proofs of it in the context of list decoding appear, for example, in [3, 4, 12]. Let us focus on the case of binary codes. Here, the Johnson bound states that in any binary code of block length n and relative distance , every Hamming ball of radius E = J( n, where J( 1 2) has at most n codewords (for a slightly smaller radius, the bound on the number of codewords ....

.... radius in terms of the minimum distance For general nonlinear codes, this was known to be the case in fact a simple random coding argument shows that there exist codes of relative distance which have exponentially many codewords in a Hamming ball of (relative) radius , for every J( [4]. However, the situation for the case of linear codes has remained open. Since the most widely studied and used codes such as Reed Solomon codes, algebraic geometric codes and low density parity check codes, are all linear, it is an important task to understand whether even with the restriction of ....

[Article contains additional citation context not shown here]

Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pages 294-303, 1995.

....half the distance of the code there may be more than one possible decoding. The problem of nding all possible decodings in such cases is called the list decoding problem. For Reed Solomon codes the number of possible decodings is constant as long as the number of errors is less than N p NK (see [GRS95]) Using techniques introduced by Sudan [Su97] and subsequently improved in [RR00] and [GS99] one can produce this list with a randomized polynomial time algorithm. In its most ecient form [GS99] the algorithm can handle up to dN p NK 1e errors. We could use this list decoding algorithm for ....

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: The highly noisy case. In Proc. 36th IEEE Symp. on Foundations of Comp. Science, pages 294-303. IEEE, 1995.

....f belongs to a certain large class of polynomials, including the polynomials of small degree. Knowing approximations to f(t) at polynomially many (in terms of log p) random points t # IF p , recover f . For another variant of polynomial interpolation problem and several relevant references see [1, 14]. The mentioned algorithm is e#cient only for su#ciently large p. In Section 2.1 we consider a di#erent situation when f(X) is a polynomial over a field IF q m of q m elements and the exact values of the trace Tr IF q m IF q (f(t) are given at polynomially many random t # IF q m . Our ....

O. Goldreich and R. Rubinfeld and M. Sudan, 'Learning polynomials with queries: the highly noisy case', Proc. of the 36th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1995, 294-- 303.

....as well as a theoretical platform. The paper is organized as follows. In Section 2 we give the preliminaries on the standard model that is used for cryptanalysis and reformulate this into a polynomial reconstruction problem. In Section 3 we review an algorithm by Goldreich, Rubinfeld and Sudan [7] that solves the polynomial reconstruction problem with queries in polynomial time. In Section 4 we derive a new algorithm for fast correlation attacks, inspired by the previous section. In Section 5 we present a sequential version of the new algorithm, i.e, this algorithm builds a tree of ....

....vector z consists of a number of noisy observations of an unknown polynomial U(x) evaluated in di erent known points fx 1 ; x 2 ; xN g. The task of the attacker is to determine the unknown polynomial U(x) 3 Learning polynomials with queries In computational learning theory (see e.g. [7] and its references) one might want to consider the following general reconstruction problem: Given: An oracle (black box) for an arbitrary unknown function f : F l F , a class of functions F and a parameter . Problem: Provide a list of all functions g 2 F that agree with f on at least a ....

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O. Goldreich, R. Rubinfeld, M. Sudan, Learning polynomials with queries: The highly noisy case, 36th Annual Symposium on Foundation of Computer Science, Milwaukee, Wisconsin, 23-25 October 1995, pp. 294303.

....m ; and thus (by Theorem 24) Corr M z;p(z) computes p on every input. Repeating the loop O(log 1 ) times ensures that every polynomial p with agreement with f is included in the output with high probability, using the well known bound that there are only O(1= such polynomials (cf. GRS98, Theorem 17] Acknowledgments We thank Oded Goldreich, Amnon Ta Shma, and Avi Wigderson for clarifying discussions and pointing us to some related work. ....

Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials with queries --- the highly noisy case. Technical Report TR98-060, Electronic Colloquium on Computational Complexity, 1998. Preliminary version in FOCS `95.

....size, thus triviliazing to full replication. 1 In case the number of errors is larger than half the code distance a ReedSolomon code can be used to recover a list of all possible decodings. The number of possible decodings is constant as long as the number of errors is less than N p NK (see [GRS95]) and the problem of nding the list is known as the list decoding problem. Using techniques introduced by Sudan [Su97] and subsequently improved in [RR00] and [GS99] such a list is produced with a randomized polynomial time algorithm. In its most ecient form [GS99] their scheme corrects up to ....

O. Goldreich, R. Rubinfeld, and M. Sudan. Learning polynomials with queries: The highly noisy case. In Proc. 36th IEEE Symp. on Foundations of Comp. Science, pages 294-303. IEEE, 1995.

....n, there exists a (non linear) n; k) q code C of distance at least d and a received word r, such that there are exponentially many codewords (with the exponent growing with ffln) within a Hamming distance of e from r. Note: The theorem above appears explicitly in Goldreich, Rubinfeld, and Sudan [13]. The crucial direction, Part (1) above, is a q ary extension of the Johnson bound in coding theory. Johnson proves this bound only for the binary case, but the extension to the q ary case seems to be implicitly known to the coding theory community [27, Chapter 4, page 301] Theorem 1 yields an ....

....of a code. 6 and reconstructs explicitly a list of all messages (note that message lengths are O(log n) for the Hadamard code) that come within this error of the received word. Subsequently, this algorithm was generalized to general q ary Hadamard codes by Goldreich, Rubinfeld, and Sudan [13]. This yields the following theorem. Theorem 3 There exists a probabilistic list decoding algorithm in the implicit input model for Hadamard codes that behaves as follows: For an (n; k) q code C, given oracle access to a received word r, the algorithm outputs a list that includes all messages ....

[Article contains additional citation context not shown here]

Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials with queries--- the highly noisy case. Technical Report TR98-060, Electronic Colloquium on Computational Complexity, 1998. Preliminary version in FOCS '95.

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O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th Annual Symposium on Foundations of Computer Science, pages 294--303, Milwaukee, Wisconsin, 23-25 October 1995.

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Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials with queries: The highly noisy case. SIAM Journal on Discrete Mathematics, 13(4):535-570, November 2000.

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O. GOLDREICH, R. RUBINFELD AND M. SUDAN. Learning polynomials with queries: The highly noisy case. FOCS, 1995.

....17, 18] Thus our algorithm ends up solving the curve fitting problem over fairly general fields. It is interesting to contrast our algorithm with results which show bounds on the number of codewords that may exist with a distance of from a received word. One such result, due to Goldreich et al. [13], shows that the number of solutions to the list decoding problem for a code with block length and minimum , is bounded by a polynomial in as long as . A similar result has also been shown by Radhakrishnan [22] Our algorithm proves this best known combinatorial bound constructively in ....

....solves the polynomial reconstruction problem provided . Proof: Follows from Lemmas 5, 6 and 7. We can also infer an upper bound on the number of codewords within radius 7E( H in a Generalized Reed Solomon code. This bound is already known even for general (even non linear codes) [13, 22]. Our result can be viewed as a constructive proof of this bound for the specific case of Generalized ReedSolomon codes. Proposition 9 The number of codewords that lie within an Hamming ball of radius 7 ( in an Generalized Reed Solomon code is STL (which is in ....

O. GOLDREICH, R. RUBINFELD AND M. SUDAN. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pages 294--303, 1995.

....we do not elaborate on the improved bound on list decoding radius that takes into account the alphabet size. highly ecient randomized list decoding algorithm for Hadamard codes, when the received vector was given implicitly. This work led to some extensions by Goldreich, Rubinfeld, and Sudan [16]. Yet no ecient list decoding algorithms were found for codes of decent rate (constant, or even slowly vanishing rate such as R(n) n 1 for some 0) The rst list decoding algorithm correcting (n) errors for 2 for codes of constant rate was due to Sudan [38] who gave such an ....

Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials with queries: The highly noisy case. SIAM Journal on Discrete Mathematics, 13(4):535-570, November 2000.

....There are a number of questions on decoding Reed Solomon codes that remain open. For instance, is there an algorithm that can decode from more errors (than (d Gamma 1) 2) when = k=n 1=3 A nice target would be a decoding algorithm that works for ffl( 1 Gamma . In this case we know (cf. [5, 10]) that the number of codewords within a distance of ffl( n is bounded by a polynomial in n. One does expect that the problem will become harder as ffl( 1 Gamma . It would be interesting to see if the problem becomes NP hard as ffl( 1 Gamma . Acknowledgments I am grateful to Dave Forney, ....

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pp. 294--303, 1995.

No context found.

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pp. 294#303, 1995.

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O. Goldreich, R. Rubinfeld, M. Sudan, \Learning polynomials with queries: the highly noisy case", in proc. FOCS 95, pp. 294-303, 1995.

....)n errors. The motivation for this particular version is that in order to solve the bounded distance decoding problem, one needs to ensure that the number of outputs (i.e. the number codewords within the given bound t) is polynomial in n. Such a bound does exist for the value of t as given above [6, 12], thus raising the hope that this problem may be solvable in polynomial time also. Similar questions may also be raised about decoding multivariate polynomials. In particular, we don t have polynomial time algorithms matching the bounded distance decoding algorithm from [16] even for the case of ....

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pp. 294--303, 1995.

....Distance of the code Next we move on to lower bounds on the list decoding radius. As mentioned earlier, it makes sense to study this as a function of the minimum distance of the code. A large minimum distance implies a large list decoding radius by existing combinatorial bounds (see for example [5]) and we want to find the smallest possible list decoding radius for a code of (at least) a certain minimum distance. This motivates the next definition. Definition 6 (Lower bound on list decoding radius) For a distance 0 1, and list size : Z Z , the lower bound on list of ....

....values are, however, mostly unknown. The main motivation for our work is the following conjecture. Conjecture 8 For every 0 1=2, L ( L ( 1 1 2) Evidence in support of the conjecture comes piecemeal. Firstly, it is known that L ( L (see for example [5]) Upper bounds on L are not as well studied. Justesen and Hoholdt [9] demonstrate some MDS code families C of distance with Rad(C; c) 1 ) but this does not apply for codes over any fixed size alphabet, and in particular for binary codes. The quantity L ( is even less well ....

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O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: the highly noisy case. Proc. of FOCS 95.

....Distance of the code Next we move on to lower bounds on the list decoding radius. As mentioned earlier, it makes sense to study this as a function of the minimum distance of the code. A large minimum distance implies a large list decoding radius by existing combinatorial bounds (see for example [9]) and we want to find the smallest possible list decoding radius for a code of (at least) a certain minimum distance. This motivates the next definition. Definition 7 (Lower bound on list decoding radius) For a distance 0 1, and list size : Z , the lower bound on list of decoding ....

....mostly unknown. The main motivation for our work is the following conjecture. Conjecture 9: For every 0 1=2, L 2 1 Evidence in support of the conjecture comes piecemeal. Firstly, it is known that ( L 1 ( and c ( 1 2 2=c (see, for example, [9], 14] for a proof of these facts) Upper bounds on L are not as well studied. Justesen and Hholdt [16] demonstrate some MDS code families C of distance with Rad(C; c) 1 1 ) for every constant c for certain values of , but this does not apply for codes over any fixed size alphabet, and ....

[Article contains additional citation context not shown here]

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: the highly noisy case. SIAM Journal on Discrete Mathematics, 13(4):535--570, November 2000.

....errors. The motivation for this particular version is that in order to solve the bounded distance decoding problem, one needs to ensure that the number of outputs (i.e. the number codewords within the given bound t) is polynomial in n. Such a bound does exist for the value of t as given above [6, 12], thus raising the hope that this problem may be solvable in polynomial time also. Similar questions may also be raised about decoding multivariate polynomials. In particular, we don t have polynomial time algorithms matching the bounded distance decoding algorithm from [16] even for the case of ....

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pp. 294-303, 1995.

....to S. Johnson (see [17] page 525) bounds the number of solutions by n(n ) 2 2 (2n) n(n d) for binary codes (i.e. codes over the alphabet f0; 1g) provided the denominator is positive. For general codes, a simple bound can be shown by an inclusion exclusion argument (see, for instance, [11]) which yields that the number of solutions to the reconstruction problem is at most 2= if = 1 ) provided n q 2n(n ) Another such bound is also known due to Goldreich, Rubinfeld and Sudan [11] We do not describe this here. However the inclusion exclusion bound is not ....

.... codes, a simple bound can be shown by an inclusion exclusion argument (see, for instance, 11] which yields that the number of solutions to the reconstruction problem is at most 2= if = 1 ) provided n q 2n(n ) Another such bound is also known due to Goldreich, Rubinfeld and Sudan [11]. We do not describe this here. However the inclusion exclusion bound is not constructive, i.e. it does not provide a list of the 2= codewords which may be the solution to the reconstruction problem. It is reasonable to ask for a solution to the reconstruction problem which runs in polynomial ....

[Article contains additional citation context not shown here]

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: The highly noisy case. Full version in preparation. Extended abstract appears in Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pp. 294-303, 1995.

....Distance of the code Next we move on to lower bounds on the list decoding radius. As mentioned earlier, it makes sense to study this as a function of the minimum distance of the code. A large minimum distance implies a large list decoding radius by existing combinatorial bounds (see for example [9]) and we want to find the smallest possible list decoding radius for a code of (at least) a certain minimum distance. This motivates the next definition. Definition 7 (Lower bound on list decoding radius) For a distance 0 1, and list size : Z Z , the lower bound on list of ....

....9: For every 0 1=2, L const ( L poly ( 1 2 1 p 1 2 . Evidence in support of the conjecture comes piecemeal. Firstly, it is known that L poly ( L poly 1 ( 1 2 1 p 1 2 and L const c ( 1 2 1 p 1 2 2=c (see, for example, [9], 14] for a proof of these facts) Upper bounds on L poly and L const are not as well studied. Justesen and Hholdt [16] demonstrate some MDS code families C of distance with Rad(C; c) 1 p 1 ) for every constant c for certain values of , but this does not apply for codes over any fixed ....

[Article contains additional citation context not shown here]

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: the highly noisy case. SIAM Journal on Discrete Mathematics, 13(4):535--570, November 2000.

....unique code word that differs in at most e coordinates from the received word hr 1 ; r n i. However, as long as e n Gamma p nk, there exists a small list containing all integers whose Chinese remainder representations differ from the vector hr 1 ; r n i in at most e coordinates [12]. In this paper we present an efficient algorithm for recovering this list in polynomial time. More precisely, the algorithm solves the following task: List Decoding (for large error) Given n relatively prime integers p 1 Delta Delta Delta p n ; n residues r 1 ; r n , with 0 r i ....

....a list of all integers m satisfying m Q k i=1 p i and (m mod p i ) r i for at least q 2n(k 2) log pn log p 1 k 3 2 2 log n = Theta( q nk log pn log p 1 ) values of i 2 f1; ng. Theorem 11. We comment that this list contains at most p 2n=k integers; cf. [12]. Coding theory context. The better known examples of asymptotically good error correcting codes with efficient algorithms can be classified in one of two categories: 1. Algebraic codes: These are codes defined using the properties of low degree polynomials over finite fields and include a wide ....

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th Annual Symposium on Foundations of Computer Science, pages 294--303, Milwaukee, Wisconsin, 23-25 October

....with these bits can be efficiently recovered. Recovering from errors and erasures: We first establish a combinatorial result proving an upper bound on the list size possible when decoding from a certain number of errors and erasures. Our result is analogous to the Johnson bound (see also [12]) and reduces to the one in [12] in presence of only errors. This places limits on the radius to which one can (currently) hope to list decode in polynomial time, and restricts qe= q Gamma 1) s, where e; s are the number of errors and erasures respectively, to be at most some quantity that is ....

....recovered. Recovering from errors and erasures: We first establish a combinatorial result proving an upper bound on the list size possible when decoding from a certain number of errors and erasures. Our result is analogous to the Johnson bound (see also [12] and reduces to the one in [12] in presence of only errors. This places limits on the radius to which one can (currently) hope to list decode in polynomial time, and restricts qe= q Gamma 1) s, where e; s are the number of errors and erasures respectively, to be at most some quantity that is a function of the minimum ....

[Article contains additional citation context not shown here]

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pp. 294-303, 1995.

....17, 18] Thus our algorithm ends up solving the curve fitting problem over fairly general fields. It is interesting to contrast our algorithm with results which show bounds on the number of codewords that may exist with a distance of e from a received word. One such result, due to Goldreich et al. [13], shows that the number of solutions to the list decoding problem for a code with block length n and minimum distance d, is bounded by a polynomial in n as long as e n Gamma p n(n Gamma d) A similar result has also been shown by Radhakrishnan [22] Our algorithm proves this best known ....

....the polynomial reconstruction problem provided t p kn. Proof: Follows from Lemmas 5, 6 and 7. We can also infer an upper bound on the number of codewords within radius e n Gamma p kn in a Generalized Reed Solomon code. This bound is already known even for general (even non linear codes) [13, 22]. Our result can be viewed as a constructive proof of this bound for the specific case of Generalized ReedSolomon codes. Proposition 9 The number of codewords that lie within an Hamming ball of radius e n Gamma p kn in an [n; k 1; d] q Generalized Reed Solomon code is O( p kn 3 ) which ....

O. GOLDREICH, R. RUBINFELD AND M. SUDAN. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pages 294--303, 1995.

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O. GOLDREICH, R. RUBINFELD AND M. SUDAN. Learning polynomials with queries: The highly noisy case. IEEE FOCS 1995

....on every input with high probability) such that every polynomial that has 1 agreement with P is computed by one of the programs. This task was left as an open problem in Ar et al. ALRS92] and no polynomial (in m, d and 1 ) time algorithm was known for this problem. Goldreich et al. GRS95] solve this task when 2 p d=q in time exponential in d. We now describe our solution that works when (md=q) for some positive , and is the rst polynomial time bounded solution for any 1=2. Given a program P , our algorithm works in two phases: First, a preprocessing phase, ....

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: The highly noisy case. SIAM Journal on Discrete Mathematics, 13(4): 535-570, November 2000.

....Distance of the code Next we move on to lower bounds on the list decoding radius. As mentioned earlier, it makes sense to study this as a function of the minimum distance of the code. A large minimum distance implies a large list decoding radius by existing combinatorial bounds (see for example [5]) and we want to find the smallest possible list decoding radius for a code of (at least) a certain minimum distance. This motivates the next definition. Definition 6 (Lower bound on list decoding radius) For a distance 0 ffi 1, and list size : Z Z , the lower bound on list of ....

....0 ffi 1=2, L const (ffi) L poly (ffi) 1 2 Delta (1 Gamma p 1 Gamma 2ffi) Evidence in support of the conjecture comes piecemeal. Firstly, it is known that L poly (ffi) L const (ffi) L const 2 (ffi) 1 2 Delta i 1 Gamma p 1 Gamma 2ffi j (see for example [5]) Upper bounds on L poly and L const are not as well studied. Justesen and Hoholdt [9] demonstrate some MDS code families C of distance ffi with Rad(C; c) 1 Gamma p 1 Gamma ffi ) but this does not apply for codes over any fixed size alphabet, and in particular for binary codes. The ....

[Article contains additional citation context not shown here]

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: the highly noisy case. Proc. of FOCS 95.

....encoding Had(z) is hw; zi = P t j=1 w j z j (mod 2) where z j ; w j 2 f0; 1g are the coordinates of w and z. Though the Hadamard code is inefficient with respect to the length of codewords, it does have good list decoding properties. Specifically, we use the following well known bound (cf. GRS98, Thm. 18] Lemma 27 For every k, Had : f0; 1g k f0; 1g 2 k is a ( 1= 4 2 ) list decodable code for all 0. Goldreich and Levin [GL89] have given an efficient list decoding algorithm for the Hadamard code, which runs in time poly(k; 1= However, for us, even brute force ....

....j is sufficiently small) For the univariate case, this constant is 1 [GS99] No inherent reason is known for the gap. Before proving Theorem 29, we recall that polynomials are sufficiently list decodable from a combinatorial standpoint (i.e. efficiency considerations aside) Theorem 31 (cf. GRS98, Thm. 17] For any f : F m F and p 2d=jF j, the number of total degree d polynomials that have (relative) agreement at least with f is less than 2= 7 We note that this problem has recently been solved by Impagliazzo, Shaltiel, and Wigderson [ISW00] though their pseudorandom ....

Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials with queries --- the highly noisy case. Technical Report TR98-060, Electronic Colloquium on Computational Complexity, 1998.

....this is an open question. Subsequent work. One of the main questions left open by this paper is the problem of reconstructing all degree d polynomials that agree with an arbitrary black box on ffl fraction of the inputs. Some recent work has addressed this question. Goldreich, Rubinfeld and Sudan [16] give an algorithm to (explicitly) reconstruct all n variate degree d polynomials agreeing with a black box over F on ffl fraction of the inputs, provided ffl 2 p d=jF j. Their algorithm runs in time O( n; 1 ffl ) poly(d) which is exponential in d. Their algorithm generalizes the ....

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th IEEE Symposium on Foundations of Computer Science, pp. 294--303, 1995.

....is used in the Elias Bassalygo upper bound on the dimension of codes with certain minimum distance, and is of interest to list decoding of codes. Proofs of the Johnson bound seem to come in one of two avors. The original proof and some of its derivates follows a linear algebra based argument [5, 6, 3, 10, 11], while more recent proofs, most notably [7, 4, 1] are more geometric. Our proof follows the latter spirit, extending these proofs to the case of general alphabets. A more technical comparison of our proof with existing ones is given later, after outlining some formal de nitions) Moreover, we ....

....qe 2(q 1) g : 2) Furthermore, if e equals the R.H.S of Condition (1) then A 0 q (n; d; e) 2n(q 1) 1. Comparison with Previous Bounds: The second upper bound on A 0 q (n; d; e) in (2) is the classical version of Johnson bound for the q ary case (cf. 9] proofs appear, for instance, in [10, 11]) The new aspect of our result is the n(q 1) upper bound. For the case q = 2, this result was known. Speci cally, Elias [3] proved that if d is odd, then A 0 2 (n; d; e) n as long as e satis es 1 We use the notation A 0 q (n; d; e) instead of the apparently more natural choice Aq (n; d; ....

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: the highly noisy case. Proc. of FOCS 95.

....= i 1 Gamma 1 q j Delta n, there exists a (non linear) n; k) q code C of distance at least d and a received word r, such that there are exponentially many codewords (with the exponent growing with ffln) within a Hamming distance of e from r. Note: The theorem above appears explicitly in [13]. The crucial direction, Part (1) above, is a q ary extension of the Johnson bound in coding theory. Johnson proves this bound only for the binary case, but the extension to the q ary case seems to be implicitly known to the coding theory community [27, Chapter 4, page 301] Proof [Sketch] 1. ....

....an application somewhere of the Cauchy Schwartz inequality) shows that the expected inner product is at least qe 2 (q Gamma1)n . This yields the inequality qe 2 (q Gamma 1)n 2e Gamma d d m which, in turn, yields the bound in Part (1) of the Theorem. 2. Following Goldreich et al.[13]) Let r be the all zeroes vector. Pick c 1 ; c m independently as follows: In each coordinate c j is chosen randomly (and independently of all else) to be 0 with probability 1 Gamma e n and chosen to be a random non zero element of Sigma otherwise. The probability that there exists a ....

[Article contains additional citation context not shown here]

Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials with queries---the highly noisy case. Technical Report TR98-060, Electronic Colloquium on Computational Complexity, 1998. Preliminary version in FOCS '95.

....field (i.e. F = GF(2) and any agreement rate that is bigger than the error rate (i.e. ffi 1 2 ) Their ideas were subsequently used by Kushilevitz and Mansour [25] to devise an algorithm for learning Boolean decision trees. 1.4. Subsequent work. At the time this work was first published [18] no algorithm other than brute force was known for reconstructing a list of degree d polynomials agreeing with an arbitrary function on a vanishing fraction of inputs, for any d 2. Our algorithm solves this problem with exponential dependence on d, but with polynomial dependence on n, the number ....

....a new black box consistent with f that takes y as an input. It may be noted that the transformation does not preserve other properties of the polynomial; e.g. its sparsity. Comment: The above solution to the above difficulty is different than the one in the original version of this paper [18]. The solution there was to pick many different suffixes (instead of 0 n Gammai ) and to argue that at least in one of them the agreement rate is preserved. However, picking many different suffixes creates additional problems, which needed to be dealt with carefully. This resulted in a more ....

Oded Goldreich, Ronitt Rubinfeld and Madhu Sudan. Learning polynomials with queries: The highly noisy case. 36th Annual Symposium on Foundations of Computer Science, pages 294--303, Milwaukee, Wisconsin, 23-25 October 1995. IEEE.

....bound, due to S. Johnson (see [17] bounds the number of solutions by n(n ) 2 2 (2n) n(n d) for binary codes (i.e. codes over the alphabet f0; 1g) provided the denominator is positive. For general codes, a simple bound can be shown by an inclusion exclusion argument (see, for instance, [11]) which yields that the number of solutions to the reconstruction problem is at most 2= if = 1 ) provided n p 2n(n ) Another such bound is also known due to Goldreich, Rubinfeld and Sudan [11] We do not describe this here. However the inclusion exclusion bound is not ....

.... codes, a simple bound can be shown by an inclusion exclusion argument (see, for instance, 11] which yields that the number of solutions to the reconstruction problem is at most 2= if = 1 ) provided n p 2n(n ) Another such bound is also known due to Goldreich, Rubinfeld and Sudan [11]. We do not describe this here. However the inclusion exclusion bound is not constructive, i.e. it does not provide a list of the 2= codewords which may be the solution to the reconstruction problem. It is reasonable to ask for a solution to the reconstruction problem which runs in polynomial ....

[Article contains additional citation context not shown here]

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: The highly noisy case. FOCS, 1995.

....a tool for improving our understanding of error correction even in adversarial models of error. Strong combinatorial results are known that bound the list decoding radius of an error correcting code as a function of its rate and its distance (See [5, 8, 40] for some of the earlier results, and [13, 15, 21] for some recent progress. However till the late 90 s no non trivial algorithms were developed to perform ecient list decoding. In [36] the author gave an algorithm to list decode Reed Solomon codes. This was shortly followed up by an algorithm by Shokrollahi and Wasserman [33] to decode ....

Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th Annual Symposium on Foundations of Computer Science, pages 294-303, Milwaukee, Wisconsin, 23-25 October 1995.

....eld (i.e. F = GF(2) and any agreement rate that is bigger than the error rate (i.e. 1 2 ) Their ideas were subsequently used by Kushilevitz and Mansour [25] to devise an algorithm for learning Boolean decision trees. 1. 4 Subsequent work At the time this work was done (and rst published [18]) no algorithm (other than brute force) was known for reconstructing a list of degree d polynomials agreeing with an arbitrary function on a vanishing fraction of inputs, for any d 2. Our algorithm solves this problem with exponential dependence on d, but with polynomial dependence on n, the ....

....a new black box consistent with f that takes y as an input. It may be noted that the transformation does not preserve other properties of the polynomial; e.g. its sparsity. Comment: The above solution to the above diculty is di erent than the one in the original version of this paper [18]. The solution there was to pick many di erent suxes (instead of 0 n i ) and to argue that at least in one of them the agreement rate is preserved. 4 Let c k be the coecient of ( Q i 1 j=1 x e j ) x k i in p 0 , and v be the coecient of Q i 1 j=1 x e j in f ( Then, v ....

Oded Goldreich, Ronitt Rubinfeld and Madhu Sudan. Learning polynomials with queries: The highly noisy case. 36th Annual Symposium on Foundations of Computer Science, pages 294-303, Milwaukee, Wisconsin, 23-25 October 1995. IEEE.

....consistent with these bits can be eciently recovered. Recovering from errors and erasures: We rst establish a combinatorial result proving an upper bound on the list size possible when decoding from a certain number of errors and erasures. Our result is analogous to the Johnson bound (see also [12]) and reduces to the one in [12] in presence of only errors. This places limits on the radius to which one can (currently) hope to list decode in polynomial time, and restricts qe= q 1) s, where e; s are the number of errors and erasures respectively, to be at most some quantity that is a ....

....eciently recovered. Recovering from errors and erasures: We rst establish a combinatorial result proving an upper bound on the list size possible when decoding from a certain number of errors and erasures. Our result is analogous to the Johnson bound (see also [12] and reduces to the one in [12] in presence of only errors. This places limits on the radius to which one can (currently) hope to list decode in polynomial time, and restricts qe= q 1) s, where e; s are the number of errors and erasures respectively, to be at most some quantity that is a function of the minimum distance of ....

[Article contains additional citation context not shown here]

O. Goldreich, R. Rubinfeld and M. Sudan. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pp. 294-303, 1995.

....can find a brief description in Section 2. 2 [Recovering from errors and erasures: We first establish a combinatorial result proving an upper bound on the list size possible when decoding from a certain number of errors and erasures. Our result is analogous to the Johnson bound (see also [8]) and reduces to the one in [8] in presence of only errors; our proof is in fact simpler than the one in [8] even though our result is more general. This places limits on the radius to which one can (currently) hope to list decode in polynomial time, and restricts qe= q 1) s, where e; s are the ....

....in Section 2. 2 [Recovering from errors and erasures: We first establish a combinatorial result proving an upper bound on the list size possible when decoding from a certain number of errors and erasures. Our result is analogous to the Johnson bound (see also [8] and reduces to the one in [8] in presence of only errors; our proof is in fact simpler than the one in [8] even though our result is more general. This places limits on the radius to which one can (currently) hope to list decode in polynomial time, and restricts qe= q 1) s, where e; s are the number of errors and erasures ....

[Article contains additional citation context not shown here]

O. GOLDREICH, R. RUBINFELD AND M. SUDAN. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pp. 294-303, 1995.

....with these bits can be efficiently recovered. Recovering from errors and erasures: We first establish a combinatorial result proving an upper bound on the list size possible when decoding from a certain number of errors and erasures. Our result is analogous to the Johnson bound (see also [8]) and reduces to the one in [8] in presence of only errors; our proof is in fact simpler than the one in [8] even though our result is more general. This places limits on the radius to which one can (currently) hope to list decode in polynomial time, and restricts qe= q Gamma 1) s, where e; s ....

....recovered. Recovering from errors and erasures: We first establish a combinatorial result proving an upper bound on the list size possible when decoding from a certain number of errors and erasures. Our result is analogous to the Johnson bound (see also [8] and reduces to the one in [8] in presence of only errors; our proof is in fact simpler than the one in [8] even though our result is more general. This places limits on the radius to which one can (currently) hope to list decode in polynomial time, and restricts qe= q Gamma 1) s, where e; s are the number of errors and ....

[Article contains additional citation context not shown here]

O. GOLDREICH, R. RUBINFELD AND M. SUDAN. Learning polynomials with queries: The highly noisy case. Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pp. 294-303, 1995.

No context found.

Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials with queries: The highly noisy case. SIAM Journal on Discrete Mathematics, 13(4):535--570, 2000.

No context found.

Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials with queries: The highly noisy case. Siam Journal on Discrete Mathematics, 13(4):535--570, 2000.

No context found.

Oded Goldreich, Madhu Sudan and Ronitt Rubinfeld, Learning Polynomials with Queries: The Highly Noisy Case. In the Proceedings of the 36th Annual Symposium on Foundations of Computer Science, 1995.

No context found.

Oded Goldreich, Madhu Sudan and Ronitt Rubinfeld, Learning Polynomials with Queries: The Highly Noisy Case, in the Proceedings of the 36th Annual Symposium on Foundations of Computer Science, IEEE Computer Society, pp. 294--303, 1995.

No context found.

O. Goldreich, R. Rubinfeld, M. Sudan, Learning polynomials with queries: The highly noisy case, 36th Annual Symposium on Foundation of Computer Science, Milwaukee, Wisconsin, 23-25 October 1995, pp. 294303.

No context found.

Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials with queries: The highly noisy case. SIAM Journal on Discrete Mathematics, 13(4):535-- 570, 2000.

No context found.

O. Goldreich, R. Rubinfeld, and M. Sudan. Learning polynomials with queries: The highly noisy case. Proc. of 36th FOCS, 1995, pp. 294--303.

No context found.

Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials with queries: The highly noisy case. SIAM Journal on Discrete Mathematics, 13, 2000.

No context found.

O. Goldreich, M. Sudan and R. Rubinfeld, Learning Polynomials with Queries: The Highly Noisy Case, Proc. 36th FOCS, pp. 294-303, 1995.

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