We provide formal definitions and efficient secure techniques for • turning noisy information into keys usable for any cryptographic application, and, in particular, • reliably and securely authenticating biometric data. Our techniques apply not just to biometric information, but to any keying material that, unlike traditional cryptographic keys, is (1) not reproducible precisely and (2) not distributed uniformly. We propose two primitives: a fuzzy extractor reliably extracts nearly uniform randomness R from its input; the extraction is error-tolerant in the sense that R will be the same even if the input changes, as long as it remains reasonably close to the original. Thus, R can be used as a key in a cryptographic application. A secure sketch produces public information about its input w that does not reveal w, and yet allows exact recovery of w given another value that is close to w. Thus, it can be used to reliably reproduce error-prone biometric inputs without incurring the security risk inherent in storing them. We define the primitives to be both formally secure and versatile, generalizing much prior work. In addition, we provide nearly optimal constructions of both primitives for various measures of “closeness” of input data, such as Hamming distance, edit distance, and set difference.

constant-weight codes, distance distribution, equidistant codes, linear programming bound New lower bounds are presented on the second moment of the distance distribution of binary codes, in terms of the first moment of the distribution. These bounds are used to obtain upper bounds on the size of codes whose maximum distance is close to their minimum distance. It is then demonstrated how such bounds can be applied to bound from below the smallest attainable ratio between the maximum distance and the minimum distance of codes. Finally, counterparts of the bounds are derived for the special case of constant-weight codes.

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We present extensions of some recent geometric proofs of the well-known Johnson bound. Our extensions apply to arbitrary alphabets (while previous proofs were given only for the binary case). Our extensions yield a "weighted " version of the Johnson bound equally easily -- the weighted version is of interest in light of the recent developments on soft-decision list decoding algorithms.

Given a graph, in the maximum clique problem one wants to find the largest number of vertices, any two of which are adjacent. In the maximum-weight clique problem, the vertices have positive, integer weights, and one wants to find a clique with maximum weight. A recent algorithm for the maximum clique problem is here used as a basis for developing an algorithm for the weighted case. Computational experiments with random graphs show that this new algorithm is faster than earlier algorithms in many cases. A set of weighted graphs obtained from the problem of constructing good constant weight error-correcting codes are proposed as test cases for maximum-weight clique algorithms

Let ( ) denote the maximum possible number of codewords in a binary code of length and minimum Hamming distance . For large values of , the best known upper bound, for fixed , is the Johnson bound. We give a new upper bound which is at least as good as the Johnson bound for all values of and , and for each there are infinitely many values of for which the new bound is better than the Johnson bound. For small values of and , the best known method to obtain upper bounds on ( ) is linear programming. We give new inequalities for the linear programming and show that with these new inequalities some of the known bounds on ( ) for 28 are improved.

Existing tables of constant weight codes are mainly confined to codes of length n ≤ 28. This paper presents tables of codes of lengths 29 ≤ n ≤ 63. The motivation for creating these tables was their application to the generation of good sets of frequency hopping lists in radio networks. The complete generation of all relevant cases by a small number of algorithms is augmented in individual cases by miscellaneous constructions. These sometimes give a larger number of codewords than the algorithms. 1

We consider bounds on codes in spherical caps and related problems in geometry and coding theory. An extension of the Delsarte method is presented that relates upper bounds on the size of spherical codes to upper bounds on codes in caps. Several new upper bounds on codes in caps are derived. Applications of these bounds to estimates of the kissing numbers and one-sided kissing numbers are considered. It is proved that the maximum size of codes in spherical caps for large dimensions is determined by the maximum size of spherical codes, so these problems are asymptotically equivalent.

Delsarte’s method and its extensions allow to consider the upper bound problem for codes in 2-point-homogeneous spaces as a linear programming problem with perhaps infinitely many variables, which are the distance distribution. We show that using as variables power sums of distances this problem can be considered as a finite semidefinite programming problem. This method allows to improve some linear programming upper bounds. In particular we obtain new bounds of one-sided kissing numbers. 1

In this paper we devise a rational curve fitting algorithm and apply it to the list decoding of Reed-Solomon and BCH codes. The proposed list decoding algorithms exhibit the following significant properties. • The algorithm corrects up to n(1 − √ 1 − D) errors for a (generalized) (n, k, d = n − k + 1) Reed-Solomon code, which matches the Johnson bound, where D △ = d n denotes the normalized minimum distance. In comparison with the Guruswami-Sudan algorithm, which exhibits the same list correction capability, the former requires multiplicity, which dictates the algorithmic complexity, O(n(1 − √ 1 − D)), whereas the latter requires multiplicity O(n 2 (1−D)). With the up-to-date most efficient implementation, the former has complexity O � n 6 (1 − √ 1 − D) 7/2 �, whereas the latter has complexity O(n 10 (1 − D) 4). • With the multiplicity set to one, the derivative list correction capability precisely sits in between the conventional hard-decision decoding and the optimal list decoding. Moreover, the number of candidate codewords is upper bounded by a constant for a fixed code rate and thus, the derivative algorithm exhibits quadratic complexity O(n 2). • By utilizing the unique properties of the Berlekamp algorithm, the algorithm corrects up to n 2 (1 − √ 1 − 2D) errors for a narrow-sense (n, k, d) binary BCH code, which matches the Johnson bound for binary codes. The algorithmic complexity is O � n 6 (1 − √ 1 − 2D) 7 �. 1 I.