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19
GraphBased Authentication of Digital Streams
 IEEE Symposium on Security and Privacy
, 2000
"... We consider the authentication of digital streams over a lossy network. The overall approach taken is graphbased, as this yields simple methods for controlling overhead, delay, and the ability to authenticate, while serving to unify many previously known hash and MACbased techniques. The loss pat ..."
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We consider the authentication of digital streams over a lossy network. The overall approach taken is graphbased, as this yields simple methods for controlling overhead, delay, and the ability to authenticate, while serving to unify many previously known hash and MACbased techniques. The loss pattern of the network is defined probabilistically, allowing both bursty and random packet loss to be modeled. Our authentication schemes are customizable by the sender of the stream; that is, within reasonable constraints on the input parameters, we provide schemes that achieve the desired authentication probability while meeting the input upper bound on the overhead per packet. In addition, we demonstrate that some of the shortcomings of previously known schemes correspond to easily identifiable properties of a graph, and hence, may be more easily avoided by taking a graphbased approach to designing authentication schemes.
Efficient Traitor Tracing Algorithms using List Decoding
 In Proceedings of ASIACRYPT ’01, volume 2248 of LNCS
, 2001
"... Abstract. We use powerful new techniques for list decoding errorcorrecting codes to efficiently trace traitors. Although much work has focused on constructing traceability schemes, the complexity of the tracing algorithm has received little attention. Because the TA tracing algorithm has a runtime o ..."
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Cited by 11 (0 self)
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Abstract. We use powerful new techniques for list decoding errorcorrecting codes to efficiently trace traitors. Although much work has focused on constructing traceability schemes, the complexity of the tracing algorithm has received little attention. Because the TA tracing algorithm has a runtime of O(N) in general, where N is the number of users, it is inefficient for large populations. We produce schemes for which the TA algorithm is very fast. The IPP tracing algorithm, though less efficient, can list all coalitions capable of constructing a given pirate. We give evidence that when using an algebraic structure, the ability to trace with the IPP algorithm implies the ability to trace with the TA algorithm. We also construct schemes with an algorithm that finds all possible traitor coalitions faster than the IPP algorithm. Finally, we suggest uses for other decoding techniques in the presence of additional information about traitor behavior. 1
Lower bounds for the noisy broadcast problem
 In Proceedings of the 46 th IEEE Symposium on Foundations of Computer Science (FOCS 2005
, 2005
"... We prove the first nontrivial (super linear) lower bound in the noisy broadcast model, defined by El Gamal in [6]. In this model there are n + 1 processors P0, P1,..., Pn, each of which is initially given a private input bit xi. The goal is for P0 to learn the value of f(x1,..., xn), for some speci ..."
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We prove the first nontrivial (super linear) lower bound in the noisy broadcast model, defined by El Gamal in [6]. In this model there are n + 1 processors P0, P1,..., Pn, each of which is initially given a private input bit xi. The goal is for P0 to learn the value of f(x1,..., xn), for some specified function f, using a series of noisy broadcasts. At each step a designated processor broadcasts one bit to all of the other processors, and the bit received by each processor is flipped with fixed probability (independently for each recipient). In 1988, Gallager [16] gave a noiseresistant protocol that allows P0 to learn the entire input with constant probability in O(n log log n) broadcasts. We prove that Gallager’s protocol is optimal, up to a constant factor. Our lower bound follows by reduction from a lower bound for generalized noisy decision trees, a new model which may be of independent interest. For this new model we show a lower bound of Ω(n log n) on the depth of a tree that learns the entire input. We also show an Ω(n log log n) lower bound for the number of broadcasts required to compute certain explicit booleanvalued functions, when the correct output must be attained with probability at least 1 − n −α for a constant parameter α> 0 (this bound applies to all threshold functions, as well as any other booleanvalued function with linear sensitivity). This bound also follows by reduction from a lower bound of Ω(n log n) on the depth of generalized noisy decision trees that compute the same functions with the same error. We also show a (nontrivial) Ω(n) lower bound on the depth of generalized noisy decision trees that compute such functions with small constant error. Finally, we show the first protocol in the noisy broadcast model that computes the Hamming weight of the input using a linear number of broadcasts.
Combinatorics with a geometric flavor: some examples
 in Visions in Mathematics Toward 2000 (Geometric and Functional Analysis, Special Volume
, 2000
"... In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound theorem for ..."
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In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound theorem for the face numbers of convex polytopes and its many extensions is the second topic. Next are general properties of subsets of the vertices of the discrete ndimensional cube and some relations with questions of extremal and probabilistic combinatorics. Our fourth topic is tree enumeration and random spanning trees, and finally, some combinatorial and geometrical aspects of the simplex method for linear programming are considered.
Codes, Graphs, and Schemes From Nonlinear Functions
, 2000
"... We consider functions on binary vector spaces which are far from linear functions in di#erent senses. We compare three existing notions: almost perfect nonlinear #APN# functions, almost bent #AB# functions, and crooked #CR# functions. Such functions are of importance in cryptography because of their ..."
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Cited by 6 (1 self)
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We consider functions on binary vector spaces which are far from linear functions in di#erent senses. We compare three existing notions: almost perfect nonlinear #APN# functions, almost bent #AB# functions, and crooked #CR# functions. Such functions are of importance in cryptography because of their resistance to linear and di#erential attacks on certain cryptosystems. We give a new combinatorial characterization of almost bent functions in terms of the number of solutions to a certain system of equations, and a characterization of crooked functions in terms of the Fourier transform. We also showhow these functions can be used to construct several combinatorial structures; such as semibiplanes, di#erence sets, distance regular graphs, symmetric association schemes, and uniformly packed #BCH and Preparata# codes. 1 Almost perfect nonlinear, almost bent, and crooked functions We consider functions on binary vector spaces which are far from linear functions in di#erent senses. We compar...
Applications of list decoding to tracing traitors
 IEEE Trans. Inform. Theory
, 2003
"... Abstract — We apply results from algebraic coding theory to solve problems in cryptography, by using recent results on list decoding of errorcorrecting codes to efficiently find traitors who collude to create pirates. We produce schemes for which the TA (traceability) traitor tracing algorithm is v ..."
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Abstract — We apply results from algebraic coding theory to solve problems in cryptography, by using recent results on list decoding of errorcorrecting codes to efficiently find traitors who collude to create pirates. We produce schemes for which the TA (traceability) traitor tracing algorithm is very fast. We compare the TA and IPP (identifiable parent property) traitor tracing algorithms, and give evidence that when using an algebraic structure, the ability to trace traitors with the IPP algorithm implies the ability to trace with the TA algorithm. We also demonstrate that list decoding techniques can be used to find all possible pirate coalitions. Finally, we raise some related open questions about linear codes, and suggest uses for other decoding techniques in the presence of additional information about traitor behavior. Index Terms — Algebraic geometry code, identifiable parent property, list decoding, traceability code, traitor tracing, ReedSolomon code. I.
Commutative association schemes
 European J. Combin
"... Abstract. Association schemes were originally introduced by Bose and his coworkers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield ..."
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Cited by 6 (3 self)
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Abstract. Association schemes were originally introduced by Bose and his coworkers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the “commutative case, ” has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and grouptheoretic symmetry, culminating in Schrijver’s SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work. 1.
The distance approach to approximate combinatorial counting
, 2000
"... We develop general methods to obtain fast (polynomial time) estimates of the cardinality of a combinatorially defined set via solving some randomly generated optimization problems on the set. Geometrically, we estimate the cardinality of a subset of the Boolean cube via the average distance from a ..."
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Cited by 5 (2 self)
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We develop general methods to obtain fast (polynomial time) estimates of the cardinality of a combinatorially defined set via solving some randomly generated optimization problems on the set. Geometrically, we estimate the cardinality of a subset of the Boolean cube via the average distance from a point in the cube to the subset. As an application, we present a new randomized polynomial time algorithm which approximates the permanent of a 01 matrix by solving a small number of Assignment problems.
Bounds on (n, r)arcs and their application to linear codes
 Finite Fields Appl
"... This article reviews some of the principal and recentlydiscovered lower and upper bounds on the maximum size of (n, r)arcs in PG(2, q), sets of n points with at most r points on a line. Some of the upper bounds are used to improve the Griesmer bound for linear codes in certain cases. Also, a table ..."
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Cited by 3 (0 self)
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This article reviews some of the principal and recentlydiscovered lower and upper bounds on the maximum size of (n, r)arcs in PG(2, q), sets of n points with at most r points on a line. Some of the upper bounds are used to improve the Griesmer bound for linear codes in certain cases. Also, a table is included showing the current best upper and lower bounds for q ≤ 19, and a number of open problems are discussed. 1
A Survey on Packing and Covering Problems in the Hamming Permutation Space
"... Consider the symmetric group Sn equipped with the Hamming metric dH. Packing and covering problems in the finite metric space (Sn,dH) are surveyed, including a combination of both. 1 ..."
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Consider the symmetric group Sn equipped with the Hamming metric dH. Packing and covering problems in the finite metric space (Sn,dH) are surveyed, including a combination of both. 1