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Algebraic Geometric Secret Sharing Schemes and Secure MultiParty Computations over Small Fields
"... We introduce algebraic geometric techniques in secret sharing and in secure multiparty computation (MPC) in particular. The main result is a linear secret sharing scheme (LSSS) de ned over a nite eld Fq, with the following properties. 1. It is ideal. The number of players n can be as large as #C(Fq ..."
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Cited by 27 (6 self)
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We introduce algebraic geometric techniques in secret sharing and in secure multiparty computation (MPC) in particular. The main result is a linear secret sharing scheme (LSSS) de ned over a nite eld Fq, with the following properties. 1. It is ideal. The number of players n can be as large as #C(Fq), where C is an algebraic curve C of genus g de ned over Fq. 2. It is quasithreshold: it is trejecting and t+1+2gaccepting, but not necessarily t + 1accepting. It is thus in particular a ramp scheme. High information rate can be achieved. 3. It has strong multiplication with respect to the tthreshold adversary structure, if t < 1 3 n 4 3 g. This is a multilinear algebraic property on an LSSS facilitating zeroerror multiparty multiplication, unconditionally
On the Connections Between Universal Hashing, Combinatorial Designs and ErrorCorrecting Codes
 In Proc. Congressus Numerantium 114
, 1996
"... In this primarily expository paper, we discuss the connections between two popular and useful tools in theoretical computer science, namely, universal hashing and pairwise independent random variables; and classical combinatorial stuctures such as errorcorrecting codes, balanced incomplete block de ..."
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Cited by 21 (1 self)
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In this primarily expository paper, we discuss the connections between two popular and useful tools in theoretical computer science, namely, universal hashing and pairwise independent random variables; and classical combinatorial stuctures such as errorcorrecting codes, balanced incomplete block designs, difference matrices and orthogonal arrays. 1 Introduction The concept known as "universal hashing" was invented by Carter and Wegman [5] in 1979. In [29, p. 18], Avi Wigderson characterizes universal hashing as being a tool which "should belong to the fundamental bag of tricks of every computer scientist". This is no exaggeration, as there are probably well in excess of fifty papers in theoretical computer science that employ universal hashing as an important tool. Several of the most attractive applications are outlined in the the lecture notes [29]. A closely related topic goes by several names: "strongly universal hashing " [27], "twopoint based sampling" [6], and "pairwise indep...
Universal hashing and multiple authentication
 In Proc. CRYPTO 96, Lecture Notes in Computer Science
, 1996
"... at,iciOcse.unl.edu ..."
Strongly Universal Hashing and Identification Codes Via Channels
 IEEE TRANS. INFORMATION THEORY
, 1999
"... This paper shows that fflalmost strongly universal classes of hash functions can yield better explicit constructions of identification codes via channels (ID codes) and identification plus transmission codes (IT codes) than the previous explicit constructions of Verdú and Wei. ..."
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Cited by 2 (0 self)
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This paper shows that fflalmost strongly universal classes of hash functions can yield better explicit constructions of identification codes via channels (ID codes) and identification plus transmission codes (IT codes) than the previous explicit constructions of Verdú and Wei.
Authentication Codes and Algebraic Curves
"... Abstract. We survey a recent application of algebraic curves over finite fields to the constructions of authentication codes. 1. ..."
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Cited by 1 (0 self)
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Abstract. We survey a recent application of algebraic curves over finite fields to the constructions of authentication codes. 1.
A Construction Method for Optimally Universal Hash Families and its Consequences for the Existence of
"... We introduce a method for constructing optimally universal hash families and equivalently RBIBDs. As a consequence of our construction we obtain minimal optimally universal hash families, if the cardinalities of the universe and the range are powers of the same prime. A corollary of this result is t ..."
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We introduce a method for constructing optimally universal hash families and equivalently RBIBDs. As a consequence of our construction we obtain minimal optimally universal hash families, if the cardinalities of the universe and the range are powers of the same prime. A corollary of this result is that the necessary conditions for the existence of an RBIBD with parameters v, k, λ, namely v ≡ 0 (mod k) and λ(v − 1) ≡ 0 (mod k − 1), are sufficient, if v and k are powers of the same prime. As an application of our construction, we show that the kMAXCUT algorithm of Hofmeister and Lefmann [9] can be implemented such that it has a polynomial running time, in the case that the number of vertices and k are powers of the same prime. 1
A Construction Method for Optimally Universal Hash Families and its Consequences for the Existence of RBIBDs (Extended Abstract)
"... We introduce a method for constructing optimally universal hash families and equivalently RBIBDs. As a consequence of our construction we obtain minimal optimally universal hash families, if the cardinalities of the universe and the range are powers of the same prime. A corollary of this result is t ..."
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We introduce a method for constructing optimally universal hash families and equivalently RBIBDs. As a consequence of our construction we obtain minimal optimally universal hash families, if the cardinalities of the universe and the range are powers of the same prime. A corollary of this result is that the necessary condition for the existence of an RBIBD with parameters (v, k, λ), namely v mod k = λ(v − 1) mod (k − 1) = 0, is sufficient, if v and k are powers of the same prime. As an application of our construction, we show that the kMAXCUT algorithm of Hofmeister and Lefmann [9] can be implemented such that it has a polynomial running time, in the case that the number of vertices and k are powers of the same prime.
Logarithm Cartesian authentication codes
, 2003
"... Chanson, Ding and Salomaa have recently constructed several classes of authentication codes using certain classes of functions. In this paper, we further extend that work by constructing two classes of Cartesian authentication codes using the logarithm functions. The codes constructed here involve t ..."
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Chanson, Ding and Salomaa have recently constructed several classes of authentication codes using certain classes of functions. In this paper, we further extend that work by constructing two classes of Cartesian authentication codes using the logarithm functions. The codes constructed here involve the theory of cyclotomy and are better than a subclass of Helleseth–Johansson’s codes and Bierbrauer’s codes in terms of the maximum success probability with respect to the substitution attack.