B. Chor, M. Gereb-Graus, and E. Kushilevitz. On the structure of the privacy hierarchy. Journal of Cryptology, 7(1):53--60, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of Bielefeld University, Germany. of the protocol no coalition of arbitrary size can get any information about the inputs of the remaining players other than what can be deduced by their own inputs and the value of f . Private computation in this model has been the subject of several papers [1, 7, 8, 15, 18, 5, 6, 16]. Chor and Kushilevitz [7] characterized the boolean functions that can be computed in a totally private way. More precisely, they proved a boolean function f(x 1 ; Delta Delta Delta ; xn ) is totally private if and only if it can be represented as the XOR of n one argument boolean functions. ....

B. Chor, M. Gereb-Graus, and E. Kushilevitz, On The Structure of the Privacy Hierarchy, J. of Cryptology 7, 1994, pp. 53--60.

....of Bielefeld University, Germany. of the protocol no coalition of arbitrary size can get any information about the inputs of the remaining players other than what can be deduced by their own inputs and the value of f . Private computation in this model has been the subject of several papers [1, 7, 8, 15, 18, 5, 6, 16]. Chor and Kushilevitz [7] characterized the boolean functions that can be computed in a totally private way. More precisely, they proved a boolean function f(x 1 ; Delta Delta Delta ; xn ) is totally private if and only if it can be represented as the XOR of n one argument boolean functions. ....

B. Chor, M. Gereb-Graus, and E. Kushilevitz, On The Structure of the Privacy Hierarchy, J. of Cryptology 7, 1994, pp. 53--60.

....way. That is, after the execution of the protocol no coalition of size at most t can get any information about the inputs of the remaining players other than what can be deduced from their own inputs and the value of F . Private computation in this model has been the subject of several papers [1, 6, 7, 8, 9, 16, 17, 20]. If t = n Gamma 1, then t private computation is referred to as totally private computation. Chor and Kushilevitz [8] characterized the boolean functions that can be computed in a totally private way. More precisely, they proved that a boolean function F(x 1 ; Delta Delta Delta ; x n ) is ....

B. Chor, M. Gereb-Graus, and E. Kushilevitz, On The Structure of the Privacy Hierarchy, J. of Cryptology, Vol. 7, 1994, pp. 53--60.

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B. Chor, M. Ger'eb-Graus, and E. Kushilevitz, "On the Structure of the Privacy Hierarchy", Journal of Cryptology, Vol. 7(1), pp. 53-60, 1994.

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B. Chor, M. Ger'eb-Graus, and E. Kushilevitz, "On the Structure of the Privacy Hierarchy", Journal of Cryptology, Vol. 7, No. 1, 1994, pp. 53-60.

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B. Chor, M. Ger'eb-Graus, and E. Kushilevitz, "On the Structure of the Privacy Hierarchy ", Journal of Cryptology, Vol. 7, No. 1, 1994, pp. 53-60.

....adversaries, have been studied in the literature. Negative results on private computation in our model hold in the more adversarial (information theoretic) models as well. The seminal works of [2, 4] showed that all n argument functions over nite domains X i can be computed b 2 c privately. In [5] it is shown that there exists a dense privacy hierarchy: for any b 2 c t n 2 there exists an n argument function which is t private (i.e. can be computed by a t private protocol) but is not (t 1) private. In the works of [10, 1] a complete characterization of 1 private two argument ....

....of characterizing the class of t private functions of n argument, for any n 3 and d 2 e t n 1, is still open. The only technique which appears in the literature for proving non t privacy of functions with n 3 arguments uses a reduction to the two party case, via a partition argument [6, 5, 7]. If f is t private, where d 2 e t n 1, then for every partition (S; S) of the parties f1; 2; ng such that jSj; jSj t, the two argument function obtained by viewing f as a function of fx i g i2S and fx i g i2S is 1 private. In [6] at the course of proving a characterization of ....

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B. Chor, M. Gereb-Graus, and E. Kushilevitz. On the structure of the privacy hierarchy. J. of Cryptology, 7:53-60, 1994.

.... in a way that no single player learns about the initial inputs of other players more than what is revealed by the value of f( x) and its own input 1 (no matter what its computational resources are) Private computations in this setting were the subject of a considerable amount of work, e.g. [5, 13, 2, 3, 15, 16, 17, 18, 21, 33, 37] 2 . In this paper, we consider An early version of this paper appeared in the Proc. of the 17th PODC conference, 1998, pp. 81 90. y Dept. of Computer Science, Technion, Haifa, Israel. e mail: eyalk cs.technion.ac.il; Part of this research was done while visiting ICSI Berkeley. Supported by ....

B. Chor, M. Ger'eb-Graus, and E. Kushilevitz, "On the Structure of the Privacy Hierarchy", Journal of Cryptology, Vol. 7(1), pp. 53-60, 1994.

....they all follow the prescribed protocol P, but they could try to get additional information from the messages they receive during the execution of the protocol. The study of private computations in this setting was initiated by [6, 13] and was the subject of a considerable amount of work, e.g. [3, 14, 31, 4, 20, 15, 16, 17, 29, 30]. 1 The use of randomness is a crucial ingredient in private protocols; without randomness only degenerate functions can be computed privately. Protocols for the xor (exclusive or) function (and, more generally, protocols for the modular sum function) are basic building blocks in most private ....

B. Chor, M. Ger eb-Graus, and E. Kushilevitz, On the structure of the privacy hierarchy, J. Cryptology, 7 (1994), pp. 53--60.

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B. Chor, M. Ger'eb-Graus, and E. Kushilevitz, On the Structure of the Privacy Hierarchy, To appear in Journal of Cryptology.

....all follow the prescribed protocol P but they could try to get additional information from the messages they receive during the execution of the protocol. The study of private computations in this setting was initiated by [BGW88, CCD88] and was the subject of a considerable amount of work, e.g. [BB89, CK89, K89, B89, FY92, CK92, CGK90, CGK92, KMO94]. The use of randomness is a crucial ingredient in private protocols; without randomness only degenerate functions can be computed privately. The xor (exclusive or) function (and more generally, modular sum function) is a basic building block in all private protocols. As a result (and due to its ....

B. Chor, M. Ger'eb-Graus, and E. Kushilevitz, "On the Structure of the Privacy Hierarchy ", Journal of Cryptology, Vol. 7, No. 1, 1994, pp. 53-60.

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B. Chor, M. Ger'eb-Graus, and E. Kushilevitz, On the Structure of the Privacy Hierarchy, Journal of Cryptology, Vol. 7, No. 1, 1994, pp. 53-60.

....secret input, x i , to compute the value of f( x) in a way that no single player learns about the initial inputs of other players more than what is revealed by the value of f( x) and its own input 1 . Private computations in this setting were the subject of a considerable amount of work, e.g. [5, 13, 2, 3, 15, 16, 17, 18, 21, 31, 35]. In this paper, we consider this setting for the basic xor function, and show quite unexpected results relating the rounds complexity and the randomness complexity of such computations. An early version of this paper appeared in the Proc. of the 17th PODC conference, 1998, pp. 81 90. y Dept. ....

B. Chor, M. Ger'eb-Graus, and E. Kushilevitz, "On the Structure of the Privacy Hierarchy", Journal of Cryptology, Vol. 7(1), pp. 53-60, 1994.

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B. Chor, M. Ger'eb-Graus, and E. Kushilevitz, On the Structure of the Privacy Hierarchy, Journal of Cryptology, Vol. 7, No. 1, 1994, pp. 53-60.

....supported by research contracts ONR N0001491 J 1981 and NSF CCR 90 07677. z Dept. of Computer Science, Tel Aviv University, Tel Aviv, Israel. e mail: adiro math.tau.ac.il 1 In the literature a more general definition of t privacy is given. The above definition is the case t = 1. work, e.g. [BGW88, CCD88, BB89, CK89, K89, B89, FY92, CK92, CGK90, CGK92, KMO94]. One crucial ingredient in private protocols is the use of randomness. Quantifying the amount of randomness needed for computing functions privately is the subject of the present work. Randomness as a resource was extensively studied in the last decade. Methods for saving random bits range over ....

B. Chor, M. Ger'eb-Graus, and E. Kushilevitz, "On the Structure of the Privacy Hierarchy", Journal of Cryptology, Vol. 7, No. 1, 1994, pp. 53-60.

....with his kind permission, in the appendix. It remains open whether the corresponding generalization of Theorem 11 is true. One of the interesting questions in the area of private computations has been to investigate the structure of the privacy hierarchy. Recently, we have resolved this question [12] for n argument functions which are defined over finite domains. We have shown that for finite domains, the privacy hierarchy has exactly bn=2c levels. For any dn=2e t n Gamma 2, an n argument function which is t private but not t 1 private was constructed. The major tool used in this ....

....to partition the n inputs into two sets of appropriate sizes, and applying the criteria for privacy of two argument functions. This approach is not useful in determining t privacy for t which is at most b(n Gamma 1) 2c, because one of the two sizes will be larger than t. Indeed, the results in [12] leave open the structure of the privacy hierarchy for functions defined over countable domains. The communication complexity techniques used in the present work are of very different nature than the partition techniques which we mentioned above. They enabled us to show that the privacy hierarchy ....

B. Chor, M. Ger'eb-Graus, and E. Kushilevitz, "On the Structure of the Privacy Hierarchy", J. Cryptology, In Press.

....all follow the prescribed protocol P but they could try to get additional information from the messages they receive during the execution of the protocol. The study of private computations in this setting was initiated by [BGW88, CCD88] and was the subject of a considerable amount of work, e.g. [BB89, CK89, K89, B89, FY92, CK92, CGK90, CGK92, KMO94, KOR96]. 1 The use of randomness is a crucial ingredient in private protocols; without randomness only degenerate functions can be computed privately. Protocols for the xor (exclusive or) function (and more generally, protocols for the modular sum function) are basic building blocks in most ....

B. Chor, M. Ger'eb-Graus, and E. Kushilevitz, "On the Structure of the Privacy Hierarchy ", Journal of Cryptology, Vol. 7, No. 1, 1994, pp. 53-60.

....all follow the prescribed protocol P but they could try to get additional information from the messages they receive during the execution of the protocol. The study of private computations in this setting was initiated by [BGW88, CCD88] and was the subject of a considerable amount of work, e.g. [BB89, CK89, K89, B89, FY92, CK92, CGK90, CGK92, KMO94, KOR96]. 1 The use of randomness is a crucial ingredient in private protocols; without randomness only degenerate functions can be computed privately. 1 This setting is different than the one studied in [Y86, GMW87] in which the computational power of the players is restricted and hence ....

B. Chor, M. Ger'eb-Graus, and E. Kushilevitz, "On the Structure of the Privacy Hierarchy", Journal of Cryptology, Vol. 7, No. 1, 1994, pp. 53-60.

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B. Chor, M. Gereb-Graus, and E. Kushilevitz. On the structure of the privacy hierarchy. Journal of Cryptology, 7(1):53--60, 1994.

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B. Chor, M. Ger eb-Graus, and E. Kushilevitz. On the structure of the privacy hierarchy. Journal of Cryptology, 7(1):53-- 60, 1994.

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B. Chor, M. Ger eb-Graus, and E. Kushilevitz. On the structure of the privacy hierarchy. Journal of Cryptology, 7(1):53-- 60, 1994.

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B. Chor, M. Gereb-Graus, and E. Kushilevitz. On the Structure of the Privacy Hierarchy. Journal of Cryptology, vol. 7, pp. 53-60, 1994. 28

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B. Chor, M. Ger eb-Graus, and E. Kushilevitz. On the structure of the privacy hierarchy. Journal of Cryptology, 7(1):53-- 60, 1994.

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