Multiparty Computation for Interval, Equality, and Comparison Without Bit-Decomposition Protocol

Damgard et al. [11] showed a novel technique to convert a polynomial sharing of secret a into the sharings of the bits of a in constant rounds, which is called the bit-decomposition protocol. The bit-decomposition protocol is a very powerful tool because it enables bitoriented operations even if shared secrets are given as elements in the field. However, the bit-decomposition protocol is relatively expensive. In this paper, we present a simplified bit-decomposition protocol by analyzing the original protocol. Moreover, we construct more efficient protocols for a comparison, interval test and equality test of shared secrets without relying on the bit-decomposition protocol though it seems essential to such bit-oriented operations. The key idea is that we do computation on secret a with c and r where c = a + r, c is a revealed value, and r is a random bitwise-shared secret. The outputs of these protocols are also shared without being revealed. The realized protocols as well as the original protocol are constantround and run with less communication rounds and less data communication than those of [11]. For example, the round complexities are reduced by a factor of approximately 3 to 10.

[1]  Jacques Stern,et al.  Sharing Decryption in the Context of Voting or Lotteries , 2000, Financial Cryptography.

[2]  Jan Camenisch,et al.  Efficient Computation Modulo a Shared Secret with Application to the Generation of Shared Safe-Prime Products , 2002, CRYPTO.

[3]  Avi Wigderson,et al.  Completeness theorems for non-cryptographic fault-tolerant distributed computation , 1988, STOC '88.

[4]  Ivan Damgård,et al.  Universally Composable Efficient Multiparty Computation from Threshold Homomorphic Encryption , 2003, CRYPTO.

[5]  Ivan Damgård,et al.  A Practical Implementation of Secure Auctions Based on Multiparty Integer Computation , 2006, Financial Cryptography.

[6]  Gita Alaghband,et al.  Fundamentals of Parallel Processing , 2002 .

[7]  Adi Shamir,et al.  How to share a secret , 1979, CACM.

[8]  Arto Salomaa,et al.  Public-Key Cryptography , 1991, EATCS Monographs on Theoretical Computer Science.

[9]  Pim Tuyls,et al.  Efficient Binary Conversion for Paillier Encrypted Values , 2006, EUROCRYPT.

[10]  Richard J. Lipton,et al.  Unbounded Fan-In Circuits and Associative Functions , 1985, J. Comput. Syst. Sci..

[11]  Moti Yung,et al.  Advances in Cryptology — CRYPTO 2002 , 2002, Lecture Notes in Computer Science.

[12]  Tal Rabin,et al.  Simplified VSS and fast-track multiparty computations with applications to threshold cryptography , 1998, PODC '98.

[13]  Judit Bar-Ilan,et al.  Non-cryptographic fault-tolerant computing in constant number of rounds of interaction , 1989, PODC '89.

[14]  Andrew Chi-Chih Yao,et al.  Protocols for secure computations , 1982, FOCS 1982.

[15]  David Chaum,et al.  Multiparty unconditionally secure protocols , 1988, STOC '88.

[16]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[17]  R. Cramer,et al.  Multiparty Computation from Threshold Homomorphic Encryption , 2000 .

[18]  Silvio Micali,et al.  How to play ANY mental game , 1987, STOC.

[19]  Aggelos Kiayias,et al.  Self Protecting Pirates and Black-Box Traitor Tracing , 2001, CRYPTO.

[20]  John Kubiatowicz,et al.  Optimizing Robustness While Generating Shared Secret Safe Primes , 2005, Public Key Cryptography.

[21]  I. Damglurd.  Unconditionally secure constant-rounds multi-party computation for equality, comparison, bits and exponentiation , 2006 .

[22]  Serge Vaudenay,et al.  Advances in Cryptology - EUROCRYPT 2006 , 2006, Lecture Notes in Computer Science.

[23]  Dan Boneh,et al.  Advances in Cryptology - CRYPTO 2003 , 2003, Lecture Notes in Computer Science.

[24]  Serge Vaudenay.  Public Key Cryptography - PKC 2005, 8th International Workshop on Theory and Practice in Public Key Cryptography, Les Diablerets, Switzerland, January 23-26, 2005, Proceedings , 2005, Public Key Cryptography.

[25]  Richard J. Lipton,et al.  Lower Bounds for Constant Depth Circuits for Prefix Problems , 1983, ICALP.

[26]  Ivan Damgård,et al.  A Generalisation, a Simplification and Some Applications of Paillier's Probabilistic Public-Key System , 2001, Public Key Cryptography.

[27]  Yuval Ishai,et al.  Share Conversion, Pseudorandom Secret-Sharing and Applications to Secure Computation , 2005, TCC.

[28]  Ivan Damgård,et al.  Secure Distributed Linear Algebra in a Constant Number of Rounds , 2001, CRYPTO.