Sufficient Conditions for Intractability over Black-Box Groups: Generic Lower Bounds for Generalized DL and DH Problems

The generic group model is a valuable methodology for analyzing the computational hardness of number-theoretic problems used in cryptography. Although generic hardness proofs exhibit many similarities, still the computational intractability of every newly introduced problem needs to be proven from scratch, a task that can easily become complicated and cumbersome when done rigorously. In this paper we make the first steps towards overcoming this problem by identifying criteria which guarantee the hardness of a problem in an extended generic model where algorithms are allowed to perform any operation representable by a polynomial function.

[1]  Eike Kiltz,et al.  A Tool Box of Cryptographic Functions Related to the Diffie-Hellman Function , 2001, INDOCRYPT.

[2]  Victor Shoup,et al.  Lower Bounds for Discrete Logarithms and Related Problems , 1997, EUROCRYPT.

[3]  Alexander W. Dent,et al.  Adapting the Weaknesses of the Random Oracle Model to the Generic Group Model , 2002, ASIACRYPT.

[4]  Matthew K. Franklin,et al.  Identity-Based Encryption from the Weil Pairing , 2001, CRYPTO.

[5]  Jung Hee Cheon,et al.  Security Analysis of the Strong Diffie-Hellman Problem , 2006, EUROCRYPT.

[6]  Yacov Yacobi,et al.  A Note on the Bilinear Diffie-Hellman Assumption , 2002, IACR Cryptol. ePrint Arch..

[7]  Ivan Damgård,et al.  Generic Lower Bounds for Root Extraction and Signature Schemes in General Groups , 2002, EUROCRYPT.

[8]  Victor Shoup,et al.  Sequences of games: a tool for taming complexity in security proofs , 2004, IACR Cryptol. ePrint Arch..

[9]  Hovav Shacham,et al.  Aggregate and Verifiably Encrypted Signatures from Bilinear Maps , 2003, EUROCRYPT.

[10]  David Pointcheval,et al.  The Gap-Problems: A New Class of Problems for the Security of Cryptographic Schemes , 2001, Public Key Cryptography.

[11]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[12]  V. Nechaev.  Complexity of a determinate algorithm for the discrete logarithm , 1994 .

[13]  Dan Boneh,et al.  Hierarchical Identity Based Encryption with Constant Size Ciphertext , 2005, EUROCRYPT.

[14]  Dan Boneh,et al.  Short Signatures Without Random Oracles , 2004, EUROCRYPT.

[15]  Dan Boneh,et al.  Breaking RSA May Not Be Equivalent to Factoring , 1998, EUROCRYPT.

[16]  J. Pollard,et al.  Monte Carlo methods for index computation () , 1978 .

[17]  Ueli Maurer,et al.  Abstract Models of Computation in Cryptography , 2005, IMACC.

[18]  Ahmad-Reza Sadeghi,et al.  Assumptions Related to Discrete Logarithms: Why Subtleties Make a Real Difference , 2001, EUROCRYPT.

[19]  Emmanuel Bresson,et al.  A Generalization of DDH with Applications to Protocol Analysis and Computational Soundness , 2007, CRYPTO.

[20]  Robert H. Deng,et al.  Variations of Diffie-Hellman Problem , 2003, ICICS.

[21]  Dan Boneh,et al.  Efficient Selective-ID Secure Identity Based Encryption Without Random Oracles , 2004, IACR Cryptol. ePrint Arch..