Algebraic (Trapdoor) One-Way Functions and Their Applications

In this paper we introduce the notion of Algebraic (Trapdoor) One Way Functions, which, roughly speaking, captures and formalizes many of the properties of number-theoretic one-way functions. Informally, a (trapdoor) one way function F: X#8594;Y is said to be algebraic if X and Y are (finite) abelian cyclic groups, the function is homomorphic i.e. F(x)·F(y)=F(x ·y), and is ring-homomorphic, meaning that it is possible to compute linear operations 'in the exponent' over some ring (which may be different from ℤp where p is the order of the underlying group X) without knowing the bases. Moreover, algebraic OWFs must be flexibly one-way in the sense that given y=F(x), it must be infeasible to compute (x′, d) such that F(x′)=yd (for d≠0). Interestingly, algebraic one way functions can be constructed from a variety of standard number theoretic assumptions, such as RSA, Factoring and CDH over bilinear groups. As a second contribution of this paper, we show several applications where algebraic (trapdoor) OWFs turn out to be useful. These include publicly verifiable secure outsourcing of polynomials, linearly homomorphic signatures and batch execution of Sigma protocols.

[1]  C. P. Schnorr,et al.  Efficient Identification and Signatures for Smart Cards (Abstract) , 1989, EUROCRYPT.

[2]  Marc Fischlin,et al.  Public Key Cryptography – PKC 2012 , 2012, Lecture Notes in Computer Science.

[3]  Henri Gilbert,et al.  Advances in Cryptology - EUROCRYPT 2010, 29th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Monaco / French Riviera, May 30 - June 3, 2010. Proceedings , 2010, EUROCRYPT.

[4]  Ivan Damgård,et al.  Zero-Knowledge Proofs for Finite Field Arithmetic or: Can Zero-Knowledge be for Free? , 1997 .

[5]  Nuttapong Attrapadung,et al.  Homomorphic Network Coding Signatures in the Standard Model , 2011, Public Key Cryptography.

[6]  Rosario Gennaro,et al.  Publicly verifiable delegation of large polynomials and matrix computations, with applications , 2012, IACR Cryptol. ePrint Arch..

[7]  Dan Boneh,et al.  Linearly Homomorphic Signatures over Binary Fields and New Tools for Lattice-Based Signatures , 2011, Public Key Cryptography.

[8]  Craig Gentry,et al.  Non-interactive Verifiable Computing: Outsourcing Computation to Untrusted Workers , 2010, CRYPTO.

[9]  Silvio Micali,et al.  CS proofs , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[10]  Jonathan Katz,et al.  Signing a Linear Subspace: Signature Schemes for Network Coding , 2009, IACR Cryptol. ePrint Arch..

[11]  Payman Mohassel,et al.  Efficient and Secure Delegation of Linear Algebra , 2011, IACR Cryptol. ePrint Arch..

[12]  Jean-Jacques Quisquater,et al.  A Practical Zero-Knowledge Protocol Fitted to Security Microprocessor Minimizing Both Transmission and Memory , 1988, EUROCRYPT.

[13]  Sean W. Smith,et al.  Building a high-performance, programmable secure coprocessor , 1999, Comput. Networks.

[14]  Fabian Monrose,et al.  Distributed Execution with Remote Audit , 1999, NDSS.

[15]  Allison Bishop,et al.  New Proof Methods for Attribute-Based Encryption: Achieving Full Security through Selective Techniques , 2012, CRYPTO.

[16]  Niklaus Wirth,et al.  Advances in Cryptology — EUROCRYPT ’88 , 2000, Lecture Notes in Computer Science.

[17]  M. Rabin.  DIGITALIZED SIGNATURES AND PUBLIC-KEY FUNCTIONS AS INTRACTABLE AS FACTORIZATION , 1979 .

[18]  Stanislaw Jarecki,et al.  Public Key Cryptography – PKC 2009 , 2009, Lecture Notes in Computer Science.

[19]  Aggelos Kiayias,et al.  Group Signatures: Provable Security, Efficient Constructions and Anonymity from Trapdoor-Holders , 2004, IACR Cryptol. ePrint Arch..

[20]  Yael Tauman Kalai,et al.  Improved Delegation of Computation using Fully Homomorphic Encryption , 2010, IACR Cryptol. ePrint Arch..

[21]  Rosario Gennaro,et al.  Public Key Cryptography - PKC 2011 - 14th International Conference on Practice and Theory in Public Key Cryptography, Taormina, Italy, March 6-9, 2011. Proceedings , 2011, Public Key Cryptography.

[22]  R. Cramer,et al.  Linear Zero-Knowledgde. A Note on Efficient Zero-Knowledge Proofs and Arguments , 1996 .

[23]  Dawn Xiaodong Song,et al.  Homomorphic Signature Schemes , 2002, CT-RSA.

[24]  Yuval Ishai,et al.  From Secrecy to Soundness: Efficient Verification via Secure Computation , 2010, ICALP.

[25]  Jonathan Katz,et al.  Secure Network Coding Over the Integers , 2010, IACR Cryptol. ePrint Arch..

[26]  Darren Leigh,et al.  Batching Schnorr Identification Scheme with Applications to Privacy-Preserving Authorization and Low-Bandwidth Communication Devices , 2004, ASIACRYPT.

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

[28]  Hovav Shacham,et al.  Short Signatures from the Weil Pairing , 2001, J. Cryptol..

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

[30]  Dan Boneh,et al.  Homomorphic Signatures for Polynomial Functions , 2011, EUROCRYPT.

[31]  Tibor Jager,et al.  Short Signatures From Weaker Assumptions , 2011, IACR Cryptol. ePrint Arch..

[32]  Yevgeniy Vahlis,et al.  Verifiable Delegation of Computation over Large Datasets , 2011, IACR Cryptol. ePrint Arch..

[33]  Bogdan Warinschi,et al.  Efficient Network Coding Signatures in the Standard Model , 2012, Public Key Cryptography.

[34]  T. Ho,et al.  On Linear Network Coding , 2010 .

[35]  Tal Rabin.  Advances in Cryptology - CRYPTO 2010, 30th Annual Cryptology Conference, Santa Barbara, CA, USA, August 15-19, 2010. Proceedings , 2010, CRYPTO.

[36]  Adi Shamir,et al.  On the generation of cryptographically strong pseudorandom sequences , 1981, TOCS.

[37]  Shafi Goldwasser,et al.  Proceedings : 35th Annual Symposium on Foundations of Computer Science, November 20-22, 1994, Santa Fe, New Mexico , 1994 .

[38]  David Cash,et al.  Bonsai Trees, or How to Delegate a Lattice Basis , 2010, Journal of Cryptology.

[39]  Vinod Vaikuntanathan,et al.  How to Delegate and Verify in Public: Verifiable Computation from Attribute-based Encryption , 2012, IACR Cryptol. ePrint Arch..

[40]  Mihir Bellare,et al.  Fast Batch Verification for Modular Exponentiation and Digital Signatures , 1998, IACR Cryptol. ePrint Arch..

[41]  NaorMoni,et al.  Number-theoretic constructions of efficient pseudo-random functions , 2004 .

[42]  Aggelos Kiayias,et al.  Multi-query Computationally-Private Information Retrieval with Constant Communication Rate , 2010, Public Key Cryptography.

[43]  Ivan Damgård,et al.  Linear zero-knowledge—a note on efficient zero-knowledge proofs and arguments , 1997, STOC '97.

[44]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

[45]  Hugo Krawczyk,et al.  Advances in Cryptology - CRYPTO '98 , 1998 .

[46]  Ivan Damgård,et al.  Zero-Knowledge Proofs for Finite Field Arithmetic; or: Can Zero-Knowledge be for Free? , 1998, CRYPTO.

[47]  Rudolf Ahlswede,et al.  Network information flow , 2000, IEEE Trans. Inf. Theory.

[48]  Bart Preneel,et al.  Topics in Cryptology — CT-RSA 2002 , 2002, Lecture Notes in Computer Science.

[49]  Shuo-Yen Robert Li,et al.  Linear network coding , 2003, IEEE Trans. Inf. Theory.

[50]  Craig Gentry,et al.  Fully homomorphic encryption using ideal lattices , 2009, STOC '09.

[51]  David Mandell Freeman,et al.  Improved Security for Linearly Homomorphic Signatures: A Generic Framework , 2012, Public Key Cryptography.

[52]  Alptekin Küpçü,et al.  Incentivizing outsourced computation , 2008, NetEcon '08.

[53]  Rosario Gennaro,et al.  Improved Publicly Verifiable Delegation of Large Polynomials and Matrix Computations , 2012, IACR Cryptol. ePrint Arch..

[54]  Silvio Micali,et al.  The knowledge complexity of interactive proof-systems , 1985, STOC '85.

[55]  Bogdan Warinschi,et al.  Adaptive Pseudo-Free Groups and Applications , 2011, IACR Cryptol. ePrint Arch..

[56]  Kenneth G. Paterson.  Advances in Cryptology - EUROCRYPT 2011 - 30th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Tallinn, Estonia, May 15-19, 2011. Proceedings , 2011, EUROCRYPT.

[57]  Pil Joong Lee,et al.  Advances in Cryptology — ASIACRYPT 2001 , 2001, Lecture Notes in Computer Science.

[58]  Moni Naor,et al.  Number-theoretic constructions of efficient pseudo-random functions , 2004, JACM.

[59]  Elaine Shi,et al.  Signatures of Correct Computation , 2013, TCC.

[60]  Craig Gentry,et al.  Trapdoors for hard lattices and new cryptographic constructions , 2008, IACR Cryptol. ePrint Arch..

[61]  Yael Tauman Kalai,et al.  Delegating computation: interactive proofs for muggles , 2008, STOC.

[62]  Elisabeth Oswald,et al.  A Comprehensive Evaluation of Mutual Information Analysis Using a Fair Evaluation Framework , 2011, CRYPTO.

[63]  Bennet S. Yee,et al.  Using Secure Coprocessors , 1994 .

[64]  Elaine Shi,et al.  Publicly Verifiable Delegation of Computation , 2011, IACR Cryptol. ePrint Arch..

[65]  Yevgeniy Dodis,et al.  On the Instantiability of Hash-and-Sign RSA Signatures , 2012, TCC.