An Algebraic Framework for Diffie–Hellman Assumptions

We put forward a new algebraic framework to generalize and analyze Diffie–Hellman like decisional assumptions which allows us to argue about security and applications by considering only algebraic properties. Our D ℓ , k - MDDH Assumption states that it is hard to decide whether a vector in G ℓ is l...

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Vydáno v:Journal of cryptology Ročník 30; číslo 1; s. 242 - 288
Hlavní autoři: Escala, Alex, Herold, Gottfried, Kiltz, Eike, Ràfols, Carla, Villar, Jorge
Médium: Journal Article Publikace
Jazyk:angličtina
Vydáno: New York Springer US 01.01.2017
Springer Nature B.V
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ISSN:0933-2790, 1432-1378
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Abstract We put forward a new algebraic framework to generalize and analyze Diffie–Hellman like decisional assumptions which allows us to argue about security and applications by considering only algebraic properties. Our D ℓ , k - MDDH Assumption states that it is hard to decide whether a vector in G ℓ is linearly dependent of the columns of some matrix in G ℓ × k sampled according to distribution D ℓ , k . It covers known assumptions such as DDH , 2 - Lin (Linear Assumption) and k - Lin (the k -Linear Assumption). Using our algebraic viewpoint, we can relate the generic hardness of our assumptions in m -linear groups to the irreducibility of certain polynomials which describe the output of D ℓ , k . We use the hardness results to find new distributions for which the D ℓ , k - MDDH Assumption holds generically in m -linear groups. In particular, our new assumptions 2 - SCasc and 2 - ILin are generically hard in bilinear groups and, compared to 2 - Lin , have shorter description size, which is a relevant parameter for efficiency in many applications. These results support using our new assumptions as natural replacements for the 2 - Lin assumption which was already used in a large number of applications. To illustrate the conceptual advantages of our algebraic framework, we construct several fundamental primitives based on any MDDH Assumption. In particular, we can give many instantiations of a primitive in a compact way, including public-key encryption, hash proof systems, pseudo-random functions, and Groth–Sahai NIZK and NIWI proofs. As an independent contribution, we give more efficient NIZK and NIWI proofs for membership in a subgroup of G ℓ . The results imply very significant efficiency improvements for a large number of schemes.
AbstractList We put forward a new algebraic framework to generalize and analyze Diffie-Hellman like Decisional Assumptions which allows us to argue about security and applications by considering only algebraic properties. Our D`,k-MDDH assumption states that it is hard to decide whether a vector in ¿ìs linearly dependent of the columns of some matrix in ¿`×k sampled according to distribution D`,k. It covers known assumptions such as DDH, 2-Lin (linear assumption), and k-Lin (the k-linear assumption). Using our algebraic viewpoint, we can relate the generic hardness of our assumptions in m-linear groups to the irreducibility of certain polynomials which describe the output of D`,k. We use the hardness results to find new distributions for which the D`,k-MDDH-Assumption holds generically in m-linear groups. In particular, our new assumptions 2-SCasc and 2-ILin are generically hard in bilinear groups and, compared to 2-Lin, have shorter description size, which is a relevant parameter for efficiency in many applications. These results support using our new assumptions as natural replacements for the 2-Lin Assumption which was already used in a large number of applications. To illustrate the conceptual advantages of our algebraic framework, we construct several fundamental primitives based on any MDDH-Assumption. In particular, we can give many instantiations of a primitive in a compact way, including public-key encryption, hash-proof systems, pseudo-random functions, and Groth-Sahai NIZK and NIWI proofs. As an independent contribution we give more efficient NIZK and NIWI proofs for membership in a subgroup of ¿` . The results imply very significant efficiency improvements for a large number of schemes. Peer Reviewed
We put forward a new algebraic framework to generalize and analyze Diffie–Hellman like decisional assumptions which allows us to argue about security and applications by considering only algebraic properties. Our D ℓ , k - MDDH Assumption states that it is hard to decide whether a vector in G ℓ is linearly dependent of the columns of some matrix in G ℓ × k sampled according to distribution D ℓ , k . It covers known assumptions such as DDH , 2 - Lin (Linear Assumption) and k - Lin (the k -Linear Assumption). Using our algebraic viewpoint, we can relate the generic hardness of our assumptions in m -linear groups to the irreducibility of certain polynomials which describe the output of D ℓ , k . We use the hardness results to find new distributions for which the D ℓ , k - MDDH Assumption holds generically in m -linear groups. In particular, our new assumptions 2 - SCasc and 2 - ILin are generically hard in bilinear groups and, compared to 2 - Lin , have shorter description size, which is a relevant parameter for efficiency in many applications. These results support using our new assumptions as natural replacements for the 2 - Lin assumption which was already used in a large number of applications. To illustrate the conceptual advantages of our algebraic framework, we construct several fundamental primitives based on any MDDH Assumption. In particular, we can give many instantiations of a primitive in a compact way, including public-key encryption, hash proof systems, pseudo-random functions, and Groth–Sahai NIZK and NIWI proofs. As an independent contribution, we give more efficient NIZK and NIWI proofs for membership in a subgroup of G ℓ . The results imply very significant efficiency improvements for a large number of schemes.
We put forward a new algebraic framework to generalize and analyze Diffie–Hellman like decisional assumptions which allows us to argue about security and applications by considering only algebraic properties. Our Dℓ,k-MDDH Assumption states that it is hard to decide whether a vector in Gℓ is linearly dependent of the columns of some matrix in Gℓ×k sampled according to distribution Dℓ,k. It covers known assumptions such as DDH,2-Lin (Linear Assumption) and k-Lin (the k-Linear Assumption). Using our algebraic viewpoint, we can relate the generic hardness of our assumptions in m-linear groups to the irreducibility of certain polynomials which describe the output of Dℓ,k. We use the hardness results to find new distributions for which the Dℓ,k-MDDH Assumption holds generically in m-linear groups. In particular, our new assumptions 2-SCasc and 2-ILin are generically hard in bilinear groups and, compared to 2-Lin, have shorter description size, which is a relevant parameter for efficiency in many applications. These results support using our new assumptions as natural replacements for the 2-Lin assumption which was already used in a large number of applications. To illustrate the conceptual advantages of our algebraic framework, we construct several fundamental primitives based on any MDDH Assumption. In particular, we can give many instantiations of a primitive in a compact way, including public-key encryption, hash proof systems, pseudo-random functions, and Groth–Sahai NIZK and NIWI proofs. As an independent contribution, we give more efficient NIZK and NIWI proofs for membership in a subgroup of Gℓ. The results imply very significant efficiency improvements for a large number of schemes.
Author Herold, Gottfried
Villar, Jorge
Escala, Alex
Ràfols, Carla
Kiltz, Eike
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Issue 1
Keywords Diffie–Hellman assumption
Public-key encryption
Groth–Sahai proofs
Hash proof systems
Generic hardness
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D. Boneh, X. Boyen, and E.-J. Goh, Hierarchical identity based encryption with constant size ciphertext. In R. Cramer, editor, EUROCRYPT 2005, vol. 3494 of LNCS, pp. 440–456, Aarhus, Denmark, May 22–26, 2005. Springer, Berlin, Germany
M. Naor and G. Segev, Public-key cryptosystems resilient to key leakage. In S. Halevi, editor, CRYPTO 2009, vol. 5677 of LNCS, pp. 18–35, Santa Barbara, CA, USA, Aug. 16–20, 2009. Springer, Berlin, Germany
A. Joux, A one round protocol for tripartite Diffie–Hellman. Journal of Cryptology, 17(4):263–276, Sept. 2004
D. Cox, J. Little, and D. O’Shea, Ideal, Varieties and Algorithms. Springer, second edition, 1996
R. Cramer and V. Shoup, A practical public key cryptosystem provably secure against adaptive chosen ciphertext attack. In H. Krawczyk, editor, CRYPTO’98, vol. 1462 of LNCS, pp. 13–25, Santa Barbara, CA, USA, Aug. 23–27, 1998. Springer, Berlin, Germany
C. S. Jutla and A. Roy, Switching lemma for bilinear tests and constant-size NIZK proofs for linear subspaces. In J. A. Garay and R. Gennaro, editors, CRYPTO 2014, Part II, vol. 8617 of LNCS, pp. 295–312, Santa Barbara, CA, USA, Aug. 17–21, 2014. Springer, Berlin, Germany
B. Libert, T. Peters, M. Joye, and M. Yung, Non-malleability from malleability: Simulation-sound quasi-adaptive NIZK proofs and CCA2-secure encryption from homomorphic signatures. In P. Q. Nguyen and E. Oswald, editors, EUROCRYPT 2014, vol. 8441 of LNCS, pp. 514–532, Copenhagen, Denmark, May 11–15, 2014. Springer, Berlin, Germany
M. Naor and O. Reingold, Number-theoretic constructions of efficient pseudo-random functions. In 38th FOCS, pp. 458–467, Miami Beach, Florida, Oct. 19–22, 1997. IEEE Computer Society Press
D. Boneh, X. Boyen, and H. Shacham, Short group signatures. In M. Franklin, editor, CRYPTO 2004, vol. 3152 of LNCS, pp. 41–55, Santa Barbara, CA, USA, Aug. 15–19, 2004. Springer, Berlin, Germany
A. B. Lewko, T. Okamoto, A. Sahai, K. Takashima, and B. Waters, Fully secure functional encryption: Attribute-based encryption and (hierarchical) inner product encryption. In H. Gilbert, editor, EUROCRYPT 2010, vol. 6110 of LNCS, pp. 62–91, French Riviera, May 30–June 3, 2010. Springer, Berlin, Germany
A. Escala, G. Herold, E. Kiltz, C. Ràfols, and J. Villar, An algebraic framework for Diffie-Hellman assumptions. In R. Canetti and J. A. Garay, editors, CRYPTO 2013, Part II, vol. 8043 of LNCS, pp. 129–147, Santa Barbara, CA, USA, Aug. 18–22, 2013. Springer, Berlin, Germany
M. Naor and M. Yung, Public-key cryptosystems provably secure against chosen ciphertext attacks. In 22nd ACM STOC, pp. 427–437, Baltimore, Maryland, USA, May 14–16, 1990. ACM Press
J. Katz and V. Vaikuntanathan, Round-optimal password-based authenticated key exchange. In Y. Ishai, editor, TCC 2011, vol. 6597 of LNCS, pp. 293–310, Providence, RI, USA, March 28–30, 2011. Springer, Berlin, Germany
E. Kiltz, A tool box of cryptographic functions related to the Diffie-Hellman function. In C. P. Rangan and C. Ding, editors, INDOCRYPT 2001, vol. 2247 of LNCS, pp. 339–350, Chennai, India, Dec. 16–20, 2001. Springer, Berlin, Germany
S. Wolf, Information-Theoretically and Computionally Secure Key Agreement in Cryptography. Ph.D. thesis, ETH Zuerich, 1999
M. Fischlin, B. Libert, and M. Manulis, Non-interactive and re-usable universally composable string commitments with adaptive security. In D. H. Lee and X. Wang, editors, ASIACRYPT 2011, vol. 7073 of LNCS, pp. 468–485, Seoul, South Korea, Dec. 4–8, 2011. Springer, Berlin, Germany
B. Libert and M. Yung, Non-interactive CCA-secure threshold cryptosystems with adaptive security: New framework and constructions. In R. Cramer, editor, TCC 2012, vol. 7194 of LNCS, pp. 75–93, Taormina, Sicily, Italy, March 19–21, 2012. Springer, Berlin, Germany
Y. Dodis, K. Haralambiev, A. López-Alt, and D. Wichs, Cryptography against continuous memory attacks. In 51st FOCS, pp. 511–520, Las Vegas, Nevada, USA, Oct. 23–26, 2010. IEEE Computer Society Press
J. H. Seo and J. H, Cheon, Beyond the limitation of prime-order bilinear groups, and round optimal blind signatures. In R. Cramer, editor, TCC 2012, vol. 7194 of LNCS, pp. 133–150, Taormina, Sicily, Italy, March 19–21, 2012. Springer, Berlin, Germany
D. Boneh and M. K. Franklin, Identity-based encryption from the Weil pairing. In J. Kilian, editor, CRYPTO 2001, vol. 2139 of LNCS, pp. 213–229, Santa Barbara, CA, USA, Aug. 19–23, 2001. Springer, Berlin, Germany
E. Kiltz, Chosen-ciphertext security from tag-based encryption. In S. Halevi and T. Rabin, editors, TCC 2006, vol. 3876 of LNCS, pp. 581–600, New York, NY, USA, March 4–7, 2006. Springer, Berlin, Germany
J. H. Seo, On the (im)possibility of projecting property in prime-order setting. In X. Wang and K. Sako, editors, ASIACRYPT 2012, vol. 7658 of LNCS, pp. 61–79, Beijing, China, Dec. 2–6, 2012. Springer, Berlin, Germany
X. Boyen, The uber-assumption family (invited talk). In S. D. Galbraith and K. G. Paterson, editors, PAIRING 2008, vol. 5209 of LNCS, pp. 39–56, Egham, UK, Sept. 1–3, 2008. Springer, Berlin, Germany
J. Camenisch, N. Chandran, and V. Shoup, A public key encryption scheme secure against key dependent chosen plaintext and adaptive chosen ciphertext attacks. In A. Joux, editor, EUROCRYPT 2009, vol. 5479 of LNCS, pp. 351–368, Cologne, Germany, April 26–30, 2009. Springer, Berlin, Germany
R. Cramer and V. Shoup, Design and analysis of practical public-key encryption schemes secure against adaptive chosen ciphertext attack. SIAM Journal on Computing, 33(1):167–226, 2003
D. Boneh and A. Silverberg, Applications of multilinear forms to cryptography. Contemporary Mathematics, 324:71–90, 2003
D. Hofheinz and T. Jager, Tightly secure signatures and public-key encryption. In R. Safavi-Naini and R. Canetti, editors, CRYPTO 2012, vol. 7417 of LNCS, pp. 590–607, Santa Barbara, CA, USA, Aug. 19–23, 2012. Springer, Berlin, Germany
D. Galindo, J. Herranz, and J. L. Villar, Identity-based encryption with master key-dependent message security and leakage-resilience. In S. Foresti, M. Yung, and F. Martinelli, editors, ESORICS 2012, vol. 7459 of LNCS, pp. 627–642, Pisa, Italy, Sept. 10–12, 2012. Springer, Berlin, Germany
O. Blazy, D. Pointcheval, and D. Vergnaud, Round-optimal privacy-preserving prot
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References_xml – reference: D. Galindo, J. Herranz, and J. L. Villar, Identity-based encryption with master key-dependent message security and leakage-resilience. In S. Foresti, M. Yung, and F. Martinelli, editors, ESORICS 2012, vol. 7459 of LNCS, pp. 627–642, Pisa, Italy, Sept. 10–12, 2012. Springer, Berlin, Germany
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– reference: M. Naor and O. Reingold, Number-theoretic constructions of efficient pseudo-random functions. In 38th FOCS, pp. 458–467, Miami Beach, Florida, Oct. 19–22, 1997. IEEE Computer Society Press
– reference: S. Meiklejohn, H. Shacham, and D. M. Freeman, Limitations on transformations from composite-order to prime-order groups: The case of round-optimal blind signatures. In M. Abe, editor, ASIACRYPT 2010, vol. 6477 of LNCS, pp. 519–538, Singapore, Dec. 5–9, 2010. Springer, Berlin, Germany
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– reference: R. Cramer and V. Shoup, Design and analysis of practical public-key encryption schemes secure against adaptive chosen ciphertext attack. SIAM Journal on Computing, 33(1):167–226, 2003
– reference: C. S. Jutla and A. Roy, Switching lemma for bilinear tests and constant-size NIZK proofs for linear subspaces. In J. A. Garay and R. Gennaro, editors, CRYPTO 2014, Part II, vol. 8617 of LNCS, pp. 295–312, Santa Barbara, CA, USA, Aug. 17–21, 2014. Springer, Berlin, Germany
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Snippet We put forward a new algebraic framework to generalize and analyze Diffie–Hellman like decisional assumptions which allows us to argue about security and...
We put forward a new algebraic framework to generalize and analyze Diffie-Hellman like Decisional Assumptions which allows us to argue about security and...
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SubjectTerms 11 Number theory
11Y Computational number theory
Algebra
Classificació AMS
Coding and Information Theory
Columns (structural)
Combinatorics
Communications Engineering
Computational Mathematics and Numerical Analysis
Computer Science
Decision analysis
Diffie-Hellman Assumption
Encryption
Functions (mathematics)
Generic Hardness
Groth-Sahai proofs
Hash Proof Systems
Matemàtiques i estadística
Mathematical analysis
Networks
Nombres, Teoria dels
Number theory
Probability Theory and Stochastic Processes
Public-key Encryption
Teoria de nombres
Àlgebra
Àrees temàtiques de la UPC
Title An Algebraic Framework for Diffie–Hellman Assumptions
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https://recercat.cat/handle/2072/267425
Volume 30
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