Abstract
We investigate the hardness of finding solutions to bivariate polynomial congruences modulo RSA composites. We establish necessary conditions for a bivariate polynomial to be one-way, second preimage resistant, and collision resistant based on arithmetic properties of the polynomial. From these conditions we deduce a new computational assumption that implies an efficient algebraic collision-resistant hash function. We explore the assumption and relate it to known computational problems. The assumption leads to (i) a new statistically hiding commitment scheme that composes well with Pedersen commitments, (ii) a conceptually simple cryptographic accumulator, and (iii) an efficient chameleon hash function.
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Aggarwal, D., Maurer, U.: Breaking RSA generically is equivalent to factoring. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 36–53. Springer, Heidelberg (2009)
Ash, A., Gross, R.: Elliptic Tales: Curves, Counting, and Number Theory. Princeton University Press (2012)
Ateniese, G., de Medeiros, B.: Identity-based chameleon hash and applications. In: Juels, A. (ed.) FC 2004. LNCS, vol. 3110, pp. 164–180. Springer, Heidelberg (2004)
Barić, N., Pfitzmann, B.: Collision-free accumulators and fail-stop signature schemes without trees. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 480–494. Springer, Heidelberg (1997)
Benaloh, J.C., de Mare, M.: One-way accumulators: A decentralized alternative to digital signatures. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 274–285. Springer, Heidelberg (1994)
Boneh, D., Gentry, C., Hamburg, M.: Space-efficient identity based encryption without pairings. In: FOCS, pp. 647–657 (2007)
Browkin, J., Brzeziński, J.: Some remarks on the abc-conjecture. Mathematics of Computation 62(206), 931–939 (1994)
Camenisch, J., Lysyanskaya, A.: Dynamic accumulators and application to efficient revocation of anonymous credentials. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 61–76. Springer, Heidelberg (2002)
Camenisch, J., Stadler, M.: Efficient group signature schemes for large groups. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 410–424. Springer, Heidelberg (1997)
Camenisch, J., Stadler, M.: Proof systems for general statements about discrete logarithms. Tech. Rep. 260, Dept. of Computer Science, ETH Zurich (March 1997)
Camenisch, J.: Group Signature Schemes and Payment Systems Based on the Discrete Logarithm Problem. Ph.D. thesis, Swiss Federal Institute of Technology Zürich (ETH Zürich) (1998)
Catalano, D., Gennaro, R., Howgrave-Graham, N., Nguyen, P.Q.: Paillier’s cryptosystem revisited. In: ACM Conference on Computer and Communications Security, pp. 206–214 (2001)
Cornelissen, G.: Stockage diophantien et hypothese abc généralisée. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics 328(1), 3–8 (1999)
Damgård, I.B.: The Application of Claw Free Functions in Cryptography. Ph.D. thesis, Aarhus University (May 1988)
Damgård, I.B.: A design principle for hash functions. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 416–427. Springer, Heidelberg (1990)
Gennaro, R., Halevi, S., Rabin, T.: Secure hash-and-sign signatures without the random oracle. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 123–139. Springer, Heidelberg (1999)
Goldwasser, S., Micali, S., Rivest, R.L.: A digital signature scheme secure against adaptive chosen-message attacks. SIAM Journal on Computing 17(2), 281–308 (1988)
Hindry, M., Silverman, J.H.: Diophantine geometry: an introduction, vol. 201. Springer (2000)
Kilian, J., Petrank, E.: Identity escrow. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 169–185. Springer, Heidelberg (1998)
Krawczyk, H., Rabin, T.: Chameleon hashing and signatures. In: NDSS, pp. 143–154 (2000)
Merkle, R.C.: A digital signature based on a conventional encryption function. In: Pomerance, C. (ed.) CRYPTO 1987. LNCS, vol. 293, pp. 369–378. Springer, Heidelberg (1988)
Merkle, R.C.: One way hash functions and DES. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 428–446. Springer, Heidelberg (1990)
Micali, S., Rabin, M., Kilian, J.: Zero-knowledge sets. In: FOCS, pp. 80–91 (2003)
Miers, I., Garman, C., Green, M., Rubin, A.D.: Zerocoin: Anonymous distributed e-cash from Bitcoin. IEEE Security and Privacy, 397–411 (2013)
Naor, M.: On cryptographic assumptions and challenges. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 96–109. Springer, Heidelberg (2003)
Ong, H., Schnorr, C.P., Shamir, A.: An efficient signature scheme based on quadratic equations. In: STOC, pp. 208–216 (1984)
Pedersen, T.P.: Non-interactive and information-theoretic secure verifiable secret sharing. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 129–140. Springer, Heidelberg (1992)
Pollard, J., Schnorr, C.: An efficient solution of the congruence. IEEE Transactions on Information Theory 33(5), 702–709 (1987)
Poonen, B.: Varieties without extra automorphisms III: hypersurfaces. Finite Fields and their Applications 11(2), 230–268 (2005)
Poonen, B.: Multivariable polynomial injections on rational numbers. arXiv preprint arXiv:0902.3961v2 (June 2010)
Schwenk, J., Eisfeld, J.: Public key encryption and signature schemes based on polynomials over ℤ n . In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 60–71. Springer, Heidelberg (1996)
Shallit, J.: An exposition of Pollard’s algorithm for quadratic congruences (October 1984)
Shamir, A.: On the generation of multivariate polynomials which are hard to factor. In: STOC, pp. 796–804. ACM (1993)
Zagier, D.: Personal communication (June 2014)
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Boneh, D., Corrigan-Gibbs, H. (2014). Bivariate Polynomials Modulo Composites and Their Applications. In: Sarkar, P., Iwata, T. (eds) Advances in Cryptology – ASIACRYPT 2014. ASIACRYPT 2014. Lecture Notes in Computer Science, vol 8873. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45611-8_3
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