Abstract
Consider the following secret-sharing problem: A file s should be distributed between n servers such that (d-1)-subsets cannot recover the file, (d+1)-subsets can recover the file, and d-subsets should be able to recover s if and only if they appear in some pre-defined list L. The goal is to minimize the information ratio—that is, the number of bits stored on a server per each bit of the secret.
We show that for any constant d and any pre-defined list L, if the file is sufficiently long (exponential in nd), the problem can be solved with a constant asymptotic information ratio of cd that does not grow with the number of servers n. This result is based on a new construction of d-party conditional disclosure of secrets for arbitrary predicates over an n-size domain in which each party communicates at most four bits per secret bit.
In both settings, previous results achieved a non-constant information ratio that grows asymptotically with n, even for the simpler special case of d = 2. Moreover, our constructions yield the first example of an access structure whose amortized information ratio is constant, whereas its best-known non-amortized information ratio is sub-exponential, thus providing a unique evidence for the potential power of amortization in the context of secret sharing.
Our main result applies to exponentially long secrets, and so it should be mainly viewed as a barrier against amortizable lower-bound techniques. We also show that in some natural simple cases (e.g., low-degree predicates), amortization kicks in even for quasi-polynomially long secrets. Finally, we prove some limited lower bounds and point out some limitations of existing lower-bound techniques.
- William Aiello, Yuval Ishai, and Omer Reingold. 2001. Priced oblivious transfer: How to sell digital goods. In Advances in Cryptology—EUROCRYPT 2001. Lecture Notes in Computer Science, Vol. 20145. Springer, 119--135. DOI:https://doi.org/10.1007/3-540-44987-6_8Google Scholar
Cross Ref
- Benny Applebaum, Barak Arkis, Pavel Raykov, and Prashant Nalini Vasudevan. 2017. Conditional disclosure of secrets: Amplification, closure, amortization, lower-bounds, and separations. In Advances in Cryptology—CRYPTO 2017. Lecture Notes in Computer Science, Vol. 10401. Springer, 727--757. DOI:https://doi.org/10.1007/978-3-319-63688-7_24Google Scholar
Cross Ref
- Benny Applebaum, Amos Beimel, Oriol Farràs, Oded Nir, and Naty Peter. 2019. Secret-sharing schemes for general and uniform access structures. In Advances in Cryptology—EUROCRYPT 2019. Lecture Notes in Computer Science, Vol. 11478. Springer, 441--471. DOI:https://doi.org/10.1007/978-3-030-17659-4_15Google Scholar
Cross Ref
- Benny Applebaum, Thomas Holenstein, Manoj Mishra, and Ofer Shayevitz. 2020. The communication complexity of private simultaneous messages, revisited. Journal of Cryptology 33, 3 (2020), 917--953. DOI:https://doi.org/10.1007/s00145-019-09334-yGoogle Scholar
Cross Ref
- Amos Beimel. 2011. Secret-sharing schemes: A survey. In Coding and Cryptology. Lecture Notes in Computer Science, Vol. 6639. Springer, 11--46. DOI:https://doi.org/10.1007/978-3-642-20901-7_2Google Scholar
- Amos Beimel, Oriol Farràs, and Yuval Mintz. 2016. Secret-sharing schemes for very dense graphs. Journal of Cryptology 29, 2 (2016), 336--362. DOI:https://doi.org/10.1007/s00145-014-9195-8Google Scholar
Digital Library
- Amos Beimel, Oriol Farràs, Yuval Mintz, and Naty Peter. 2017. Linear secret-sharing schemes for forbidden graph access structures. In Theory of Cryptography. Lecture Notes in Computer Science, Vol. 10678. Springer, 394--423. DOI:https://doi.org/10.1007/978-3-319-70503-3_13Google Scholar
- Amos Beimel and Yuval Ishai. 2001. On the power of nonlinear secrect-sharing. In Proceedings of the 16th Annual IEEE Conference on Computational Complexity. IEEE, Los Alamitos, CA, 188--202. DOI:https://doi.org/10.1109/CCC.2001.933886Google Scholar
Digital Library
- Amos Beimel, Yuval Ishai, Ranjit Kumaresan, and Eyal Kushilevitz. 2014. On the cryptographic complexity of the worst functions. In Theory of Cryptography. Lecture Notes in Computer Science, Vol. 8349. Springer, 317--342. DOI:https://doi.org/10.1007/978-3-642-54242-8_14Google Scholar
- Amos Beimel, Eyal Kushilevitz, and Pnina Nissim. 2018. The complexity of multiparty PSM protocols and related models. In Advances in Cryptology—EUROCRYPT 2018. Lecture Notes in Computer Science, Vol. 10821. Springer, 287--318. DOI:https://doi.org/10.1007/978-3-319-78375-8_10Google Scholar
Cross Ref
- Amos Beimel and Ilan Orlov. 2011. Secret sharing and non-Shannon information inequalities. IEEE Transactions on Information Theory 57, 9 (2011), 5634--5649. DOI:https://doi.org/10.1109/TIT.2011.2162183Google Scholar
Digital Library
- Josh Cohen Benaloh and Jerry Leichter. 1988. Generalized secret sharing and monotone functions. In Advances in Cryptology—CRYPTO 1988. Lecture Notes in Computer Science, Vol. 403. Springer, 27--35. DOI:https://doi.org/10.1007/0-387-34799-2_3Google Scholar
- G. R. Blakley. 1979. Safeguarding cryptographic keys. In Proceedings of the AFIPS 1979 National Computer Conference. 313--317.Google Scholar
Cross Ref
- Andrej Bogdanov, Siyao Guo, and Ilan Komargodski. 2016. Threshold secret sharing requires a linear size alphabet. In Theory of Cryptography. Lecture Notes in Computer Science, Vol. 9986. Springer, 471--484. DOI:https://doi.org/10.1007/978-3-662-53644-5_18Google Scholar
- Renato M. Capocelli, Alfredo De Santis, Luisa Gargano, and Ugo Vaccaro. 1993. On the size of shares for secret sharing schemes. Journal of Cryptology 6, 3 (1993), 157--167. DOI:https://doi.org/10.1007/BF00198463Google Scholar
Digital Library
- Benny Chor, Eyal Kushilevitz, Oded Goldreich, and Madhu Sudan. 1998. Private information retrieval. Journal of the ACM 45, 6 (1998), 965--981. DOI:https://doi.org/10.1145/293347.293350Google Scholar
Digital Library
- László Csirmaz. 1997. The size of a share must be large. Journal of Cryptology 10, 4 (1997), 223--231. DOI:https://doi.org/10.1007/s001459900029Google Scholar
Digital Library
- Romain Gay, Iordanis Kerenidis, and Hoeteck Wee. 2015. Communication complexity of conditional disclosure of secrets and attribute-based encryption. In Advances in Cryptology—CRYPTO 2015. Lecture Notes in Computer Science, Vol. 9216. Springer, 485--502. DOI:https://doi.org/10.1007/978-3-662-48000-7_24Google Scholar
Cross Ref
- Yael Gertner, Yuval Ishai, Eyal Kushilevitz, and Tal Malkin. 2000. Protecting data privacy in private information retrieval schemes. Journal of Computer and System Sciences 60, 3 (2000), 592--629. DOI:https://doi.org/10.1006/jcss.1999.1689Google Scholar
Digital Library
- Vipul Goyal, Omkant Pandey, Amit Sahai, and Brent Waters. 2006. Attribute-based encryption for fine-grained access control of encrypted data. In Proceedings of the 13th ACM Conference on Computer and Communications Security (CCS’06). ACM, New York, NY, 89--98. DOI:https://doi.org/10.1145/1180405.1180418Google Scholar
Digital Library
- M. Ito, A. Saito, and T. Nishizeki. 1987. Secret sharing scheme realizing general access structure. In Proceedings of the 1987 IEEE GLOBECOM Conference. IEEE, Los Alamitos, CA, 99--102.Google Scholar
- Mauricio Karchmer and Avi Wigderson. 1993. On span programs. In Proceedings of the 8th Annual Structure in Complexity Theory Conference. IEEE, Los Alamitos, CA, 102--111. DOI:https://doi.org/10.1109/SCT.1993.336536Google Scholar
Cross Ref
- Ehud D. Karnin, J. W. Greene, and Martin E. Hellman. 1983. On secret sharing systems. IEEE Transactions on Information Theory 29, 1 (1983), 35--41. DOI:https://doi.org/10.1109/TIT.1983.1056621Google Scholar
Digital Library
- Jonathan Katz and Hovav Shacham (Eds.). 2017. Advances in Cryptology—CRYPTO 2017: 37th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 20--24, 2017, Proceedings, Part I. Lecture Notes in Computer Science, Vol. 10401. Springer. DOI:https://doi.org/10.1007/978-3-319-63688-7Google Scholar
- Tianren Liu and Vinod Vaikuntanathan. 2018. Breaking the circuit-size barrier in secret sharing. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC’18). https://eprint.iacr.org/2018/333.Google Scholar
Digital Library
- Tianren Liu, Vinod Vaikuntanathan, and Hoeteck Wee. 2017. Conditional disclosure of secrets via non-linear reconstruction. In Advances in Cryptology—CRYPTO 2017. Lecture Notes in Computer Science, Vol. 10401. Springer, 758--790. DOI:https://doi.org/10.1007/978-3-319-63688-7_25Google Scholar
Cross Ref
- Tianren Liu, Vinod Vaikuntanathan, and Hoeteck Wee. 2018. Towards breaking the exponential barrier for general secret sharing. In Advances in Cryptology—EUROCRYPT 2018. Lecture Notes in Computer Science, Vol. 10820. Springer, 567--596. DOI:https://doi.org/10.1007/978-3-319-78381-9_21Google Scholar
Cross Ref
- Yuval Mintz. 2012. Information Ratios of Graph Secret-Sharing Schemes. Master’s Thesis. Department of Computer Science, Ben Gurion University.Google Scholar
- Sebastià Martín Molleví, Carles Padró, and An Yang. 2016. Secret sharing, rank inequalities, and information inequalities. IEEE Transactions on Information Theory 62, 1 (2016), 599--609. DOI:https://doi.org/10.1109/TIT.2015.2500232Google Scholar
Digital Library
- Amit Sahai and Brent Waters. 2005. Fuzzy identity-based encryption. In Advances in Cryptology—EUROCRYPT 2005. Lecture Notes in Computer Science, Vol. 3494. Springer, 457--473. DOI:https://doi.org/10.1007/11426639_27Google Scholar
Digital Library
- Adi Shamir. 1979. How to share a secret. Communications of the ACM 22, 11 (1979), 612--613. DOI:https://doi.org/10.1145/359168.359176Google Scholar
Digital Library
- Douglas R. Stinson. 1994. Decomposition constructions for secret-sharing schemes. IEEE Transactions on Information Theory 40, 1 (1994), 118--125. DOI:https://doi.org/10.1109/18.272461Google Scholar
Digital Library
- Hung-Min Sun and Shiuh-Pyng Shieh. 1997. Secret sharing in graph-based prohibited structures. In Proceedings of the 16th Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM’97). IEEE, Los Alamitos, CA, 718--724. http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=4979.Google Scholar
- Vinod Vaikuntanathan and Prashant Nalini Vasudevan. 2015. Secret sharing and statistical zero knowledge. In Advances in Cryptology—ASIACRYPT 2015. Lecture Notes in Computer Science, Vol. 9452. Springer, 656--680. DOI:https://doi.org/10.1007/978-3-662-48797-6_27Google Scholar
Digital Library
Index Terms
On the Power of Amortization in Secret Sharing: d-Uniform Secret Sharing and CDS with Constant Information Rate
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