research-article

Arithmetic Cryptography: Extended Abstract

Authors:

Benny Applebaum,

Jonathan Avron,

Christina BrzuskaAuthors Info & Claims

ITCS '15: Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science

Pages 143 - 151

https://doi.org/10.1145/2688073.2688114

Published: 11 January 2015 Publication History

Abstract

We study the possibility of computing cryptographic primitives in a fully-black-box arithmetic model over a finite field $\F$. In this model, the input to a cryptographic primitive (e.g., encryption scheme) is given as a sequence of field elements, the honest parties are implemented by arithmetic circuits which make only a black-box use of the underlying field, and the adversary has a full (non-black-box) access to the field. This model captures many standard information-theoretic constructions.

We prove several positive and negative results in this model for various cryptographic tasks. On the positive side, we show that, under reasonable assumptions, computational primitives like commitment schemes, public-key encryption, oblivious transfer, and general secure two-party computation can be implemented in this model. On the negative side, we prove that garbled circuits, homomorphic encryption, and secure computation with low online complexity cannot be achieved in this model. Our results reveal a qualitative difference between the standard model and the arithmetic model, and explain, in retrospect, some of the limitations of previous constructions.

References

[1]

S. Aaronson and A. Wigderson. Algebrization: a new barrier in complexity theory. In R. E. Ladner and C. Dwork, editors, 40th ACM STOC, pages 731--740. ACM Press, May 2008.

Digital Library

[2]

D. Aggarwal and U. Maurer. Breaking RSA generically is equivalent to factoring. In A. Joux, editor, EUROCRYPT 2009, volume 5479 of LNCS, pages 36--53. Springer, Apr. 2009.

Digital Library

[3]

M. Alekhnovich. More on average case vs approximation complexity. In 44th FOCS, pages 298--307. IEEE Computer Society Press, Oct. 2003.

Digital Library

[4]

B. Applebaum, Y. Ishai, and E. Kushilevitz. Cryptography by cellular automata or how fast can complexity emerge in nature? In A. C.-C. Yao, editor, ICS 2010, pages 1--19. Tsinghua University Press, Jan. 2010.

[5]

B. Applebaum, Y. Ishai, and E. Kushilevitz. How to garble arithmetic circuits. In R. Ostrovsky, editor, 52nd FOCS, pages 120--129. IEEE Computer Society Press, Oct. 2011.

Digital Library

[6]

B. Applebaum, Y. Ishai, E. Kushilevitz, and B. Waters. Encoding functions with constant online rate or how to compress garbled circuits keys. In R. Canetti and J. A. Garay, editors, CRYPTO 2013, Part II, volume 8043 of LNCS, pages 166--184. Springer, Aug. 2013.

[7]

Baker, Gill, and Solovay. Relativizations of the P =? NP question. SICOMP: SIAM Journal on Computing, 4, 1975.

[8]

D. Beaver. Precomputing oblivious transfer. In D. Coppersmith, editor, CRYPTO'95, volume 963 of LNCS, pages 97--109. Springer, Aug. 1995.

Digital Library

[9]

M. Bellare, V. T. Hoang, and P. Rogaway. Foundations of garbled circuits. In T. Yu, G. Danezis, and V. D. Gligor, editors, ACM CCS 12, pages 784--796. ACM Press, Oct. 2012.

Digital Library

[10]

M. Ben-Or, S. Goldwasser, and A. Wigderson. Completeness theorems for non-cryptographic fault-tolerant distributed computation (extended abstract). In 20th ACM STOC, pages 1--10. ACM Press, May 1988.

Digital Library

[11]

R. Bendlin, I. Damgård, C. Orlandi, and S. Zakarias. Semi-homomorphic encryption and multiparty computation. In K. G. Paterson, editor, EUROCRYPT 2011, volume 6632 of LNCS, pages 169--188. Springer, May 2011.

Digital Library

[12]

A. Blum, M. L. Furst, M. J. Kearns, and R. J. Lipton. Cryptographic primitives based on hard learning problems. In D. R. Stinson, editor, CRYPTO'93, volume 773 of LNCS, pages 278--291. Springer, Aug. 1993.

Digital Library

[13]

M. Blum and S. Micali. How to generate cryptographically strong sequences of pseudo random bits. In 23rd FOCS, pages 112--117. IEEE Computer Society Press, Nov. 1982.

Digital Library

[14]

Z. Brakerski and V. Vaikuntanathan. Efficient fully homomorphic encryption from (standard) LWE. In R. Ostrovsky, editor, 52nd FOCS, pages 97--106. IEEE Computer Society Press, Oct. 2011.

Digital Library

[15]

D. Chaum, C. Crépeau, and I. Damg\r ard. Multiparty unconditionally secure protocols (extended abstract). In 20th ACM STOC, pages 11--19. ACM Press, May 1988.

Digital Library

[16]

R. Cramer, I. Damgård, and J. B. Nielsen. Multiparty computation from threshold homomorphic encryption. In B. Pfitzmann, editor, EUROCRYPT 2001, volume 2045 of LNCS, pages 280--299. Springer, May 2001.

Digital Library

[17]

R. Cramer and S. Fehr. Optimal black-box secret sharing over arbitrary Abelian groups. In M. Yung, editor, CRYPTO 2002, volume 2442 of LNCS, pages 272--287. Springer, Aug. 2002.

Digital Library

[18]

R. Cramer, S. Fehr, Y. Ishai, and E. Kushilevitz. Efficient multi-party computation over rings. In E. Biham, editor, EUROCRYPT 2003, volume 2656 of LNCS, pages 596--613. Springer, May 2003.

Digital Library

[19]

I. Damgård, V. Pastro, N. P. Smart, and S. Zakarias. Multiparty computation from somewhat homomorphic encryption. In R. Safavi-Naini and R. Canetti, editors, CRYPTO 2012, volume 7417 of LNCS, pages 643--662. Springer, Aug. 2012.

[20]

A. W. Dent. Adapting the weaknesses of the random oracle model to the generic group model. In Y. Zheng, editor, ASIACRYPT 2002, volume 2501 of LNCS, pages 100--109. Springer, Dec. 2002.

Digital Library

[21]

Y. Desmedt and Y. Frankel. Shared generation of authenticators and signatures (extended abstract). In J. Feigenbaum, editor, CRYPTO'91, volume 576 of LNCS, pages 457--469. Springer, Aug. 1991.

Digital Library

[22]

Z. Dvir, A. Gabizon, and A. Wigderson. Extractors and rank extractors for polynomial sources. In 48th FOCS, pages 52--62. IEEE Computer Society Press, Oct. 2007.

Digital Library

[23]

T. ElGamal. A public key cryptosystem and a signature scheme based on discrete logarithms. In G. R. Blakley and D. Chaum, editors, CRYPTO'84, volume 196 of LNCS, pages 10--18. Springer, Aug. 1984.

Digital Library

[24]

U. Feige, J. Kilian, and M. Naor. A minimal model for secure computation (extended abstract). In 26th ACM STOC, pages 554--563. ACM Press, May 1994.

Digital Library

[25]

M. K. Franklin and S. Haber. Joint encryption and message-efficient secure computation. In D. R. Stinson, editor, CRYPTO'93, volume 773 of LNCS, pages 266--277. Springer, Aug. 1993.

Digital Library

[26]

S. Garg, C. Gentry, and S. Halevi. Candidate multilinear maps from ideal lattices. In T. Johansson and P. Q. Nguyen, editors, EUROCRYPT 2013, volume 7881 of LNCS, pages 1--17. Springer, May 2013.

[27]

C. Gentry. Fully homomorphic encryption using ideal lattices. In M. Mitzenmacher, editor, 41st ACM STOC, pages 169--178. ACM Press, May / June 2009.

Digital Library

[28]

H. Gilbert, M. J. B. Robshaw, and Y. Seurin. How to encrypt with the LPN problem. In L. Aceto, I. Damgård, L. A. Goldberg, M. M. Halldórsson, A. Ingólfsdóttir, and I. Walukiewicz, editors, ICALP 2008, Part II, volume 5126 of LNCS, pages 679--690. Springer, July 2008.

Digital Library

[29]

O. Goldreich, S. Goldwasser, and S. Micali. How to construct random functions. Journal of the ACM, 33:792--807, 1986.

Digital Library

[30]

O. Goldreich, H. Krawczyk, and M. Luby. On the existence of pseudorandom generators. In S. Goldwasser, editor, CRYPTO'88, volume 403 of LNCS, pages 146--162. Springer, Aug. 1988.

Digital Library

[31]

O. Goldreich and L. A. Levin. A hard-core predicate for all one-way functions. In 21st ACM STOC, pages 25--32. ACM Press, May 1989.

Digital Library

[32]

O. Goldreich, S. Micali, and A. Wigderson. How to play any mental game or A completeness theorem for protocols with honest majority. In A. Aho, editor, 19th ACM STOC, pages 218--229. ACM Press, May 1987.

Digital Library

[33]

S. Goldwasser and S. Micali. Probabilistic encryption. Journal of Computer and System Sciences, 28(2):270--299, 1984.

[34]

J. Håstad, R. Impagliazzo, L. A. Levin, and M. Luby. A pseudorandom generator from any one-way function. SIAM Journal on Computing, 28(4):1364--1396, 1999.

Digital Library

[35]

Y. Ishai. Randomization techniques for secure computation. In M. Prabhakaran and A. Sahai, editors, Secure Multi-Party Computation, volume 10 of Cryptology and Information Security Series, pages 222--248. IOS press, Amsterdam, 2012.

[36]

Y. Ishai and E. Kushilevitz. Randomizing polynomials: A new representation with applications to round-efficient secure computation. In 41st FOCS, pages 294--304. IEEE Computer Society Press, Nov. 2000.

Digital Library

[37]

Y. Ishai, E. Kushilevitz, S. Meldgaard, C. Orlandi, and A. Paskin-Cherniavsky. On the power of correlated randomness in secure computation. In A. Sahai, editor, TCC 2013, volume 7785 of LNCS, pages 600--620. Springer, Mar. 2013.

Digital Library

[38]

Y. Ishai, M. Prabhakaran, and A. Sahai. Founding cryptography on oblivious transfer - efficiently. In D. Wagner, editor, CRYPTO 2008, volume 5157 of LNCS, pages 572--591. Springer, Aug. 2008.

Digital Library

[39]

Y. Ishai, M. Prabhakaran, and A. Sahai. Secure arithmetic computation with no honest majority. In O. Reingold, editor, TCC 2009, volume 5444 of LNCS, pages 294--314. Springer, Mar. 2009.

Digital Library

[40]

E. Kiltz, K. Pietrzak, D. Cash, A. Jain, and D. Venturi. Efficient authentication from hard learning problems. In K. G. Paterson, editor, EUROCRYPT 2011, volume 6632 of LNCS, pages 7--26. Springer, May 2011.

Digital Library

[41]

U. M. Maurer. Abstract models of computation in cryptography (invited paper). In N. P. Smart, editor, 10th IMA International Conference on Cryptography and Coding, volume 3796 of LNCS, pages 1--12. Springer, Dec. 2005.

Digital Library

[42]

U. M. Maurer and S. Wolf. Lower bounds on generic algorithms in groups. In K. Nyberg, editor, EUROCRYPT'98, volume 1403 of LNCS, pages 72--84. Springer, May / June 1998.

[43]

M. Naor and B. Pinkas. Oblivious transfer and polynomial evaluation. In 31st ACM STOC, pages 245--254. ACM Press, May 1999.

Digital Library

[44]

K. Pietrzak. Cryptography from learning parity with noise. In SOFSEM 2012: Theory and Practice of Computer Science - 38th Conference on Current Trends in Theory and Practice of Computer Science, Spindleruv Mlýn, Czech Republic, January 21--27, 2012. Proceedings, pages 99--114, 2012.

Digital Library

[45]

A. A. Razborov and S. Rudich. Natural proofs. In 26th ACM STOC, pages 204--213. ACM Press, May 1994.

Digital Library

[46]

O. Regev. On lattices, learning with errors, random linear codes, and cryptography. In H. N. Gabow and R. Fagin, editors, 37th ACM STOC, pages 84--93. ACM Press, May 2005.

Digital Library

[47]

A. Shamir. How to share a secret. Communications of the Association for Computing Machinery, 22(11):612--613, Nov. 1979.

Digital Library

[48]

V. Shoup. Lower bounds for discrete logarithms and related problems. In W. Fumy, editor, EUROCRYPT'97, volume 1233 of LNCS, pages 256--266. Springer, May 1997.

Digital Library

[49]

A. C.-C. Yao. Theory and applications of trapdoor functions (extended abstract). In 23rd FOCS, pages 80--91. IEEE Computer Society Press, Nov. 1982.

[50]

A. C.-C. Yao. How to generate and exchange secrets (extended abstract). In 27th FOCS, pages 162--167. IEEE Computer Society Press, Oct. 1986.

Digital Library

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Index Terms

Arithmetic Cryptography: Extended Abstract
1. Theory of computation
  1. Models of computation
    1. Interactive computation

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cover image ACM Conferences

ITCS '15: Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science

January 2015

404 pages

ISBN:9781450333337

DOI:10.1145/2688073

Program Chair:
Tim Roughgarden
Stanford University

Copyright © 2015 ACM.

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Published: 11 January 2015

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Israel Science Foundation
German-Israeli Foundation for Scientific Research and Development
Israel Ministry of Science and Technology

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ITCS'15

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ITCS'15: Innovations in Theoretical Computer Science

January 11 - 13, 2015

Rehovot, Israel

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ITCS '15 Paper Acceptance Rate 45 of 159 submissions, 28%;

Overall Acceptance Rate 172 of 513 submissions, 34%

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Cited By

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https://dl.acm.org/doi/10.1007/978-3-031-58723-8_9
Beck GGoel AHegde AJain AJin ZKaptchuk GMeng WJensen CCremers CKirda E(2023)Scalable Multiparty GarblingProceedings of the 2023 ACM SIGSAC Conference on Computer and Communications Security10.1145/3576915.3623132(2158-2172)Online publication date: 15-Nov-2023
https://dl.acm.org/doi/10.1145/3576915.3623132
Ball MLi HLin HLiu T(2023)New Ways to Garble Arithmetic CircuitsAdvances in Cryptology – EUROCRYPT 202310.1007/978-3-031-30617-4_1(3-34)Online publication date: 15-Apr-2023
https://doi.org/10.1007/978-3-031-30617-4_1
Jain ALin HSahai A(2022)Indistinguishability Obfuscation from LPN over $$\mathbb {F}_p$$, DLIN, and PRGs in NC$$^0$$Advances in Cryptology – EUROCRYPT 202210.1007/978-3-031-06944-4_23(670-699)Online publication date: 25-May-2022
https://doi.org/10.1007/978-3-031-06944-4_23
Jain ALin HSahai AKhuller SVassilevska Williams V(2021)Indistinguishability obfuscation from well-founded assumptionsProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451093(60-73)Online publication date: 15-Jun-2021
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Yu YZhang J(2021)Smoothing Out Binary Linear Codes and Worst-Case Sub-exponential Hardness for LPNAdvances in Cryptology – CRYPTO 202110.1007/978-3-030-84252-9_16(473-501)Online publication date: 16-Aug-2021
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Boyle ECouteau GGilboa NIshai YKohl LScholl P(2020)Correlated Pseudorandom Functions from Variable-Density LPN2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS46700.2020.00103(1069-1080)Online publication date: Nov-2020
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Fleischhacker NMalavolta GSchröder D(2019)Arithmetic Garbling from Bilinear MapsComputer Security – ESORICS 201910.1007/978-3-030-29962-0_9(172-192)Online publication date: 15-Sep-2019
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Bartusek JLepoint TMa FZhandry M(2019)New Techniques for Obfuscating ConjunctionsAdvances in Cryptology – EUROCRYPT 201910.1007/978-3-030-17659-4_22(636-666)Online publication date: 24-Apr-2019
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Applebaum BDamgård IIshai YNielsen MZichron L(2017)Secure Arithmetic Computation with Constant Computational OverheadAdvances in Cryptology – CRYPTO 201710.1007/978-3-319-63688-7_8(223-254)Online publication date: 29-Jul-2017
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