Abstract
Functional encryption (FE) and predicate encryption (PE) can be utilized in deploying and executing machine learning (ML) algorithms to improve efficiency. However, most of existing FE and PE algorithms only consider generic functions. Actually, quadratic-functions-based FE and PE can be used to further reduce the computation costs significantly. In this paper, we present a functional encryption scheme for quadratic functions from those for generic functions. In our constructions, ciphertexts are associated with a pair of vectors \((\mathsf {x},\mathsf {y})\in \mathbb {Z}^{n}_{q}\times \mathbb {Z}^{m}_{q}\), private keys are associated with a quadratic function, and the decryption of ciphertexts CT(x,y) with a private key skF, where F is a n × m-dimensional matrix, recovers \((\mathsf {x})^{\top }\mathsf {F}\mathsf {y}\in \mathbb {Z}_{q}\). Compared with Baltico et al.’s FEs for quadratic functions (at Crypto 2017), our schemes could obtain almost the same ciphertexts size of \(O((n+m)\log q)\) as their schemes (in contrast to O(n) in Baltico et al.’s schemes), and the computation for quadratic functions in our scheme does not rely on bilinear maps, while their schemes must rely on this assumption. In particular, our schemes under the standard assumptions achieve adaptive security, while Baltico et al.’s scheme only obtains selective security. Moreover, beyond the MDDH and GGM assumptions, our schemes allow for instantiations under standard assumptions such as LWE, LPN, and etc.
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References
Agrawal S, Kumarasubramanian A, Prabhakaran M, Sahai A (2015) On the practical security of inner product functional encryption. In: Katz J. (ed) Advance in PKC 2015, vol 9020. Springer, Berlin Heidelberg, pp 777–798
Ananth P, Brakerski Z, Segev G, Vaikuntanathan V (2015) From selective to adaptive security in functional encryption. In: Gennaro R, Robshaw M (eds) Advance in CRYPTO, vol 2015, pp 657–677
Ananth P, Sahai A (2017) Projective Arithmetic Functional Encryption and Indistinguishability Obfuscation from Degree-5 Multilinear Maps. In: Coron JS., Nielsen J (eds) Advances in EUROCRYPT 2017, vol 10210, Springer, Berlin Heidelberg. pp 152–181
Baltico CEZ, Catalano D, Fiore D, Gay R (2017) Practical functional encryption for quadratic functions with applications to predicate encryption. In CRYPTO 2017:67–98
Bellare M, Rogaway P (2006) The Security of Triple Encryption and a Framework for Code-Based Game-Playing Proofs. In: Vaudenay S (ed) EUROCRYPT 2006. vol 4004. Springer, Heidelberg, pp 409–426
Boneh D (1998) The decision di e-hellman problem. In: proceedings of the 3rd Algorithmic Number Theory Symposium volume 1423, pages 48–63 Lecture Notes in Computer Science
Boneh D (1999) Twenty years of attacks on the rsa cryptosystem. In: Notices of the American Mathematical Society, pp 203–213
Ehsan H, Hassan T, Mehdi G (2017) Cryptodl: Deep neural networks over encrypted data. arXiv:1711.05189
Garg S, Gentry C, Halevi S, Zhandry M (2016) Functional encryption without obfuscation, In TCC2016, 480–511
Gorbunov S, Vaikuntanathan V, Wee H (2012) Functional encryption with bounded collusions via multi-party computation. In: Reihaneh Safavi-Naini, Ran Canetti (eds) editors, Advances in Cryptology CRYPTO 2012, vol 7417. Springer, Berlin Heidelberg, pp 162–179
Graepel T, Kristin L, Michael N (2012) Ml confidential: Machine learning on encrypted data. In: International Conference on Information Security and Cryptology, pages 1–21. Springer
Wang H, Chen K, Qin B et al (2018) LR-RRA-CCA secure functional encryption for randomized functionalities from trapdoor HPS and LAF. Sci China Inf Sci 61:058101. https://doi.org/10.1007/s11432-017-9120-4
Hao M, Li H, Luo X, Xu G, Yang H, Liu S (2019) Efficient and privacy-enhanced federated learning for industrial artificial intelligence 1–1. IEEE Trans Industrial Inform. to appear, https://doi.org/10.1109/TII.2019.2945367
Jiang W, Li H, Xu G, Wen M, Dong G, Lin X (2019) Ptas: Privacy-preserving thin-client authentication scheme in blockchain-based pki. Future Generation Comput Syst 96:185–195
Jiang XQ, Kim M, Lauter K, Song YS (2018) Secure outsourced matrix computation, and Application to neural networks. In: Proceedings of the ACM SIGSAC Conference on Computer and Communications security pages 1209–1222
Keith B, Vladimir I, Ben K, Antonio M, Brendan MH, Sarvar P, Daniel R, Aaron S, Karn S (2017) Practical secure aggregation for privacy-preserving machine learning. In: Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, pp 1175–1191
Li H, Liu D, Dai Y, Luan TH, Yu S (2018) Personalized search over encrypted data with efficient and secure updates in mobile clouds. IEEE Trans Emerg Topic Comput 6(1):97–109
Li H, Yang Y, Dai Y, Yu S, Xiang Y Achieving secure and efficient dynamic searchable symmetric encryption over medical cloud data, pages 1–1, 2017, to appear. https://doi.org/10.1109/TCC.2017.2769645https://doi.org/10.1109/TCC.2017.2769645 IEEE Transactions on Cloud Computing
Li HW, Yang Y, Dai YS, Bai J, Yu S, Xiang Y (2017) Achieving secure and efficient dynamic searchable symmetric encryption over medical cloud data IEEE Transactions on Cloud Computing accepted
Lin H (2016) Indistinguishability obfuscation from ddh on 5-linear maps and locality-5 prgs, In Cryptology ePrint Archive, Report 2016/1096., 2016. http://eprint.iacr.org/2016/1096
Miao YB, Liu XM, Choo KKR, Deng H, Li JG, Li HW, Ma JF (2019) Privacy-preserving attribute-based keyword search in shared multi-owner setting. IEEE Trans Dependable Sec Comput Accepted. https://doi.org/10.1109/TDSC.2019.2897675
Mohassel P, Zhang YP (2017) Secureml: a system for scalable privacy-preserving machine learning. In: In 2017 IEEE Symposium on Security and Privacy (S&P), pp 19–38
Ran G, Nathan D, Kim L, Kristin L, Michael N, John W (2016) Cryptonets: Applying neural networks to encrypted data with high throughput and accuracy. In: International Conference on Machine Learning, pp 201–210
Ren H, Li HW, Dai YS, Yang K, Lin XD (2018) Querying in internet of things with privacy preserving: Challenges, solutions and opportunities. IEEE Network 32(6):144–151
Sun XC, Li B, Lu XH, Fang FY (2015) Cca secure public key encryption scheme based on lwe without gaussian sampling. In Inscrypt 2015:361–378
Shacham H, Ristenpart T, Shrimpton T (2011) Careful with composition: Limitations of the indiferentiability framework. In: Paterson KG (ed) EUROCRYPT 2011, volume 6632 of LNCS, pages 487–506 pringer
Wang HG, Chen KF, Joseph KL, Hu ZY (2018) Leakage-resilient chosen-ciphertext secure functional encryption from garbled circuits, In ISPEC2018, 119–140
Wang HG, Zhang Y, Chen K, Sui GY, Zhao YL, huang XY (2019) Functional broadcast encryption with applications to data sharing for cloud storage Information Sciences
Waters B (2015) A punctured programming approach to adaptively secure functional encryption. In Advances in CRYPTO 2015:678–697
Li GWXHW, Liu S, Yang K, Lin XD (2019) Verifynet: Secure and verifiable federated learning. IEEE Trans Inform Forensics Secur Accepted. https://doi.org/10.1109/TIFS.2019.2929409
Xu G, Li H, Liu S, Wen M, Lu R (2019) Efficient and privacy-preserving truth discovery in mobile crowd sensing systems. IEEE Trans Vehicular Technol 68(4):3854–3865
Xu G, Li H, Ren H, Yang K, Deng RH (2019) Data security issues in deep learning: Attacks, countermeasures and opportunities. IEEE Commun Mag 57(11):116–122
Xu GW, Li HW, Dai YS, Yang K, Lin XD (2018) Enabling efficient and geometric range query with access control over encrypted spatial data. IEEE Trans Inform Forensics Secur 14(4):870–885
Xu RH, James JB, Lin C (2019) Cryptonn: Training neural networks over encrypted data. arXiv:1904.07303
Yu Y, Zhang J (2016) Cryptography with auxiliary input and trapdoor from constant-noise lpn. In CRYPTO 2016:214–243
Zhang Y, Xu CX, Li HW, Yang K, Zhou JY, Lin XD (2018) Healthdep: An efficient and secure deduplication scheme for cloud-assisted ehealth systems. IEEE Trans Industrial Inform 14(9):4101–4112
Zhang Y, Xu CX, Lin XD, Shen XM (2019) Blockchain-based public integrity verification for cloud storage against procrastinating auditors. IEEE Transactions on Cloud Computing accepted
Zhang Y, Xu CX, Ni JB, Li HW, Shen XM (2019) Blockchain-assisted public-key encryption with keyword search against keyword guessing attacks for cloud storage. IEEE Trans Cloud Comput. https://doi.org/10.1109/TCC.2019.2923222
Acknowledgments
The first author is supported by the National Key Research and Development Program of China (Grant No. 2017YFB0802000) and the National Natural Science Foundation of China (Grant Nos. NSFC61702007, NSFC61572318) and Other Foundations (Grant Nos. 2019M661360, gxbjZD27,KJ2018A0533, XWWD201801, ahnis20178002, KJ2017A519, 16ZB0140). The second author is supported by the National Key Research and Development Program of China (Grant No. 2017YFB0802000) and the National Natural Science Foundation of China (Grant No. U1705264). The fourth author is supported in part by National Key Research and Development Program of China (Grant No. 2017YFB0802000), the National Natural Science Foundation of China (Grant Nos. 61877011, 61472084, U1536205), Shanghai Innovation Action Project (Grant No. 16DZ1100200), Shanghai Science and Technology Development Funds (Grant No. 16JC1400801), and Shandong Provincial Key Research and Development Program of China (Grant Nos. 2017CXG0701, 2018CXGC0701).
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This article is part of the Topical Collection: Special Issue on Security and Privacy in Machine Learning Assisted P2P Networks
Guest Editors: Hongwei Li, Rongxing Lu and Mohamed Mahmoud
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Wang, H., Chen, K., Zhang, Y. et al. Functional encryption with application to machine learning: simple conversions from generic functions to quadratic functions. Peer-to-Peer Netw. Appl. 13, 2334–2341 (2020). https://doi.org/10.1007/s12083-020-00907-4
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DOI: https://doi.org/10.1007/s12083-020-00907-4