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A Survey of Homomorphic Encryption for Nonspecialists

Abstract

Processing encrypted signals requires special properties of the underlying encryption scheme. A possible choice is the use of homomorphic encryption. In this paper, we propose a selection of the most important available solutions, discussing their properties and limitations.

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References

  1. 1.

    Rivest R, Adleman L, Dertouzos M: On data banks and privacy homomorphisms. In Foundations of Secure Computation. Academic Press; 1978:169-177.

    Google Scholar 

  2. 2.

    Brickell E, Yacobi Y: On privacy homomorphisms. In Advances in Cryptology (EUROCRYPT '87), Lecture Notes in Computer Science. Volume 304. Springer, New York, NY, USA; 1987:117-126.

    Google Scholar 

  3. 3.

    Rappe D: Homomorphic cryptosystems and their applications, Ph.D. thesis. University of Dortmund, Dortmund, Germany; 2004.http://www.rappe.de/doerte/Diss.pdf

    Google Scholar 

  4. 4.

    Cramer R, Damgård I: Zero-knowledge for finite field arthmetic, or: can zeroknowledge be for free? In Advances in Cryptology (CRYPTO '98), Lecture Notes in Computer Science. Volume 1462. Springer, New York, NY, USA; 1998:424-441. 10.1007/BFb0055745

    Google Scholar 

  5. 5.

    Lipmaa H: Verifiable homomorphic oblivious transfer and private equality test. In Advances in Cryptology (ASIACRYPT '03), Lecture Notes in Computer Science. Volume 2894. Springer, New York, NY, USA; 2003:416-433. 10.1007/978-3-540-40061-5_27

    Google Scholar 

  6. 6.

    Fouque P-A, Poupard G, Stern J: Sharing decryption in the context of voting or lotteries. Proceedings of the 4th International Conference on Financial Cryptography, 2000, Anguilla, British West Indies, Lecture Notes in Computer Science 1962: 90-104.

    Google Scholar 

  7. 7.

    Sander T, Tschudin C: Protecting mobile agents against malicious hosts. In Mobile Agents and Security, Lecture Notes in Computer Science. Volume 1419. Springer, New York, NY, USA; 1998:44-60.

    Google Scholar 

  8. 8.

    Golle P, Jakobsson M, Juels A, Syverson P: Universal re-encryption for mixnets. Proceedings of the RSA Conference Cryptographer's (Track '04), 2004, San Francisco, Calif, USA, Lecture Notes in Computer Science 2964: 163-178.

    MathSciNet  Google Scholar 

  9. 9.

    Damgård I, Jurik M: A length-flexible threshold cryptosystem with applications. Proceedings of the 8th Australian Conference on Information Security and Privacy (ACISP '03), 2003, Wollongong, Australia, Lecture Notes in Computer Science 2727:

    Google Scholar 

  10. 10.

    Adelsbach A, Katzenbeisser S, Sadeghi A: Cryptology meets watermarking: detecting watermarks with minimal or zero-knowledge disclosures. Proceedings of the European Signal Processing Conference (EUSIPCO '02), September 2002, Toulouse, France

    Google Scholar 

  11. 11.

    Pfitzmann B, Waidner W: Anonymous fingerprinting. In Advances in Cryptology (EUROCRYPT '97), Lecture Notes in Computer Science. Volume 1233. Springer, New York, NY, USA; 1997:88-102. 10.1007/3-540-69053-0_8

    Google Scholar 

  12. 12.

    Memon N, Wong P: A buyer-seller watermarking protocol. IEEE Transactions on Image Processing 2001, 10(4):643-649. 10.1109/83.913598

    MATH  Article  Google Scholar 

  13. 13.

    Lei C-L, Yu P-L, Tsai P-L, Chan M-H: An efficient and anonymous buyer-seller watermarking protocol. IEEE Transactions on Image Processing 2004, 13(12):1618-1626. 10.1109/TIP.2004.837553

    Article  Google Scholar 

  14. 14.

    Kuribayashi M, Tanaka H: Fingerprinting protocol for images based on aditive homomorphic property. IEEE Transactions on Image Processing 2005, 14(12):2129-2139.

    Article  Google Scholar 

  15. 15.

    Shoup V: A Computational Introduction to Number Theory and Algebra. Cambridge University Press; 2005.http://www.shoup.net/ntb/

    Google Scholar 

  16. 16.

    Menezes A, Van Orschot P, Vanstone S: Handbook of applied cryptography. CRC Press; 1997.http://www.cacr.math.uwaterloo.ca/hac/

    Google Scholar 

  17. 17.

    Van Tilborg H (Ed): Encyclopedia of Cryptography and Security. Springer, New York, NY, USA; 2005.

    Google Scholar 

  18. 18.

    Kerckhoffs A: La cryptographie militaire (part i). Journal des Sciences Militaires 1883, 9(1):5-38.

    Google Scholar 

  19. 19.

    Kerckhoffs A: La cryptographie militaire (part ii). Journal des Sciences Militaires 1883, 9(2):161-191.

    Google Scholar 

  20. 20.

    Daemen J, Rijmen V: The block cipher RIJNDAEL. In (CARDIS '98), Lecture Notes in Computer Science. Volume 1820. Springer, New York, NY, USA; 2000:247-256.

    Google Scholar 

  21. 21.

    Daemen J, Rijmen V: The design of Rijndael. In AES—the Advanced Encryption Standard, Informtion Security and Cryptography. Springer, New York, NY, USA; 2002.

    Google Scholar 

  22. 22.

    Vernam G: Cipher printing telegraph systems for secret wire and radio telegraphic communications. Journal of the American Institute of Electrical Engineers 1926, 45: 109-115.

    Google Scholar 

  23. 23.

    Ekdahl P, Johansson T: A new version of the stream cipher SNOW. In Selected Areas in Cryptography (SAC '02), Lecture Notes in Computer Science. Volume 2595. Springer, New York, NY, USA; 2002:47-61.

    Google Scholar 

  24. 24.

    Rivest R, Shamir A, Adleman L: A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM 1978, 21(2):120-126. 10.1145/359340.359342

    MATH  MathSciNet  Article  Google Scholar 

  25. 25.

    ElGamal T: A prublic key cryptosystem and a signature scheme based on discrete logarithms. In Advances in Cryptology (CRYPTO '84), Lecture Notes in Computer Science. Volume 196. Springer, New York, NY, USA; 1985:10-18. 10.1007/3-540-39568-7_2

    Google Scholar 

  26. 26.

    Shannon C: Communication theory of secrecy systems. Bell System Technical Journal 1949, 28: 656-715.

    MATH  MathSciNet  Article  Google Scholar 

  27. 27.

    Ajtai M, Dwork C: A public key cryptosystem with worst-case/average-case equivalence. Proceedings of the 29th ACM Symposium on Theory of Computing (STOC '97), 1997 284-293.

    Google Scholar 

  28. 28.

    Nguyen P, Stern J: Cryptanalysis of the Ajtai-Dwork cryptosystem. In Advances in Cryptology (CRYPTO '98), Lecture Notes in Computer Science. Volume 1462. Springer, New York, NY, USA; 1999:223-242.

    Google Scholar 

  29. 29.

    Canetti R, Goldreich O, Halevi S: The random oracle model, revisited. Proceedings of the 30th ACM Symposium on Theory of Computing (STOC '98), 1998, Berkeley, Calif, USA 209-218.

    Google Scholar 

  30. 30.

    Paillier P: Impossibility proofs for RSA signatures in the standard model. Proceedings of the RSA Conference 2007, Cryptographers' (Track), 2007, San Fancisco, Calif, USA, Lecture Notes in Computer Science 4377: 31-48.

    MathSciNet  Google Scholar 

  31. 31.

    Diffie W, Hellman M: New directions in cryptography. IEEE Transactions on Information Theory 1976, 22(6):644-654. 10.1109/TIT.1976.1055638

    MATH  MathSciNet  Article  Google Scholar 

  32. 32.

    Kahn D: The Codebreakers: The Story of Secret Writing. Macmillan, New York, NY, USA; 1967.

    Google Scholar 

  33. 33.

    Bellare M, Rogaway P: Optimal asymmetric encryption—how to encrypt with RSA. In Advances in Cryptology (EUROCRYPT '94), Lecture Notes in Computer Science. Volume 950. Springer, New York, NY, USA; 1995:92-111. 10.1007/BFb0053428

    Google Scholar 

  34. 34.

    Goldwasser S, Micali S: Probabilistic encryption & how to play mental poker keeping secret all partial information. Proceedings of the 14th ACM Symposium on the Theory of Computing (STOC '82), 1982, New York, NY, USA 365-377.

    Google Scholar 

  35. 35.

    Blum M, Goldwasser S: An efficient probabilistic public-key encryption scheme which hides all partial information. In Advances in Cryptology (EUROCRYPT '84), Lecture Notes in Computer Science. Volume 196. Springer, New York, NY, USA; 1985:289-299.

    Google Scholar 

  36. 36.

    Goldreich O: A uniform complexity treatment of encryption and zero-knowledge. Journal of Cryptology 1993, 6(1):21-53. 10.1007/BF02620230

    MATH  MathSciNet  Article  Google Scholar 

  37. 37.

    Naor M, Yung M: Public-key cryptosystems provably secure against chosen ciphertext attacks. Proceedings of the 22nd ACM Annual Symposium on the Theory of Computing (STOC '90), 1990, Baltimore, Md, USA 427-437.

    Google Scholar 

  38. 38.

    Rackoff C, Simon D: Non-interactive zero-knowledge proof of knowledge and chosen ciphertext attack. In Advances in Cryptology (CRYPTO '91), Lecture Notes in Computer Science. Volume 576. Springer, New York, NY, USA; 1991:433-444.

    Google Scholar 

  39. 39.

    Dolev D, Dwork C, Naor M: Non-malleable cryptography. Proceedings of the 23rd ACM Annual Symposium on the Theory of Computing —(STOC '91), 1991 542-552.

    Google Scholar 

  40. 40.

    Dolev D, Dwork C, Naor M: Non-malleable cryptography. SIAM Journal of Computing 2000, 30(2):391-437. 10.1137/S0097539795291562

    MATH  MathSciNet  Article  Google Scholar 

  41. 41.

    Bellare M, Desai A, Pointcheval D, Rogaway P: Relations among notions of security for public-key encryption schemes. In Advances in Cryptology (CRYPTO '98), Lecture Notes in Computer Science. Volume 1462. Springer, New York, NY, USA; 1998:26-45. 10.1007/BFb0055718

    Google Scholar 

  42. 42.

    Bellare M, Sahai A: Non-malleable encryption: equivalence between two notions, and an indistinguishability-based characterization. In Advances in Cryptology (CRYPTO '99), Lecture Notes in Computer Science. Volume 1666. Springer, New York, NY, USA; 1999:519-536. 10.1007/3-540-48405-1_33

    Google Scholar 

  43. 43.

    Watanabe Y, Shikata J, Imai H: Equivalence between semantic security and indistinguishability against chosen ciphertext attacks. In Public Key Cryptography (PKC '03), Lecture Notes in Computer Science. Volume 2567. Springer, New York, NY, USA; 2003:71-84.

    Google Scholar 

  44. 44.

    Ahituv N, Lapid Y, Neumann S: Processing encrypted data. Communications of the ACM 1987, 30(9):777-780. 10.1145/30401.30404

    MATH  Article  Google Scholar 

  45. 45.

    Boneh D, Lipton R: Algorithms for black box fields and their application to cryptography. In Advances in Cryptology (CRYPTO '96), Lecture Notes in Computer Science. Volume 1109. Springer, New York, NY, USA; 1996:283-297. 10.1007/3-540-68697-5_22

    Google Scholar 

  46. 46.

    Goldwasser S, Micali S: Probabilistic encryption. Journal of Computer and System Sciences 1984, 28(2):270-299. 10.1016/0022-0000(84)90070-9

    MATH  MathSciNet  Article  Google Scholar 

  47. 47.

    Paillier P: Public-key cryptosystems based on composite degree residuosity classes. In Advances in Cryptology (EUROCRYPT '99), Lecture Notes in Computer Science. Volume 1592. Springer, New York, NY, USA; 1999:223-238. 10.1007/3-540-48910-X_16

    Google Scholar 

  48. 48.

    Cramer R, Gennaro R, Schoenmakers B: A secure and optimally efficient multiauthority election scheme. In Advances in Cryptology (EUROCRYPT '97), Lecture Notes in Computer Science. Volume 1233. Springer, New York, NY, USA; 1997:103-118. 10.1007/3-540-69053-0_9

    Google Scholar 

  49. 49.

    McEliece R: A public-key cryptosystem based on algebraic coding theory. Dsn progress report 1978.

    Google Scholar 

  50. 50.

    Benaloh J: Verifiable secret-ballot elections, Ph.D. thesis. Yale University, Department of Computer Science, New Haven, Conn, USA; 1988.

    Google Scholar 

  51. 51.

    Naccache D, Stern J: A new public-key cryptosystem based on higher residues. Proceedings of the 5th ACM Conference on Computer and Communications Security, November 1998, San Francisco, Calif, USA 59-66.

    Google Scholar 

  52. 52.

    Okamoto T, Uchiyama S: A new public-key cryptosystem as secure as factoring. In Advances in Cryptology (EUROCRYPT '98), Lecture Notes in Computer Science. Volume 1403. Springer, New York, NY, USA; 1998:308-318. 10.1007/BFb0054135

    Google Scholar 

  53. 53.

    Okamoto T, Uchiyama S, Fujisaki E: Epoc: efficient probabilistic publickey encryption. 2000.Proposal to IEEE P1363a, http://grouper.ieee.org/groups/1363/P1363a/draft.htmlhttp://grouper.ieee.org/groups/1363/P1363a/draft.html

    Google Scholar 

  54. 54.

    Joye M, Quisquater J-J, Yung M: On the power of misbehaving adversaries and security analysis of the original EPOC. In Topics in Cryptology CT-RSA 2001, Lecture Notes in Computer Science. Volume 2020. Springer, New York, NY, USA; 2001.

    Google Scholar 

  55. 55.

    Cramer R, Shoup V: Universal hash proofs and a paradigm for adaptive chosen ciphertext secure public-key encryption. In Advances in Cryptology (EUROCRYPT '02), Lecture Notes in Computer Science. Volume 2332. Springer, New York, NY, USA; 2002:45-64. 10.1007/3-540-46035-7_4

    Google Scholar 

  56. 56.

    Bresson E, Catalano D, Pointcheval D: A simple public-key cryptosystem with a double trapdoor decryption mechanism and its applications. In Advances in Cryptology (ASIACRYPT '03), Lecture Notes in Computer Science. Volume 2894. Springer, New York, NY, USA; 2003:37-54. 10.1007/978-3-540-40061-5_3

    Google Scholar 

  57. 57.

    Damgård I, Jurik M: A generalisation, a simplification and some applications of Paillier's probabilistic public-key system. In 4th International Workshop on Practice and Theory in Public-Key Cryptography, Lecture Notes in Computer Science. Volume 1992. Springer, New York, NY, USA; 2001:119-136.

    Google Scholar 

  58. 58.

    Galbraith S: Elliptic curve paillier schemes. Journal of Cryptology 2002, 15(2):129-138.

    MATH  MathSciNet  Article  Google Scholar 

  59. 59.

    Castagnos G: An efficient probabilistic public-key cryptosystem over quadratic fields quotients. 2007.Finite Fields and Their Applications, paper version in press, http://users.info.unicaen.fr/~gcastagn/http://users.info.unicaen.fr/~gcastagn/

    Google Scholar 

  60. 60.

    Castagnos G: Quelques schémas de cryptographie asymétrique probabiliste, Ph.D. thesis. , Bochum, Germany; 2006.http://users.info.unicaen.fr/~gcastagn/

    Google Scholar 

  61. 61.

    Boneh D, Franklin M: Identity-based encryption from the Weil pairing. In Advances in Cryptology (CRYPTO '01), Lecture Notes in Computer Science. Volume 2139. Springer, New York, NY, USA; 2001:213-229. 10.1007/3-540-44647-8_13

    Google Scholar 

  62. 62.

    Boneh D, Boyen X, Goh E-J: Hierarchical identity based encryption with constant size ciphertext. In Advances in Cryptology (EUROCRYPT '05), Lecture Notes in Computer Science. Volume 3494. Springer, New York, NY, USA; 2005:440-456. 10.1007/11426639_26

    Google Scholar 

  63. 63.

    Domingo-Ferrer J: A provably secure additive and multiplicative privacy homomorphism. Proceedings of the 5th International Conference on Information Security (ISC '02), 2002, Sao Paulo, Brazil, Lecture Notes in Computer Science 2433: 471-483.

    Google Scholar 

  64. 64.

    Wagner D: Cryptanalysis of an algebraic privacy homomorphism. Proceedings of the 6th International Conference on Information Security (ISC '03), 2003, Bristol, UK, Lecture Notes in Computer Science 2851:

    Google Scholar 

  65. 65.

    Bao F: Cryptanalysis of a provable secure additive and multiplicative privacy homomorphism. International Workshop on Coding and Cryptograhy (WCC '03), 2003, Versailles, France 43-49.

    Google Scholar 

  66. 66.

    Domingo-Ferrer J: A new privacy homomorphism and applications. Information Processing Letters 1996, 60(5):277-282. 10.1016/S0020-0190(96)00170-6

    MathSciNet  Article  Google Scholar 

  67. 67.

    Cheon J, Kim W-H, Nam H: Known-plaintext cryptanalysis of the domingo-ferrer algebraic privacy homomorphism scheme. Information Processing Letters 2006, 97(3):118-123.

    MATH  MathSciNet  Article  Google Scholar 

  68. 68.

    Castelluccia C, Mykletun E, Tsudik G: Efficient aggregation of encrypted data in wireless sensor networks. ACM/IEEE Mobile and Ubiquitous Systems: Networking and Services (Mobiquitous '05) 2005, 109-117.

    Google Scholar 

  69. 69.

    Fellows M, Koblitz N: Combinatorial cryptosystems galore! Contemporary Mathematics, Finite Fields: Theory, Applications, and Algorithms, FQ2 1993, 168: 51-61.

    MathSciNet  Article  Google Scholar 

  70. 70.

    Ly L: A public-key cryptosystem based on Polly Cracker, Ph.D. thesis. 2002.

    Google Scholar 

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Correspondence to Caroline Fontaine.

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Fontaine, C., Galand, F. A Survey of Homomorphic Encryption for Nonspecialists. EURASIP J. on Info. Security 2007, 013801 (2007). https://doi.org/10.1155/2007/13801

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Keywords

  • Special Property
  • Encryption Scheme
  • Data Security
  • Homomorphic Encryption
  • Encrypt Signal