Private Visual Share-Homomorphic Computation and Randomness Reduction in Visual Cryptography

  • Paolo D’ArcoEmail author
  • Roberto De Prisco
  • Yvo Desmedt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10015)


Secure computation through non standard methods, suitable for users who have to perform the computation without the aid of a computer, or for settings in which the degree of trustworthiness of the hardware and software equipments is very low, are an interesting, very challenging and quite unexplored research topic. In this paper we put forward a collection of ideas and some techniques which could be useful in order to make some progress in designing protocols with such properties. Our contribution is twofold: we explore the power of visual cryptography as a computing tool, exploiting alternative uses and share manipulations, and we address the central issue of randomness reduction in visual schemes, by showing a strict relation with existing results in secure multiparty computation. More specifically, we prove that:
  • by properly defining operations on the shares, we show that visual shares are homomorphic with respect to some functions f. More precisely, in the two-party case, each user, by applying to his two shares \(a_i, b_i\) of the secrets ab the operation, gets a share \(g_i(a_i,b_i)\), \(i=1,2\), such that the superposition of \(g_1(a_1,b_1)\) and \(g_2(a_2,b_2)\) visually provides, applying the standard Naor and Shamir superposition reconstruction strategy, the value of the function f;

  • we link our analysis to a general known result on private two-party computation, and we classify all the boolean functions of two input bits which admit homomorphic visual share computation;

  • we prove that by encoding pixels in groups, instead of encoding each pixel independently, and exploiting dependencies, some randomness can be saved if and only if the pixel dependencies can be expressed through some specific boolean functions. For example, given three pixels, if the third one is the and or the or of the first two, randomness reduction is impossible, while if it is the xor of the first two, randomness reduction can be achieved.


Information theory Visual cryptography Secure computation Unconditional privacy 


  1. 1.
    International Business Times: Cisco Faces Challenges as Chinese Media Urge Switching to Domestic Products for National Security Reasons in Wake of NSA Surveillance Leaks. Accessed 25 June 2013
  2. 2.
    The Telegraph: Russia spied on G20 leaders with USB sticks. Accessed 29 October 2013
  3. 3.
    The Telegraph: Indian High Commission returns to typewriters. Accessed 27 September 2013
  4. 4.
    Asharov, G., Lindell, Y.: A Full Proof of the BGW Protocol for Perfectly-Secure Multiparty Computation, September 2014.
  5. 5.
    Benaloh, J.C.: Secret sharing homomorphisms: keeping shares of a secret secret (extended abstract). In: Odlyzko, A.M. (ed.) Crypto 1986. LNCS, vol. 263, pp. 251–260. Springer, Heidelberg (1987). doi: 10.1007/3-540-47721-7_19 Google Scholar
  6. 6.
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation. In: Proceedings of STOC (1988)Google Scholar
  7. 7.
    Chaum, D.: Secret-Ballot Receipts and Transparent Integrity.
  8. 8.
    Cimato, S., Yang, C.-N.: Visual Cryptography and Secret Image Sharing. CRC Press, Boca Raton (2012)Google Scholar
  9. 9.
    Cimato, S., De Prisco, R., De Santis, A.: Probabilistic visual cryptography schemes. Comput. J. 49(1), 97–107 (2006)CrossRefGoogle Scholar
  10. 10.
    Chaum, D. , Crépeau, C., Damgaard, I.: Multiparty unconditionally secure protocols. In: Proceedings of STOC (1988)Google Scholar
  11. 11.
    D’Arco, P., Prisco, R.: Secure Two-Party Computation: A Visual Way. In: Padró, C. (ed.) ICITS 2013. LNCS, vol. 8317, pp. 18–38. Springer, Heidelberg (2014). doi: 10.1007/978-3-319-04268-8_2 CrossRefGoogle Scholar
  12. 12.
    De Bonis, A., De Santis, A.: Randomness in secret sharing and visual cryptography schemes. Theor. Comput. Sci. 314(3), 351–374 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    De Prisco, R., De Santis, A.: On the relation of random grid and deterministic visual cryptography. IEEE Trans. Inf. Forensics Secur. 9(4), 653–665 (2014)CrossRefGoogle Scholar
  14. 14.
    Kafri, O., Keren, E.: Encryption of pictures and shapes by random grids. Opt. Lett. 12(6), 377–379 (1987)CrossRefGoogle Scholar
  15. 15.
    Kushilevitz, E.: Privacy and communication complexity. In: Proceedings of IEEE Symposium on Foundations of Computer Science (FOCS), pp. 416–421 (1989)Google Scholar
  16. 16.
    Maeng, Y.-J., Mohaisen, A., Lee, M.-K., Nyang, D.: Transaction authentication using complementary colors. Comput. Secur. 48, 167–181 (2015)CrossRefGoogle Scholar
  17. 17.
    Naor, M., Shamir, A.: Visual cryptography. In: Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 1–12. Springer, Heidelberg (1995). doi: 10.1007/BFb0053419 Google Scholar
  18. 18.
    Yang, C.-N.: New visual secret sharing schemes using probabilistic method. Pattern Recogn. Lett. 25, 481–494 (2004)CrossRefGoogle Scholar
  19. 19.
    Zhang, Y., Steele, A., Blanton, M.: PICCO: a general-purpose compiler for private distributed computation. In: Proceedings of the Conference on Computer and Communications Security, CCS13, Berlin, Germany, November 4–8, pp. 813–826 (2013)Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Paolo D’Arco
    • 1
    Email author
  • Roberto De Prisco
    • 1
  • Yvo Desmedt
    • 2
  1. 1.Dipartimento di InformaticaUniversity of SalernoFiscianoItaly
  2. 2.Department of Computer ScienceThe University of Texas at DallasRichardsonUSA

Personalised recommendations