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Secure Two-Party Computation: A Visual Way

Part of the Lecture Notes in Computer Science book series (LNSC,volume 8317)

Abstract

In this paper we propose a novel method for performing secure two-party computation. By merging together in a suitable way two beautiful ideas of the 80’s and the 90’s, Yao’s garbled circuit construction and Naor and Shamir’s visual cryptography, respectively, we enable Alice and Bob to securely evaluate a function \(f(\cdot ,\cdot )\) of their inputs, \(x\) and \(y\), through a pure physical process. Indeed, once Alice has prepared a set of properly constructed transparencies, Bob computes the function value \(f(x,y)\) by applying a sequence of simple steps which require the use of a pair of scissors, superposing transparencies, and the human visual system. A crypto-device for the function evaluation process is not needed any more.

Keywords

  • Yao’s construction
  • Visual cryptography
  • Secure computation

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Notes

  1. 1.

    The introduction of [4] offers a brief history of the construction and a nice accounting of the research efforts which followed.

  2. 2.

    A detailed description of Yao’s protocol can be found in [28].

  3. 3.

    A random variable is sufficient to represent the input, the output or any intermediate computation of a randomized entity in a single protocol execution. However, since it is of interest analyzing the behavior of protocol executions, according to input sizes depending on the security parameter \(k\), collections of random variables are needed: an ensemble is exactly a family of random variables, where each of them, say \(X_s,\) is uniquely identified by an index \(s,\) related to the security parameter \(k\).

  4. 4.

    We deal in the following with a deterministic functionality.Hence, we state the simplified versions of the definitions in [20, 28]. Moreover, we also state the definition for the unconditionally secure case. As we will show later, by using an unconditionally secure physical implementation of the oblivious transfer, known to be possible [29], the definition in the physical world is achieved by our protocol.

  5. 5.

    We do not follow the traditional entropy-based characterization, e.g., [8, 25], since in our analysis we are not going to use the entropy function. A comprehensive study of secret sharing schemes which does not use the language of information theory can be found in [5]. See also a recent survey [3].

  6. 6.

    In this abstract, to simplify the presentation of our approach, instead of providing general definitions, we concentrate on specific definitions of VCS for the tools we need in our construction.

  7. 7.

    We stress that the scheme is the same, and it is used twice with independent and fresh randomness.

  8. 8.

    If Alice thinks that Bob has had a career as illusionist, in order to be sure that Bob does not substitute the envelope that will be destroyed with an identical but fake one, might requests that Bob shows up in swimsuit.

  9. 9.

    An alternative could be that the envelope is burned in front of Alice. The key-property that need to be satisfied is that the physical process should be irreversible, the secret cannot be even partially recovered.

  10. 10.

    Notice that, for the permutation bit, we are using a deterministic \((2,2)\)-VCS with pixel expansion \(m=2\). We have used this solution for the permutation bit because, first of all it is possible to use a scheme with pixel expansion since each permutation bit propagates only from one level of the circuit to the subsequent one, and secondly because a scheme with pixel expansion allows a deterministic reconstruction.

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Acknowledgment

We would like to thank Alfredo De Santis for discussions and for pointing out to our attention [29], Carlo Blundo for comments on a preliminary version of this paper, and an anonymous referee for hints and suggestions.

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D’Arco, P., De Prisco, R. (2014). Secure Two-Party Computation: A Visual Way. In: Padró, C. (eds) Information Theoretic Security. ICITS 2013. Lecture Notes in Computer Science(), vol 8317. Springer, Cham. https://doi.org/10.1007/978-3-319-04268-8_2

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