Secure Computation with Constant Communication Overhead Using Multiplication Embeddings

  • Alexander R. BlockEmail author
  • Hemanta K. Maji
  • Hai H. Nguyen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11356)


Secure multi-party computation (MPC) allows mutually distrusting parties to compute securely over their private data. The hardness of MPC, essentially, lies in performing secure multiplications over suitable algebras.

There are several cryptographic resources that help securely compute one multiplication over a large finite field, say \({\mathbb G} {\mathbb F} \left[ 2^n\right] \), with linear communication complexity. For example, the computational hardness assumption like noisy Reed-Solomon codewords are pseudorandom. However, it is not known if we can securely compute, say, a linear number of \(\mathsf {AND}\)-gates from such resources, i.e., a linear number of multiplications over the base field \({\mathbb G} {\mathbb F} \left[ 2\right] \). Before our work, we could only perform o(n) secure \(\mathsf {AND}\)-evaluations.

Technically, we construct a perfectly secure protocol that realizes a linear number of multiplication gates over the base field using one multiplication gate over a degree-n extension field. This construction relies on the toolkit provided by algebraic function fields.

Using this construction, we obtain the following results. We provide the first construction that computes a linear number of oblivious transfers with linear communication complexity from the computational hardness assumptions like noisy Reed-Solomon codewords are pseudorandom, or arithmetic-analogues of LPN-style assumptions. Next, we highlight the potential of our result for other applications to MPC by constructing the first correlation extractor that has 1 / 2 resilience and produces a linear number of oblivious transfers.


Secure computation Multiplication embeddings Oblivious transfer Basis-independent circuit compututation Leakage-resilient cryptography Randomness extractors 


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Alexander R. Block
    • 1
    Email author
  • Hemanta K. Maji
    • 1
  • Hai H. Nguyen
    • 1
  1. 1.Department of Computer SciencePurdue UniversityWest LafayetteUSA

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