The CRYPTO 2016 Best Paper Award went to Boyle et al [1]. The paper provides several new protocols based on a DDH assumption with applications to 2PC (2 party-computation), private information retrieval as well as function secret sharing.

Even more interesting, the authors present a protocol where 2PC for branching programs is realized in a way that communication complexity depends only on the input size and the computation is linear in circuit size.

The central idea develops around building efficient evaluation of RMS (restricted multiplication straight line) programs. The special feature of RMS is that they allow multiplications only with memory and input values; the additions come for free between memory values. Although this class seems quite restrictive it covers the class of branching programs (logaritmic depth boolean circuits with polynomial size and bounded input).

In the 2PC evaluation of RMS, suppose there is a linear shared memory value $[y]$ between the parties $P_1$ and $P_2$. When $P_1$ wants to share an input value $x$ to $P_2$ it sends an ElGamal encryption of $x$, $g^{xc}$ where $c$ is a symmetric ElGamal key. Clearly, the encryption is homomorphic with respect to multiplication, but how can we make any operations between a linear SS (secret shared) value and an ElGamal encryption?

This is solved by introducing a distributive DLog procedure which converts the El-Gamal ciphertexts into linear SS values. The method uses a truncated PRF which counts the number of steps until the PRF evaluated in the ElGamal encryption equals to $0$. Unfortunately this algorithm has a probability of outputting an incorrect result but it can be fixed by evaluating multiple instances of the same protocol in parallel and then use an MPC protocol to select the result majority.

Of course, there are some caveats at the beginning of the scheme such as converting the key generation procedure to a public key one and removing circularity key assumptions. These are gradually presented by the authors so that it can ease the reader's understanding of the ideas.

What I find neat is that at the end of the paper we can see easily how to reduce the communication for general 'dense' arithmetic circuits by splitting them in multiple reduced depth chunks and then apply the RMS programs for each gate (because an addition or multiplication gate can be represented as a branching program).

Of course we can spot some open problems left as future work such as:

[1]: Boyle, Elette, Niv Gilboa, and Yuval Ishai. "Breaking the Circuit Size Barrier for Secure Computation Under DDH."

[2]: Ishai, Yuval, et al. "Cryptography with constant computational overhead."

Even more interesting, the authors present a protocol where 2PC for branching programs is realized in a way that communication complexity depends only on the input size and the computation is linear in circuit size.

The central idea develops around building efficient evaluation of RMS (restricted multiplication straight line) programs. The special feature of RMS is that they allow multiplications only with memory and input values; the additions come for free between memory values. Although this class seems quite restrictive it covers the class of branching programs (logaritmic depth boolean circuits with polynomial size and bounded input).

In the 2PC evaluation of RMS, suppose there is a linear shared memory value $[y]$ between the parties $P_1$ and $P_2$. When $P_1$ wants to share an input value $x$ to $P_2$ it sends an ElGamal encryption of $x$, $g^{xc}$ where $c$ is a symmetric ElGamal key. Clearly, the encryption is homomorphic with respect to multiplication, but how can we make any operations between a linear SS (secret shared) value and an ElGamal encryption?

This is solved by introducing a distributive DLog procedure which converts the El-Gamal ciphertexts into linear SS values. The method uses a truncated PRF which counts the number of steps until the PRF evaluated in the ElGamal encryption equals to $0$. Unfortunately this algorithm has a probability of outputting an incorrect result but it can be fixed by evaluating multiple instances of the same protocol in parallel and then use an MPC protocol to select the result majority.

Of course, there are some caveats at the beginning of the scheme such as converting the key generation procedure to a public key one and removing circularity key assumptions. These are gradually presented by the authors so that it can ease the reader's understanding of the ideas.

What I find neat is that at the end of the paper we can see easily how to reduce the communication for general 'dense' arithmetic circuits by splitting them in multiple reduced depth chunks and then apply the RMS programs for each gate (because an addition or multiplication gate can be represented as a branching program).

Of course we can spot some open problems left as future work such as:

- Extend the protocols for larger classes other than branching programs.
- Protocol only works for $2$ parties. Can we find something with constant communication for multiple parties without using FHE?
- Can we convert the protocol for malicious parties in some other way rather than a generic complier as in [2]?

[1]: Boyle, Elette, Niv Gilboa, and Yuval Ishai. "Breaking the Circuit Size Barrier for Secure Computation Under DDH."

[2]: Ishai, Yuval, et al. "Cryptography with constant computational overhead."

*Proceedings of the fortieth annual ACM symposium on Theory of computing*. ACM, 2008.
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