Thursday, December 3, 2015

Garbling with constant gate size

In the final MPC session of Asiacrypt, Carmen Kempka from NTT presented a Garbling scheme for formulas with constant size of garbled gates. The talk described an interesting new technique for garbling Boolean formulas (i.e. circuits with one output) based on trapdoor permutations and "backwards" garbling that, remarkably, achieves a constant size of just 4 bits per garbled gate.

Garbling schemes are a popular technique for constructing secure two-party computation schemes, and most practical approaches are based on the classic technique of Yao, where for each gate of the circuit, the resulting garbled circuit contains several ciphertexts that must be transmitted in the 2-PC protocol, where each ciphertext is of size $O(k)$ bits. At Eurocrypt this year, Zahur, Rosulek and Evans proposed the half-gate technique for garbling with just two ciphertexts per gate, and also showed that any "linear" garbling scheme cannot possibly do better than this. So, to reduce gate size any more, new non-linear techniques are needed; in Carmen's talk, some possible such techniques were given.

The main idea of the scheme is to use randomly generated, independent ciphertexts for each gate, so that the circuit evaluator can reconstruct these from just a single seed. The main difficulty is to ensure that the evaluator cannot then do the same as the circuit constructor, and create additional garbled circuits other than the one specified. To overcome this problem, they use trapdoor permutations (as opposed to typical garbling schemes, which just use hash functions) combined with a backwards garbling technique, where the input wire keys for each gate are chosen by solving a system of equations in the output keys and the ciphertexts.

The overall size of a garbled circuit is 4 bits per garbled gate, plus a ciphertext for each input, so this is the first scheme to achieve a constant gate size without a huge expansion in the input key size (as in Kolesnikov's scheme from 2005). However, since each input wire is uniquely determined by the output wires, gates can only have fan-out one, so the scheme is restricted to garbling formulas only. Extension to general circuits is possible, but at the cost of including 2 ciphertexts per gate.

Overall, the idea is neat, and it seems like a very interesting open problem to overcome the limitations of the scheme for general circuits, and reduce the size of the TDP-based ciphertexts for input gates.

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