# Improved Delegation of Computation

using Fully Homomorphic Encryption

Kai-Min Chung, Yael Kalai, Salil Vadhan

### Abstract

Following Gennaro, Gentry, and Parno (Cryptology ePrint Archive 2009/547), we use fully homomorphic encryption to design improved schemes for delegating computation. In such schemes, a *delegator* outsources the computation of a function F on many, dynamically chosen inputs x_i to a *worker* in such a way that it is infeasible for the worker to make the delegator accept a result other than F(x_i). The "online stage" of the Gennaro et al. scheme is very efficient: the parties exchange two messages, the delegator runs in time poly(log T), and the worker runs in time poly(T), where T is the time complexity of F. However, the "offline stage" (which depends on the function F but not the inputs to be delegated) is inefficient: the delegator runs in time poly(T) and generates a public key of length poly(T) that needs to be accessed by the worker during the online stage.

Our first construction eliminates the large public key from the Gennaro et al. scheme. The delegator still invests poly(T) time in the offline stage, but does not need to communicate or publish anything. Our second construction reduces the work of the delegator in the offline stage to poly(log T) at the price of a 4-message (offline) interaction with a poly(T)-time worker (which need not be the same as the workers used in the online stage). Finally, we describe a ``pipelined'' implementation of the second construction that avoids the need to re-run the offline construction after errors are detected (assuming errors are not too frequent).

### Versions

*Cryptology ePrint Archive, *Report 210/241, April 2010. [pdf][ePrint page]
- In
*Advances in Cryptology---CRYPTO `10,* Lecture Notes in Computer Science. Springer-Verlag, 15-19 August 2010. [pdf][Springer page]

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