# Multi-identity and Multi-key Leveled FHE from Learning with Errors

## Abstract

Gentry, Sahai and Waters recently presented the first (leveled) identity-based fully homomorphic (IBFHE) encryption scheme (CRYPTO 2013). Their scheme however only works in the single-identity setting; that is, homomorphic evaluation can only be performed on ciphertexts created with the same identity. In this work, we extend their results to the multi-identity setting and obtain a multi-identity IBFHE scheme that is selectively secure in the random oracle model under the hardness of Learning with Errors (LWE). We also obtain a multi-key fully-homomorphic encryption (FHE) scheme that is secure under LWE in the standard model. This is the first multi-key FHE based on a well-established assumption such as standard LWE. The multi-key FHE of López-Alt, Tromer and Vaikuntanathan (STOC 2012) relied on a non-standard assumption, referred to as the Decisional Small Polynomial Ratio assumption.

## Notes

### Acknowledgments

We would like to thank the anonymous reviewers of for their helpful comments. The authors would like to thank Fuqun Wang for pointing out errors in an earlier version of this paper.

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