I'm new to lattice-based cryptography and have trouble understanding if the GSW13 (Gentry, Sahai, Waters) scheme works only on single bits. But is it also possible to encrypt integers with this scheme and add them together?
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3Does this answer your question? Representing a function as FHE circuit – kelalaka Feb 03 '21 at 20:20
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Short answer: represent integers in binary and encrypt bit by bit. Integer addition can be reduced to operations on the individual bits of the terms; e.g. you can just use the standard schoolbook addition methods (adapted to the basis 2) – Geoffroy Couteau Feb 03 '21 at 22:24
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@GeoffroyCouteau but it is possible to encrypt non-binary messages directly and operate homomorphically on them using GSW. – Hilder Vitor Lima Pereira Feb 04 '21 at 12:01
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that would have been part of a long answer if I had written one :) – Geoffroy Couteau Feb 04 '21 at 12:07
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Can't you use a GSW variant defined by a "powers of $p$" gadget vector rather than "powers of $2$"? My vague impression is that this gets you (at least addition) in $\mathbb{Z}/p\mathbb{Z}$ – Mark Schultz-Wu Feb 04 '21 at 18:16