Udhay Shankar N quotes wikipedia:
> The question was finally resolved in 2009 with the development of the
> first true fully homomorphic cryptosystem. The scheme, constructed by
> Craig Gentry, employs lattice based encryption and allows evaluation
> of both addition and multiplication operations without restriction.[2]
>    2. ^ Craig Gentry. On homomorphic encryption over circuits of
>       arbitrary depth. In the 41st ACM Symposium on Theory of Computing
>       (STOC), 2009. 

A URL for this paper is
http://portal.acm.org/citation.cfm?id=1536414.1536440 but you will have
to be an ACM member to download it. I was able to get a copy this morning
and quickly skimmed it.

This is IMO one of the most remarkable crypto papers ever. Not only
does it solve one of the oldest open problems in cryptography, the
construction of a fully homomorphic encryption system, it does so by
means of a self-embedding technique reminiscent of Godel's theorem.

Craig Gentry starts off by inventing a limited homomorphic encryption
system based on lattice encryption. For full homomorphism you want to do
both add and multiply - or expressed on bits, XOR and AND.  Think of your
operation as a circuit made up of XOR and AND gates, then the limiting
factor in previous work has been the number of ANDs.  While many schemes
have been found that do just XORs, and at least one that could do a very
limited number of ANDs, the new scheme allows you to go deeper.

In lattice encryption, there is a multi-dimensional lattice of points
that have a hidden structure. Encryption puts you "near" a point on
the lattice, and decryption involves finding that point. But without
knowing the hidden structure, attackers can't tell which one is closest.
In the new homomorphic lattice encryption, AND operations cause the
error term to increase. After too many of them, too deep a circuit,
the error term grows up to roughly half the distance between lattice
points, and decryption is no longer possible. So you have a limited
depth homomorphic encryption system.

By itself this is a substantial advance. But now for the amazing
Godelian trick. The error term has grown and any more operations will
make decryption impossible. So Craig Gentry proposes to allow the server
(which is working on data encrypted under a key controlled by the client)
to decrypt the data - *homomorphically*. If the data is encrypted by
client key pk1, the client has also supplied the server a second key pk2,
and also a version of the pk1 *secret key* encrypted under pk2. Since
pk2 is homomorphic, the server can compute a circuit using the encrypted
secret pk1 key just like it can compute any other circuit on encrypted
data. The result is that the server ends up with an encryption of the
original ciphertext under pk2 instead of under pk1, and because lattice
decryption in effect performs error correction, the error term is reduced
and we are ready for more.

The key idea here is that the homomorphic encryption system has to
allow enough homomorphic depth that *its own decryption algorithm*
can be expressed as a circuit that "fits" within what can be handled
homomorphically. This is quite difficult and most of the paper is taken
up with constructing such a cryptosystem and proving its properties.
The resulting scheme is apparently not practical (at one point he
mentions that the secret key bits have to be expressed in *unary*) but
it is still amazing that it is even possible. Again I have to go back
to Godel's and Turing's work to think of a comparable example exploiting
the power of self-embedding.

In its most basic form, then, the client must supply a set of public keys
pk1, pk2, ... with each key's private part encrypted under the next key;
the number of such keys would be proportional to the depth of the circuit
to be evaluated. However the paper then points out that given reasonable
assumptions, you can dispense with the whole set and make it a loop,
even a loop of one: that is, you encrypt pk1's private key under pk1,
and stick with pk1 through the whole encryption, including the magical
homomorphic decryption circuit evaluation. In this form it is the pure,
fully homomorphic encryption system which has been so long sought.

Hal Finney

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