# Biological vision as a problem of Fully Homomorphic Encryption

One revelation after another! After learning about MOOC, I am reading now the Craig Gentry’s PhD thesis on fully homomorphic encryption.

Explanation: the problem of biological vision is the following. We have an organism, say a human or a fly.  By vision, the outer space is encrypted as a physical dynamical system in the brain, in a way which is basically unknown. However, the  encrypted information  is so good that the brain can compute, based on it, some function, which is then sent to the motor system, which in turn modifies the outer space efficiently (human kills fly or fly avoids human).

During this process there is no decryption, because there is no space, or image, in the brain (see the homunculus fallacy).

Therefore, the encryption used by the brain has to be a fully homomorphic encryption!

You may imagine how amazed I am by reading Gentry’s description, which I quote, with holes and my emphasis, from page 2 of his thesis:

Imagine you have an encryption scheme with a “noise parameter” attached to each ciphertext, where encryption outputs a ciphertext with small noise – say, less than $n$  – but decryption works as long as the noise is less than some threshold $N \gg n$. Furthermore,imagine you have algorithms […] that can take ciphertexts $E(a)$  and $E(b)$  and compute $E(a+b)$ and $E(a*b)$, but at the cost of adding or multiplying the noise parameters. This immediately gives a “somewhat homomorphic” encryption scheme […]. Now suppose that you have an algorithm Recrypt that takes a ciphertext $E(a)$  with noise $N' < N$  and outputs a “fresh” ciphertext $E(a)$  that also encrypts $a$, but which has noise parameter smaller than $N^{1/2}$ .

[…] It turns out that a somewhat homomorphic encryption scheme that has this self-referential property of being able to handle circuits that are deeper than its own decryption circuit – in which case we say the somewhat homomorphic encryption scheme is “bootstrappable” – is enough to obtain the Recrypt algorithm, and thereby fully homomorphic encryption!

In order to understand my enthusiasm, here is again a link to exploring space slides, see also the posts concerning approximate structures.