[kortex]

Have you looked at reversible computing? The ideas are only vague in my mind at the moment, but the fragments are there. Disclaimer: out of my wheelhouse. Any conditional statement will throw some information away, "knowing" a==b means provably you have less complexity. Not a good look for encryption.

But instead let's say you have some register which gets rotated/xor'd in some way based on the relationship of a to b.

I don't think you can do branching logic though.

[garganzol]

Good suggestions, thank you.

> I don't think you can do branching logic though.

You can. If we would be able to introduce a special equality operator, let's call it SEQ that works like so:

SEQ(a, b) = 1 when a == b; SEQ(a, b) = 0 when a != b

Then we would be able to do conditionals by discarding the result of unmatched branch by multiplying it by 0.

Thanks to the fact that all HE calculations are pure, no observable side effects would be produced.

For example, here is a simple program with a conditional statement:

P(x) = x == 10 ? 42 : sin(x)

This is how it may look in HE domain:

P(x) = SEQ(x, 10) * 42 + (1 - SEQ(x, 10)) * sin(x)

No branching is involved here but the result is equivalent to a program with branching. Eureka!

Once again, thank you for the fruitful suggestions.

EDIT: by the way, it paves the way for implementing > and < operators as well due to the fact that operator result remains in encrypted domain. This is a serious wow moment.