3.0 KiB
Multi Precision
Each integer in the circuit has a certain bit-width, which is determined by the inputset. These bit-widths can be observed when graphs are printed:
%0 = x # EncryptedScalar<uint3> ∈ [0, 7]
%1 = y # EncryptedScalar<uint4> ∈ [0, 15]
%2 = add(%0, %1) # EncryptedScalar<uint5> ∈ [2, 22]
return %2 ^ these are ^^^^^^^
the assigned based on
bit-widths these bounds
However, it's not possible to add 3-bit and 4-bit numbers together because their encoding is different:
D: data
N: noise
3-bit number
------------
D2 D1 D0 0 0 0 ... 0 0 0 N N N N
4-bit number
------------
D3 D2 D1 D0 0 0 0 ... 0 0 0 N N N N
Furthermore, the result is a 5-bit number, so it has a different encoding as well:
5-bit number
------------
D4 D3 D2 D1 D0 0 0 0 ... 0 0 0 N N N N
Because of this encoding difference, we do a graph processing step called bit-width assignment, which takes the graph and updates bit-widths in the graph to be compatible with FHE.
After this graph processing pass, the graph would look like:
%0 = x # EncryptedScalar<uint5>
%1 = y # EncryptedScalar<uint5>
%2 = add(%0, %1) # EncryptedScalar<uint5>
return %2
Most operations cannot change the encoding, so they need to share bit-width between their inputs and their outputs but there is a very important operation which can change the encoding, and it's the table lookup operation.
Let's say you have this graph:
%0 = x # EncryptedScalar<uint2> ∈ [0, 3]
%1 = y # EncryptedScalar<uint5> ∈ [0, 31]
%2 = 2 # ClearScalar<uint2> ∈ [2, 2]
%3 = power(%0, %2) # EncryptedScalar<uint4> ∈ [0, 9]
%4 = add(%3, %1) # EncryptedScalar<uint6> ∈ [1, 39]
return %4
This is the graph for (x**2) + y where x is 2-bits and y is `5-bits. If the table lookup operation wasn't able to change the encoding, we'd need to make everything 6-bits but because they can, bit-widths can be assigned like so:
%0 = x # EncryptedScalar<uint2> ∈ [0, 3]
%1 = y # EncryptedScalar<uint6> ∈ [0, 31]
%2 = 2 # ClearScalar<uint2> ∈ [2, 2]
%3 = power(%0, %2) # EncryptedScalar<uint6> ∈ [0, 9]
%4 = add(%3, %1) # EncryptedScalar<uint6> ∈ [1, 39]
return %4
In this case, we kept x 2-bit, but set the table lookup result and y to 6-bits, so the addition can be performed.
This style of bit-width assignment is called multi-precision. Unfortunately, it's disabled by default at the moment.
To enable it, you can use single_precision=False configuration option.
{% hint style="info" %} Multi precision will become the default at one point in the near future! {% endhint %}