Multi precision
This document explains the multi-precision option for bit-width assignment for integers.
The multi-precision option enables the frontend to use the smallest bit-width possible for each operation in Fully Homomorphic Encryption (FHE), improving computation efficiency.
Bit-width and encoding differences
Each integer in the circuit has a certain bit-width, which is determined by the input-set. These bit-widths are visible when graphs are printed, for example:
%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, adding integers with different bit-widths (for example, 3-bit and 4-bit numbers) directly isn't possible due to differences in encoding, as shown below:
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
When you add a 3-bit number and a 4-bit number, the result is a 5-bit number with a different encoding:
5-bit number
------------
D4 D3 D2 D1 D0 0 0 0 ... 0 0 0 N N N N
Bit-width assignment with graph processing
To address these encoding differences, a graph processing step called bit-width assignment is performed. This step updates the graph's bit-widths to ensure compatibility with Fully Homomorphic Encryption (FHE).
After this step, the graph might look like this:
%0 = x # EncryptedScalar<uint5>
%1 = y # EncryptedScalar<uint5>
%2 = add(%0, %1) # EncryptedScalar<uint5>
return %2
Encoding flexibility with Table Lookup
Most operations cannot change the encoding, requiring the input and output bit-widths to remain the same. However, the table lookup operation can change the encoding. For example, consider the following 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 graph represents the computation (x**2) + y
where x
is 2-bits and y
is 5-bits. Without the ability to change encodings, all bit-widths would need to be adjusted to 6-bits. However, since the encoding can change, bit-widths are assigned more efficiently:
%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, x
remains a 2-bit integer, but the Table Lookup result and y
are set to 6-bits to allow for the addition.
Enabling and disabling multi-precision
This approach to bit-width assignment is known as multi-precision and is enabled by default. To disable multi-precision and enforce a single precision across the circuit, use the single_precision=True
configuration option.
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