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# 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

The result of such an addition is a 5-bit number, which also has a different encoding:

5-bit number

------------

D4 D3 D2 D1 D0 0 0 0 ... 0 0 0 N N N N

Because of these encoding differences, we perform a graph processing step called bit-width assignment, which takes the graph and updates the bit-widths to be compatible with FHE.

After this graph processing step, 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, which means that the input and output bit-widths need to be the same. However, there is an operation which can change the encoding: 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. However, since the encoding can be changed, the 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`

as 2-bits, but set the table lookup result and `y`

to be 6-bits, so that the addition can be performed.This style of bit-width assignment is called multi-precision, and it is enabled by default. To disable it and use a single precision across the circuit, you can use the

`single_precision=True`

configuration option.Last modified 1mo ago