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Co-authored-by: David Cook <dcook@divviup.org>
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cjpatton and divergentdave authored Aug 22, 2024
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Expand Up @@ -3190,8 +3190,8 @@ For example, the simplest circuit specified in this document is the following
C(x) = x * (x-1)
~~~

This circuit contains one subtraction gate (`x -1`) and one multiplication
gate (`x * (x -1)`). Observe that `C(x) = 0` if and only if `x in range(2)`.
This circuit contains one subtraction gate (`x - 1`) and one multiplication
gate (`x * (x - 1)`). Observe that `C(x) = 0` if and only if `x in range(2)`.

Our goal is to allow each Aggregator, who holds a secret share of `x`, to
correctly compute a secret share of `C(x)`. This allows the Aggregators to
Expand All @@ -3200,7 +3200,7 @@ determine validity by combining their shares of the output.
Suppose for a moment that the validity circuit `C` is affine, meaning its only
operations are addition, subtraction, and multiplication-by-constant. (The
circuit above is non-affine because it contains a multiplication gate with
non-constant inputs.) Then each Aggregator can compute its share locally, since
two non-constant inputs.) Then each Aggregator can compute its share locally, since

~~~
C(x_shares[0] + ... + x_shares[SHARES-1]) =
Expand All @@ -3209,7 +3209,7 @@ C(x_shares[0] + ... + x_shares[SHARES-1]) =

(Note that, for this equality to hold, it is necessary to scale any addition of
a constant in the circuit by `1/SHARES`.) However, this is not the case if `C`
contains multiplication gates with non-constant inputs. Thus our goal is to
contains multiplication gates with two non-constant inputs. Thus our goal is to
transform these multiplication gates into computations on secret shared data
that each Aggregator can perform locally.

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