diff --git a/background/upa.html b/background/upa.html index 1f10f484..14840385 100644 --- a/background/upa.html +++ b/background/upa.html @@ -178,8 +178,9 @@ every row (i.e. at every element in the multiplicative subgroup).

Columns

-

and we tried to implement adding an equality constraint just using step 4 of the above +

and we tried to implement adding an equality constraint just using step 5 of the above algorithm, then we would end up constructing the cycle , rather than the correct .

Argument specification

-

We need to represent permutations over columns, represented by polynomials .

-

We first assign a unique element of as an "extended domain" element for each cell -that can participate in the permutation argument.

+

We need to check a permutation of cells in columns, represented in Lagrange basis by +polynomials .

+

We first label each cell in those columns with a unique element of .

Let be a root of unity and let be a root of unity, where with odd and . -We will use as the extended domain element for the +We will use as the label for the cell in the th row of the th column of the permutation argument.

If we have a permutation , we can represent it as a vector of polynomials such that .

Notice that the identity permutation can be represented by the vector of polynomials such that .

-

Now given our permutation represented by , over advice columns represented by +

Now given our permutation represented by over columns represented by , we want to ensure that:

Let be such that and for : diff --git a/print.html b/print.html index e1735878..078f3566 100644 --- a/print.html +++ b/print.html @@ -1122,8 +1122,9 @@ nothing to do.

  • Otherwise, and belong to different cycles. Make the larger cycle and the smaller one, by swapping them iff .
  • +
  • Set .
  • Following the mapping around the right (smaller) cycle, for each element set -.
  • +.
  • Splice the smaller cycle into the larger one by swapping with .
  • @@ -1159,22 +1160,22 @@ following constraints:

  • -

    and we tried to implement adding an equality constraint just using step 4 of the above +

    and we tried to implement adding an equality constraint just using step 5 of the above algorithm, then we would end up constructing the cycle , rather than the correct .

    Argument specification

    -

    We need to represent permutations over columns, represented by polynomials .

    -

    We first assign a unique element of as an "extended domain" element for each cell -that can participate in the permutation argument.

    +

    We need to check a permutation of cells in columns, represented in Lagrange basis by +polynomials .

    +

    We first label each cell in those columns with a unique element of .

    Let be a root of unity and let be a root of unity, where with odd and . -We will use as the extended domain element for the +We will use as the label for the cell in the th row of the th column of the permutation argument.

    If we have a permutation , we can represent it as a vector of polynomials such that .

    Notice that the identity permutation can be represented by the vector of polynomials such that .

    -

    Now given our permutation represented by , over advice columns represented by +

    Now given our permutation represented by over columns represented by , we want to ensure that:

    Let be such that and for : @@ -3050,8 +3051,9 @@ every row (i.e. at every element in the multiplicative subgroup).

    Columns