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g1C{#&wfkd_I5`_QI{#yTa4@oRGDDG)ipq)oF9L0{*8l(j diff --git a/src/sonic/discussion.html b/src/sonic/discussion.html new file mode 100644 index 0000000..abe1c98 --- /dev/null +++ b/src/sonic/discussion.html @@ -0,0 +1,40 @@ +

S polynomial transformation for permutation argument

+

Decomposition

+

Following the original suggestion + s(X, Y) = X^{-N-1}Y^{N} s_{1}(X, Y) - X^{N}s2(X, Y)

+

 s_{1}(X,Y) = \sum_{i=1}^{N}u'_{i}(Y) X^{-i+N+1} + \sum_{i=1}^{N}v'_{i}(Y) X^{i+N+1} + \sum_{i=1}^{N}w'_{i}(Y) X^{i+2N+1}

+

 s_2(X,Y) is not important for this discussion.  s_1(X,Y) is in total a polynomial of degree 3N + 1.

+

 u'_{i}(Y) = \sum_{q=1}^{Q} Y^{q} u(q, i)

+

and with a similar form for  v'(Y) and  w'(Y)

+

 u(q, i) by itself is a constant in q-th linear constraint in front of a variable a(i).  v(q, i) and  w(q, i) have the same meaning for b(i) and c(i).

+

In total  s_1(X,Y) can be represented as a large convolution in a form  M_{q,i} N^{q} K^{i} where summing is over the same index that is placed up and down. Vectors are  N^{q} = [Y, Y^{1}, ..., Y^{Q}] and  K^{i} = [X, X^{2}, ..., X^{3N+1}] , so the matrix  M_{q,i} is sparse and q-th row is formed by the concatenation of coefficients of  u(q, i) ,  v(q, i) and  w(q, i) (i notation is abused). For two multiplication gates (giving variables a(1), a(2),..., c(2)) and a linear constraint 10a(1) - b(1) - c(2) = 0 a first row would look like

+

 [10, 0, -1, 0, 0, -1]

+

There are three questions:

+
    +
  • Original paper states that  s_1(X,Y) can be represented as a sum of three polynomials, each of those being a permutation by itself. Why three? One could try to transform a whole matrix  M_{q,i} to have one permutation argument.
  • +
  • If s_{1} is split into sum of three polynomials, are those polynomials each form an individual permutation argument for components like  \sum_{i=1}^{N}u'_{i}(Y) X^{-i+N+1} ?
  • +
  • What would be the most efficient procedure to do such a reduction? Just from an example above with a single constraint in a form  [10, 0, -1, 0, 0, -1] a first element will contribute in a summand  10X^{2}Y^{1}, while to make a permutation argument one has to first create a term  10X^{2}Y^{2}.
  • +
+

Continuing discussion

+

Implementation of a permutation argument requires to have some diagonal matrix  D_{q,i} to first commit to the combination like  \sum_{i=1}^{N} d_{i}X^{i}Y^{i} and later make a permutation argument to prove evaluation of  \sum_{i=1}^{N} d_{\sigma(i)}X^{i}Y^{\sigma(i)} for a fixed permutation  \sigma(i) .

+

In principle such requirement means that decomposition of our  M_{q,i} matrix into the sum of  j matrixes (let’s call them  J_{q,i}^{j} should have only a single coefficient in every row, so one can define a proper diagonal  D_{q,i}^{j}. Such decomposition and reduction needs to be done only once per circuit, cause  D_{q,i}^{j} and corresponding \sigma^{j}(i) will become fixed as a part of the specialized common reference.

+

One can not directly guess how many linear constraints and multiplication gates will be in a system. For example, trivial (w/o optimizing run, as given in the original SONICs implementation) reduction of R1CS will have number of multiplication gates equal to the  m/2 + n where  m is a number of variables and n is a number of constraints in R1CS.

+

Let’s take an assumption that  N > Q , so a final constraint system will have more multiplication gates that linear constraints. In this case one can propose the following reduction procedure:

+
    +
  • Forbid constraints that have a form  A(1) + A(1) + ... , basically require a deduplication step.
  • +
  • Each constraints may have one variable of flavors  A ,  B and  C to ensure that  M_{q,i} has only one coefficient for variable of each kind. In this case it can already be decomposed at three matrixes  J_{q,i}^{j} .
  • +
  • Every constraint that breaks this rule will give raise to a new linear constraint(s) and new multiplication gate(s).
  • +
  • Final constraints can NOT have zero coefficients in front of any flavor of variables, otherwise a permutation argument can not be made (it’s a necessary assumption for grand product argument).
  • +
  • Reduction itself is not trivial! Let’s do it step by step.
  • +
  • Constraint where  NumVar(A) > NumVars(B) > NumVars(C), so in a linear term number of contributions from variables of flavor A is greater than from flavor B, that is in term is greater than for a flavor C. In this case one can reduce  NumVar(A) and  NumVars(B) by one, and increase  NumVar(C) by one, through the introduction of a constraint  A + B - C = 0. This also gives one extra multiplication gate. One can continue this procedure until there is a linear constraint in one of the following forms: +
      +
    •  A + B + C = 0, then reduction is over
    • +
    •  A + B = 0.
    • +
    •  A = 0 (this in a case of public inputs also)
    • +
    +
  • +
  •  A + B = 0 allows to try to make a constraint system in the form  A(1) + B(1) - C(2) = 0,  A(1) + B(1) + C(2) = 0 with those two constraints going into the different  J_{q,i}^{j} .
  • +
  •  A = 0 if inflated into constraints A(1) + B(2) - C(2) = 0, A(1) - B(2) + C(2) = 0 (TODO: check the prefactors).
  • +
  • To have j = 3 one can not allow any variable of any flavor to happen more than 3 times in all the linear constraints. Otherwise for any permutation \sigma^{j}(i) one can not "choose" a corresponding linear constraint index.
  • +
+

Now this "simple" list of rules can be implemented :)

diff --git a/src/sonic/helped/helper.rs b/src/sonic/helped/helper.rs index f8823e7..b755a36 100644 --- a/src/sonic/helped/helper.rs +++ b/src/sonic/helped/helper.rs @@ -214,6 +214,7 @@ pub fn create_aggregate_on_srs_using_information, S: Sy let mut poly_positive = vec![E::Fr::zero(); 2*n]; let mut expected_value = E::Fr::zero(); + // TODO: this part can be further parallelized due to synthesis of S(X, y) being singlethreaded for (y, c_opening) in y_values.iter().zip(c_openings.iter()) { // Compute s(X, y_i) let (s_poly_negative, s_poly_positive) = {