sumcheck formula

syntax: initial claim = sumcheck formula

phase 1 out (gate x wit_out )

$$\sum _i{\alpha }^i{\text{wit\_out}}_i[r{t}_i||r{y}_i]=\sum _i{\alpha }^i\sum _{g\in [0,G]}\sum _{s||x}\text{eq(}r{t}_i\text{, s)}{\text{copy\_to}}_{\text{g ->i}}(r{y}_i,x){\text{V}}_g[s||x]$$

phase 2 (gates), support linear/non linear/custom gate

$$\sum _{g\in [0,G]}{\alpha }^g{\text{V}}_g[r{t}_g||r{y}_g]=\sum _{g\in [0,G]}{\alpha }^g\sum _{s||x}\text{eq(}r{t}_g\text{, s)}{\text{gate}}_g(r{y}_g,x){\text{Gate}}_g(V_g(s||x||c0),V_g(s||x||c1),...)$$

here right-hand-side

$V_g$ can come from either next layer, or previous layers as subset. In subset case,

$Gat{e}_g$ function as

$\text{paste\_from}(ry,x)$. c0, c1, c2, c3 … are binary constant to access fixed fanin e.g. (0, 0), (0, 1), (1, 0), (1, 1)

phase 1 (gate x gate full shuffle), linear operation

$$\sum _{g1\in [0,G]}{\alpha }^{g1}{\text{V}}_{g1}[r{t}_{g1}||r{y}_{g1}]=\sum _{g1\in [0,G]}{\alpha }^{g1}\sum _{g2\in [0,G]}\sum _{s||x}\text{eq(}r{t}_{g1}\text{, s)}{\text{copy\_to}}_{\text{g2 ->g1}}(r{y}_{g1},x){\text{V}}_{g2}[s||x]$$

here left-hand-side

$V_{g1}$ can be subset copied to early layers.

phase 1 input (gate x wit_in full shuffle)

$$\sum _{g\in [0,G]}{\alpha }^g{\text{V}}_g[r{t}_g||r{y}_g]=\sum _{g\in [0,G]}{\alpha }^g\sum _i\sum _{s||x}\text{eq(}r{t}_g\text{, s)}{\text{paste\_from}}_{\text{i ->g}}(r{y}_g,x){\text{wit\_in}}_i[s||x]$$