Stage 3 — Shift & Instruction Input

Degree 2

Rounds $\log_2 T$

$\displaystyle\sum$ over: $X_j \in \{0,1\}^{\log_2 T}$

Opening point $(r_{\text{cycle}}^{(3)})$

\textcolor{BurntOrange}{\textsf{NextUnexpandedPC}}(r_{\text{cycle}}^{(1)}) + \gamma \cdot \textcolor{BurntOrange}{\textsf{NextPC}}(r_{\text{cycle}}^{(1)}) + \gamma^{2} \cdot \textcolor{BurntOrange}{\textsf{NextIsVirtual}}(r_{\text{cycle}}^{(1)}) + \gamma^{3} \cdot \textcolor{BurntOrange}{\textsf{NextIsFirstInSequence}}(r_{\text{cycle}}^{(1)}) + \gamma^{4} \cdot \left(1 - \textcolor{BurntOrange}{\textsf{NextIsNoop}}(r_{\text{cycle}}^{(2)})\right)

\widetilde{\textsf{EqPlusOne}}(r_{\text{cycle}}^{(1)}, X_j) \cdot \left(\textcolor{BurntOrange}{\textsf{UnexpandedPC}}(X_j) + \gamma \cdot \textcolor{BurntOrange}{\textsf{PC}}(X_j) + \gamma^{2} \cdot \textcolor{BurntOrange}{\textsf{OpFlags}(\text{VirtualInstruction})}(X_j) + \gamma^{3} \cdot \textcolor{BurntOrange}{\textsf{OpFlags}(\text{IsFirstInSequence})}(X_j)\right) + \gamma^{4} \cdot \widetilde{\textsf{EqPlusOne}}(r_{\text{cycle}}^{(2)}, X_j) \cdot \left(1 - \textcolor{BurntOrange}{\textsf{InstructionFlags}(\text{IsNoop})}(X_j)\right)

$\textcolor{BurntOrange}{\textsf{UnexpandedPC}}(r_{\text{cycle}}^{(3)})$
$\textcolor{BurntOrange}{\textsf{PC}}(r_{\text{cycle}}^{(3)})$
$\textcolor{BurntOrange}{\textsf{OpFlags}(\text{VirtualInstruction})}(r_{\text{cycle}}^{(3)})$
$\textcolor{BurntOrange}{\textsf{OpFlags}(\text{IsFirstInSequence})}(r_{\text{cycle}}^{(3)})$
$\textcolor{BurntOrange}{\textsf{InstructionFlags}(\text{IsNoop})}(r_{\text{cycle}}^{(3)})$

Degree 2

Rounds $\log_2 T$

$\displaystyle\sum$ over: $X_j \in \{0,1\}^{\log_2 T}$

Opening point $(r_{\text{cycle}}^{(3)})$

\textcolor{BurntOrange}{\textsf{RightInstructionInput}}(r_{\text{cycle}}^{(1)}) + \gamma \cdot \textcolor{BurntOrange}{\textsf{LeftInstructionInput}}(r_{\text{cycle}}^{(1)}) + \gamma^{2} \cdot \textcolor{BurntOrange}{\textsf{RightInstructionInput}}(r_{\text{cycle}}^{(2)}) + \gamma^{3} \cdot \textcolor{BurntOrange}{\textsf{LeftInstructionInput}}(r_{\text{cycle}}^{(2)})

\left(\widetilde{\text{eq}}(r_{\text{cycle}}^{(1)}, X_j) + \gamma^{2} \cdot \widetilde{\text{eq}}(r_{\text{cycle}}^{(2)}, X_j)\right) \cdot \left(\textcolor{BurntOrange}{\textsf{InstructionFlags}(\text{RightOperandIsRs2Value})}(X_j) \cdot \textcolor{BurntOrange}{\textsf{Rs2Value}}(X_j) + \textcolor{BurntOrange}{\textsf{InstructionFlags}(\text{RightOperandIsImm})}(X_j) \cdot \textcolor{BurntOrange}{\textsf{Imm}}(X_j) + \gamma \cdot \left(\textcolor{BurntOrange}{\textsf{InstructionFlags}(\text{LeftOperandIsRs1Value})}(X_j) \cdot \textcolor{BurntOrange}{\textsf{Rs1Value}}(X_j) + \textcolor{BurntOrange}{\textsf{InstructionFlags}(\text{LeftOperandIsPC})}(X_j) \cdot \textcolor{BurntOrange}{\textsf{UnexpandedPC}}(X_j)\right)\right)

$\textcolor{BurntOrange}{\textsf{InstructionFlags}(\text{LeftOperandIsRs1Value})}(r_{\text{cycle}}^{(3)})$
$\textcolor{BurntOrange}{\textsf{Rs1Value}}(r_{\text{cycle}}^{(3)})$
$\textcolor{BurntOrange}{\textsf{InstructionFlags}(\text{LeftOperandIsPC})}(r_{\text{cycle}}^{(3)})$
$\textcolor{BurntOrange}{\textsf{UnexpandedPC}}(r_{\text{cycle}}^{(3)})$
$\textcolor{BurntOrange}{\textsf{InstructionFlags}(\text{RightOperandIsRs2Value})}(r_{\text{cycle}}^{(3)})$
$\textcolor{BurntOrange}{\textsf{Rs2Value}}(r_{\text{cycle}}^{(3)})$
$\textcolor{BurntOrange}{\textsf{InstructionFlags}(\text{RightOperandIsImm})}(r_{\text{cycle}}^{(3)})$
$\textcolor{BurntOrange}{\textsf{Imm}}(r_{\text{cycle}}^{(3)})$

Degree 2

Rounds $\log_2 T$

$\displaystyle\sum$ over: $X_j \in \{0,1\}^{\log_2 T}$

Opening point $(r_{\text{cycle}}^{(3)})$

\textcolor{BurntOrange}{\textsf{RdWriteValue}}(r_{\text{cycle}}^{(1)}) + \gamma \cdot \textcolor{BurntOrange}{\textsf{Rs1Value}}(r_{\text{cycle}}^{(1)}) + \gamma^{2} \cdot \textcolor{BurntOrange}{\textsf{Rs2Value}}(r_{\text{cycle}}^{(1)})

\widetilde{\text{eq}}(r_{\text{cycle}}^{(1)}, X_j) \cdot \left(\textcolor{BurntOrange}{\textsf{RdWriteValue}}(X_j) + \gamma \cdot \textcolor{BurntOrange}{\textsf{Rs1Value}}(X_j) + \gamma^{2} \cdot \textcolor{BurntOrange}{\textsf{Rs2Value}}(X_j)\right)