mirror of https://github.com/zcash/halo2.git
Merge pull request #108 from zcash/book-patch-sha256
[book] Patch message scheduling table
This commit is contained in:
ying tong
GitHub
commit
2fe7daa6f4
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@ -500,7 +500,7 @@ sr|ss|s22|s23| $a_0$ | $a_1$ | $a_2$ | $a_3$
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0| 0| 1 | 0 |{0,1,2,3,4,5}|$R_0^{even}$|$\texttt{spread}(R_0^{even})$| $b_0$ | $b'_0$ | $b_1$ | $b'_1$ |
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0| 1| 0 | 0 |{0,1,2,3,4,5}|$R_0^{odd}$ |$\texttt{spread}(R_0^{odd})$ |$\texttt{spread}(R_1^{odd})$ |$\texttt{spread}(c)$|$\texttt{spread}(d)$|
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0| 0| 0 | 1 |{0,1,2,3,4,5}|$R_1^{even}$|$\texttt{spread}(R_1^{even})$| $b_1$ | $b'_1$ | $a$ | $a'$ |
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1| 0| 0 | 0 |{0,1,2,3,4,5}|$R_1^{odd}$ |$\texttt{spread}(R_1^{odd})$ | $R_1$ | $R_0$ | $R_0^{even}$ | $R_0^{odd}$ |
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1| 0| 0 | 0 |{0,1,2,3,4,5}|$R_1^{odd}$ |$\texttt{spread}(R_1^{odd})$ | $R_0$ | $R_1$ | $R_0^{even}$ | $R_0^{odd}$ |
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Constraints:
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- `ss` ($\sigma_0$ v1 constraint): $LHS - RHS = 0$
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@ -538,7 +538,7 @@ sr|ss|s23| $a_0$ | $a_1$ | $a_2$ | $a_3$ |
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0 |0 | 1 |{0,1,2,3,4,5}|$R_0^{even}$|$\texttt{spread}(R_0^{even})$| $b_0$ | $b'_0$ | $a$ | $a'$ | $e$ |
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0 |1 | 0 |{0,1,2,3,4,5}|$R_0^{odd}$ |$\texttt{spread}(R_0^{odd})$ | $\texttt{spread}(R_1^{odd})$| $\texttt{spread}(d)$| $\texttt{spread}(g)$ |
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0 |0 | 1 |{0,1,2,3,4,5}|$R_1^{even}$|$\texttt{spread}(R_1^{even})$| $b_1$ | $b'_1$ | $c$ | $c'$ | $f$ |
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1 |0 | 0 |{0,1,2,3,4,5}|$R_1^{odd}$ |$\texttt{spread}(R_1^{odd})$ | $R_1$ | $R_0$ |$R_0^{even}$|$R_0^{odd}$ | |
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1 |0 | 0 |{0,1,2,3,4,5}|$R_1^{odd}$ |$\texttt{spread}(R_1^{odd})$ | $R_0$ | $R_1$ |$R_0^{even}$|$R_0^{odd}$ | |
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Constraints:
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- `ss` ($\sigma_0$ v2 constraint): $LHS - RHS = 0$
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@ -603,12 +603,12 @@ $$
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#### v2
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v2 of the $\sigma_1$ gate takes in a word that's split into $(3, 4, 3, 7, 1, 1, 13)$-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as $(a, b, c, d, e, f, g).$ We already have $\mathtt{spread}(e), \mathtt{spread}(g)$ from the message scheduling. The 1-bit $e,f$ remain unchanged by the spread operation and can be used directly. We further split $b$ into two 2-bit chunks $b_0, b_1.$ $a', b'_0, b'_1, c'$ are the spread versions of the small chunks.
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ss|s23| $a_0$ | $a_1$ | $a_2$ | $a_3$ | $a_4$ | $a_5$ | $a_6$ | $a_7$ |
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--|---|-------------|------------|-----------------------------|------------|------------|------------|------------|------------|
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0 | 1 |{0,1,2,3,4,5}|$R_0^{even}$|$\texttt{spread}(R_0^{even})$| $b_0$ | $b'_0$ | $a$ | $a'$ | $e$
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1 | 0 |{0,1,2,3,4,5}|$R_0^{odd}$ |$\texttt{spread}(R_0^{odd})$ | $\texttt{spread}(R_1^{odd})$| $\texttt{spread}(d)$| $\texttt{spread}(g)$ |
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0 | 1 |{0,1,2,3,4,5}|$R_1^{even}$|$\texttt{spread}(R_1^{even})$| $b_1$ | $b'_1$ | $c$ | $c'$ | $f$
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0 | 0 |{0,1,2,3,4,5}|$R_1^{odd}$ |$\texttt{spread}(R_1^{odd})$ | $R_0$ | $R_1$ |$R_0^{even}$|$R_0^{odd}$ |
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sr|ss|s23| $a_0$ | $a_1$ | $a_2$ | $a_3$ | $a_4$ | $a_5$ | $a_6$ | $a_7$ |
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--|--|---|-------------|------------|-----------------------------|------------|------------|------------|------------|------------|
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0 |0 | 1 |{0,1,2,3,4,5}|$R_0^{even}$|$\texttt{spread}(R_0^{even})$| $b_0$ | $b'_0$ | $a$ | $a'$ | $e$
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0 |1 | 0 |{0,1,2,3,4,5}|$R_0^{odd}$ |$\texttt{spread}(R_0^{odd})$ | $\texttt{spread}(R_1^{odd})$| $\texttt{spread}(d)$| $\texttt{spread}(g)$ |
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0 |0 | 1 |{0,1,2,3,4,5}|$R_1^{even}$|$\texttt{spread}(R_1^{even})$| $b_1$ | $b'_1$ | $c$ | $c'$ | $f$
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1 |0 | 0 |{0,1,2,3,4,5}|$R_1^{odd}$ |$\texttt{spread}(R_1^{odd})$ | $R_0$ | $R_1$ |$R_0^{even}$|$R_0^{odd}$ |
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Constraints:
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- `ss` ($\sigma_1$ v2 constraint): $LHS - RHS = 0$
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@ -746,40 +746,40 @@ For each block $M \in \{0,1\}^{512}$ of the padded message, $64$ words of $32$ b
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- the remaining $48$ words are constructed using the formula:
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$$W_i = \sigma_1(W_{i-2}) \boxplus W_{i-7} \boxplus \sigma_0(W_{i-15}) \boxplus W_{i-16},$$ for $i = 16, \ldots, 63$.
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sw|sd0|sd1|sd2|sd3|sr |ss0|ss0_v2|ss1|ss1_v2|s22|s23| $a_0$ | $a_1$ | $a_2$ | $a_3$ | $a_4$ | $a_5$ | $a_6$ | $a_7$ | $a_8$ |
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--|---|---|---|---|---|---|------|---|------|---|---|---------------|------------------|-----------------------------------|------------------------------|----------------------------------|---------------------------------|--------------------------------- |------------------------|------------------------|
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0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $W_{0}^{lo}$ | $\texttt{spread}(W_{0}^{lo})$ | | | $W_{0}$ |$\sigma_0(W_1)^{lo}$ |$\sigma_1(W_{14})^{lo}$ | $W_{9}^{lo}$ |
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1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $W_{0}^{hi}$ | $\texttt{spread}(W_{0}^{hi})$ | $W_{0}^{lo}$ | $W_{0}^{hi}$ | $W_{16}$ |$\sigma_0(W_1)^{hi}$ |$\sigma_1(W_{14})^{hi}$ | $W_{9}^{hi}$ |
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0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4} | $W_{1}^{d(14)}$ | $\texttt{spread}(W_{1}^{d(14)})$ | $W_{1}^{a(3)}$ | $W_{1}^{b(4)}$ | $W_{1}$ |$\sigma_0(W_2)^{lo}$ |$\sigma_1(W_{15})^{lo}$ | $W_{10}^{lo}$ |
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1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2} | $W_{1}^{c(11)}$ | $\texttt{spread}(W_{1}^{c(11)})$ | $W_{1}^{lo}$ | $W_{1}^{hi}$ | $W_{17}$ |$\sigma_0(W_2)^{hi}$ |$\sigma_1(W_{15})^{hi}$ | $W_{10}^{hi}$ |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | {0,1,2,3,4,5} | $R_0^{even}$ | $\texttt{spread}(R_0^{even})$ | $W_{1}^{b(4)lo}$ |$\texttt{spread}(W_{1}^{b(4)lo})$ | $W_{1}^{b(4) hi}$ |$\texttt{spread}(W_{1}^{b(4)hi})$ | | |
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0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $R_1^{odd}$ | $\texttt{spread}(R_1^{odd})$ | $\sigma_0 v1 R_1$ | $\sigma_0 v1 R_0$ | $\sigma_0 v1 R_0^{even}$ | $\sigma_0 v1 R_0^{odd}$ | | |
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0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $R_0^{odd}$ | $\texttt{spread}(R_0^{odd})$ | $\texttt{spread}(R_1^{odd})$ |$\texttt{spread}(W_{1}^{c(11)})$ |$\texttt{spread}(W_{1}^{d(14)})$ | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | {0,1,2,3,4,5} | $R_1^{even}$ | $\texttt{spread}(R_1^{even})$ | $W_{1}^{b(4)hi}$ |$\texttt{spread}(W_{1}^{b(4)hi})$ | $W_{1}^{a(3)}$ |$\texttt{spread}(W_{1}^{a(3)})$ | | |
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..|...|...|...|...|...|...|... |...|... |...|...| ... | ... | ... | ... | ... | ... | ... | ... | ... |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3} | $W_{14}^{g(13)}$ | $\texttt{spread}(W_{14}^{g(13)})$ | $W_{14}^{a(3)}$ | $W_{14}^{c(3)}$ | | | | |
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0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $W_{14}^{d(7)}$ | $\texttt{spread}(W_{14}^{d(7)})$ | $W_{14}^{e(1)}$ | $W_{14}^{f(1)}$ | $W_{14}$ |$\sigma_0(W_{15})^{lo}$ |$\sigma_1(W_{28})^{lo}$ | $W_{23}^{lo}$ |
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1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $W_{14}^{b(4)}$ | $\texttt{spread}(W_{14}^{b(4)})$ | $W_{14}^{lo}$ | $W_{14}^{hi}$ | $W_{30}$ |$\sigma_0(W_{15})^{hi}$ |$\sigma_1(W_{28})^{hi}$ | $W_{23}^{hi}$ |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | {0,1,2,3,4,5} | $R_0^{even}$ | $\texttt{spread}(R_0^{even})$ | $W_{14}^{b(4)lo}$ |$\texttt{spread}(W_{14}^{b(4)lo})$| $W_{14}^{a(3)}$ |$\texttt{spread}(W_{14}^{a(3)})$ | $W_{14}^{e(1)}$ | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $R_0^{odd}$ | $\texttt{spread}(R_0^{odd})$ | $\texttt{spread}(R_1^{odd})$ |$\texttt{spread}(W_{14}^{d(7)})$ |$\texttt{spread}(W_{14}^{g(13)})$| | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | {0,1,2,3,4,5} | $R_1^{even}$ | $\texttt{spread}(R_1^{even})$ | $W_{14}^{b(4) hi}$ |$\texttt{spread}(W_{14}^{b(4)hi})$| $W_{14}^{c(3)}$ |$\texttt{spread}(W_{14}^{c(3)})$ | $W_{14}^{f(1)}$ | |
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0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $R_1^{odd}$ | $\texttt{spread}(R_1^{odd})$ | $\sigma_0 v2 R_0$ | $\sigma_0 v2 R_1$ |$\sigma_0 v2 R_0^{even}$ |$\sigma_0 v2 R_0^{odd}$ | | |
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0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | {0,1,2,3,4,5} | $R_0^{even}$ | $\texttt{spread}(R_0^{even})$ | $W_{14}^{b(4)lo}$ |$\texttt{spread}(W_{14}^{b(4)lo})$| $W_{14}^{a(3)}$ |$\texttt{spread}(W_{14}^{a(3)})$ | $W_{14}^{e(1)}$ | |
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0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | {0,1,2,3,4,5} | $R_0^{odd}$ | $\texttt{spread}(R_0^{odd})$ | $\texttt{spread}(R_1^{odd})$ | $\texttt{spread}(d)$ | $\texttt{spread}(g)$ | | | |
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0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | {0,1,2,3,4,5} | $R_1^{even}$ | $\texttt{spread}(R_1^{even})$ | $W_{14}^{b(4) hi}$ |$\texttt{spread}(W_{14}^{b(4)hi})$| $W_{14}^{c(3)}$ |$\texttt{spread}(W_{14}^{c(3)})$ | $W_{14}^{f(1)}$ | |
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0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $R_1^{odd}$ | $\texttt{spread}(R_1^{odd})$ | $\sigma_1 v2 R_0$ | $\sigma_1 v2 R_1$ |$\sigma_1 v2 R_0^{even}$ |$\sigma_1 v2 R_0^{odd}$ | | |
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..|...|...|...|...|...|...|... |...|... |...|...| ... | ... | ... | ... | ... | ... | ... | ... | ... |
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0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3} | $W_{49}^{d(13)}$ | $\texttt{spread}(W_{49}^{d(13)})$ | $W_{49}^{c(2}$ | $W_{49}^{b(7)}$ | $W_{49}$ | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1} | $W_{49}^{a(10)}$ | $\texttt{spread}(W_{49}^{a(10)})$ | $W_{49}^{lo}$ | $W_{49}^{hi}$ | | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |{0,1,2,3,4,5} | $R_0^{even}$ | $\texttt{spread}(R_0^{even})$ | $W_{49}^{b(7)lo}$ |$\texttt{spread}(W_{49}^{b(7)lo})$| $W_{49}^{b(7)mid}$ |$\texttt{spread}(W_{49}^{b(7)mid})$| | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |{0,1,2,3,4,5} | $R_0^{odd}$ | $\texttt{spread}(R_0^{odd})$ | $\texttt{spread}(R_1^{odd})$ | $\texttt{spread}(a)$ | $\texttt{spread}(d)$ | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |{0,1,2,3,4,5} | $R_1^{even}$ | $\texttt{spread}(R_1^{even})$ | $W_{49}^{b(7)hi}$ |$\texttt{spread}(W_{49}^{b(7)hi})$| $W_{49}^{c(2)}$ |$\texttt{spread}(W_{49}^{c(2)})$ | | |
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0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 |{0,1,2,3,4,5} | $R_1^{odd}$ | $\texttt{spread}(R_1^{odd})$ | $\sigma_1 v1 R_0$ | $\sigma_1 v1 R_1$ |$R_0^{even}$ |$R_0^{odd}$ | | |
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..|...|...|...|...|...|...|... |...|... |...|...| ... | ... | ... | ... | ... | ... | ... | ... | ... |
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0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $W_{62}^{lo}$ | $\texttt{spread}(W_{62}^{lo})$ | | | $W_{62}$ | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $W_{62}^{hi}$ | $\texttt{spread}(W_{62}^{hi})$ | | | | | | |
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0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $W_{63}^{lo}$ | $\texttt{spread}(W_{63}^{lo})$ | | | $W_{63}$ | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $W_{63}^{hi}$ | $\texttt{spread}(W_{63}^{hi})$ | | | | | | |
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sw|sd0|sd1|sd2|sd3|sr |ss0|ss0_v2|ss1|ss1_v2|s22|s23| $a_0$ | $a_1$ | $a_2$ | $a_3$ | $a_4$ | $a_5$ | $a_6$ | $a_7$ | $a_8$ | $a_9$ |
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--|---|---|---|---|---|---|------|---|------|---|---|---------------|------------------|-----------------------------------|------------------------------|----------------------------------|---------------------------------|--------------------------------- |------------------------|------------------------|--------------|
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0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $W_{0}^{lo}$ | $\texttt{spread}(W_{0}^{lo})$ | $W_{0}^{lo}$ | $W_{0}^{hi}$ | $W_{0}$ |$\sigma_0(W_1)^{lo}$ |$\sigma_1(W_{14})^{lo}$ | $W_{9}^{lo}$ | |
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1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $W_{0}^{hi}$ | $\texttt{spread}(W_{0}^{hi})$ | | | $W_{16}$ |$\sigma_0(W_1)^{hi}$ |$\sigma_1(W_{14})^{hi}$ | $W_{9}^{hi}$ | $carry_{16}$ |
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0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4} | $W_{1}^{d(14)}$ | $\texttt{spread}(W_{1}^{d(14)})$ | $W_{1}^{lo}$ | $W_{1}^{hi}$ | $W_{1}$ |$\sigma_0(W_2)^{lo}$ |$\sigma_1(W_{15})^{lo}$ | $W_{10}^{lo}$ | |
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1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2} | $W_{1}^{c(11)}$ | $\texttt{spread}(W_{1}^{c(11)})$ | $W_{1}^{a(3)}$ | $W_{1}^{b(4)}$ | $W_{17}$ |$\sigma_0(W_2)^{hi}$ |$\sigma_1(W_{15})^{hi}$ | $W_{10}^{hi}$ | $carry_{17}$ |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | {0,1,2,3,4,5} | $R_0^{even}$ | $\texttt{spread}(R_0^{even})$ | $W_{1}^{b(4)lo}$ |$\texttt{spread}(W_{1}^{b(4)lo})$ | $W_{1}^{b(4) hi}$ |$\texttt{spread}(W_{1}^{b(4)hi})$ | | | |
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0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $R_1^{odd}$ | $\texttt{spread}(R_1^{odd})$ | $\sigma_0 v1 R_0$ | $\sigma_0 v1 R_1$ | $\sigma_0 v1 R_0^{even}$ | $\sigma_0 v1 R_0^{odd}$ | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $R_0^{odd}$ | $\texttt{spread}(R_0^{odd})$ | $\texttt{spread}(R_1^{odd})$ |$\texttt{spread}(W_{1}^{c(11)})$ |$\texttt{spread}(W_{1}^{d(14)})$ | | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | {0,1,2,3,4,5} | $R_1^{even}$ | $\texttt{spread}(R_1^{even})$ | $W_{1}^{b(4)hi}$ |$\texttt{spread}(W_{1}^{b(4)hi})$ | $W_{1}^{a(3)}$ |$\texttt{spread}(W_{1}^{a(3)})$ | | | |
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..|...|...|...|...|...|...|... |...|... |...|...| ... | ... | ... | ... | ... | ... | ... | ... | ... | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3} | $W_{14}^{g(13)}$ | $\texttt{spread}(W_{14}^{g(13)})$ | $W_{14}^{a(3)}$ | $W_{14}^{c(3)}$ | | | | | |
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0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $W_{14}^{d(7)}$ | $\texttt{spread}(W_{14}^{d(7)})$ | $W_{14}^{lo}$ | $W_{14}^{hi}$ | $W_{14}$ |$\sigma_0(W_{15})^{lo}$ |$\sigma_1(W_{28})^{lo}$ | $W_{23}^{lo}$ | |
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1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $W_{14}^{b(4)}$ | $\texttt{spread}(W_{14}^{b(4)})$ | $W_{14}^{e(1)}$ | $W_{14}^{f(1)}$ | $W_{30}$ |$\sigma_0(W_{15})^{hi}$ |$\sigma_1(W_{28})^{hi}$ | $W_{23}^{hi}$ | $carry_{30}$ |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | {0,1,2,3,4,5} | $R_0^{even}$ | $\texttt{spread}(R_0^{even})$ | $W_{14}^{b(4)lo}$ |$\texttt{spread}(W_{14}^{b(4)lo})$| $W_{14}^{a(3)}$ |$\texttt{spread}(W_{14}^{a(3)})$ | $W_{14}^{e(1)}$ | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $R_0^{odd}$ | $\texttt{spread}(R_0^{odd})$ | $\texttt{spread}(R_1^{odd})$ |$\texttt{spread}(W_{14}^{d(7)})$ |$\texttt{spread}(W_{14}^{g(13)})$| | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | {0,1,2,3,4,5} | $R_1^{even}$ | $\texttt{spread}(R_1^{even})$ | $W_{14}^{b(4) hi}$ |$\texttt{spread}(W_{14}^{b(4)hi})$| $W_{14}^{c(3)}$ |$\texttt{spread}(W_{14}^{c(3)})$ | $W_{14}^{f(1)}$ | | |
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0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $R_1^{odd}$ | $\texttt{spread}(R_1^{odd})$ | $\sigma_0 v2 R_0$ | $\sigma_0 v2 R_1$ |$\sigma_0 v2 R_0^{even}$ |$\sigma_0 v2 R_0^{odd}$ | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | {0,1,2,3,4,5} | $R_0^{even}$ | $\texttt{spread}(R_0^{even})$ | $W_{14}^{b(4)lo}$ |$\texttt{spread}(W_{14}^{b(4)lo})$| $W_{14}^{a(3)}$ |$\texttt{spread}(W_{14}^{a(3)})$ | $W_{14}^{e(1)}$ | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | {0,1,2,3,4,5} | $R_0^{odd}$ | $\texttt{spread}(R_0^{odd})$ | $\texttt{spread}(R_1^{odd})$ | $\texttt{spread}(d)$ | $\texttt{spread}(g)$ | | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | {0,1,2,3,4,5} | $R_1^{even}$ | $\texttt{spread}(R_1^{even})$ | $W_{14}^{b(4) hi}$ |$\texttt{spread}(W_{14}^{b(4)hi})$| $W_{14}^{c(3)}$ |$\texttt{spread}(W_{14}^{c(3)})$ | $W_{14}^{f(1)}$ | | |
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0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $R_1^{odd}$ | $\texttt{spread}(R_1^{odd})$ | $\sigma_1 v2 R_0$ | $\sigma_1 v2 R_1$ |$\sigma_1 v2 R_0^{even}$ |$\sigma_1 v2 R_0^{odd}$ | | | |
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..|...|...|...|...|...|...|... |...|... |...|...| ... | ... | ... | ... | ... | ... | ... | ... | ... | |
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0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3} | $W_{49}^{d(13)}$ | $\texttt{spread}(W_{49}^{d(13)})$ | $W_{49}^{lo}$ | $W_{49}^{hi}$ | $W_{49}$ | | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1} | $W_{49}^{a(10)}$ | $\texttt{spread}(W_{49}^{a(10)})$ | $W_{49}^{c(2)}$ | $W_{49}^{b(7)}$ | | | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |{0,1,2,3,4,5} | $R_0^{even}$ | $\texttt{spread}(R_0^{even})$ | $W_{49}^{b(7)lo}$ |$\texttt{spread}(W_{49}^{b(7)lo})$| $W_{49}^{b(7)mid}$ |$\texttt{spread}(W_{49}^{b(7)mid})$| | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |{0,1,2,3,4,5} | $R_0^{odd}$ | $\texttt{spread}(R_0^{odd})$ | $\texttt{spread}(R_1^{odd})$ | $\texttt{spread}(a)$ | $\texttt{spread}(d)$ | | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |{0,1,2,3,4,5} | $R_1^{even}$ | $\texttt{spread}(R_1^{even})$ | $W_{49}^{b(7)hi}$ |$\texttt{spread}(W_{49}^{b(7)hi})$| $W_{49}^{c(2)}$ |$\texttt{spread}(W_{49}^{c(2)})$ | | | |
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0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 |{0,1,2,3,4,5} | $R_1^{odd}$ | $\texttt{spread}(R_1^{odd})$ | $\sigma_1 v1 R_0$ | $\sigma_1 v1 R_1$ |$\sigma_1 v1 R_0^{even}$ |$\sigma_1 v1 R_0^{odd}$ | | | |
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..|...|...|...|...|...|...|... |...|... |...|...| ... | ... | ... | ... | ... | ... | ... | ... | ... | |
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0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $W_{62}^{lo}$ | $\texttt{spread}(W_{62}^{lo})$ | $W_{62}^{lo}$ | $W_{62}^{hi}$ | $W_{62}$ | | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $W_{62}^{hi}$ | $\texttt{spread}(W_{62}^{hi})$ | | | | | | | |
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0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $W_{63}^{lo}$ | $\texttt{spread}(W_{63}^{lo})$ | $W_{63}^{lo}$ | $W_{63}^{hi}$ | $W_{63}$ | | | | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | {0,1,2,3,4,5} | $W_{63}^{hi}$ | $\texttt{spread}(W_{63}^{hi})$ | | | | | | | |
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Constraints:
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- `sw`: construct word using $reduce_4$
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Loading…
Reference in New Issue