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EIP 101: big-int precompile #258

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EIP 101: big-int precompile #258

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18ed628
EIP101: addition precompile
pirapira Mar 2, 2017
9ee3c9a
EIP101: add the subtraction precompile and the multiplication precompile
pirapira Mar 2, 2017
3f707fb
EIP 101: add division
pirapira Mar 3, 2017
db41732
EIP 101: Add EXPMOD precompile
pirapira Mar 3, 2017
c025e12
EIP 100: add EXPMOD precompile
pirapira Mar 3, 2017
80c3045
EIP 101: simplify a formula A \/ B into B,
pirapira Mar 3, 2017
8d1e99e
EIP 101: simplification
pirapira Mar 3, 2017
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108 changes: 107 additions & 1 deletion Paper.tex
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Original file line number Diff line number Diff line change
Expand Up @@ -779,6 +779,11 @@ \section{Message Call} \label{ch:call}
\Xi_{\mathtt{SHA256}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad r = 2 \\
\Xi_{\mathtt{RIP160}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad r = 3 \\
\Xi_{\mathtt{ID}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad r = 4 \\
\Xi_{\mathtt{ADD}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad r = 5 \\
\Xi_{\mathtt{SUB}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad r = 6 \\
\Xi_{\mathtt{MUL}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad r = 7 \\
\Xi_{\mathtt{DIV}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad r = 8 \\
\Xi_{\mathtt{EXPMOD}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad r = 9 \\
\Xi(\boldsymbol{\sigma}_1, g, I) & \text{otherwise} \end{cases} \\
I_a & \equiv & r \\
I_o & \equiv & o \\
Expand Down Expand Up @@ -1393,13 +1398,109 @@ \section{Precompiled Contracts}\label{app:precompiled}
\mathtt{\small RIPEMD160}(\mathbf{i} \in \mathbb{B}) & \equiv & o \in \mathbb{B}_{20}
\end{eqnarray}

Finally, the fourth contract, the identity function $\Xi_{\mathtt{ID}}$ simply defines the output as the input:
The fourth contract, the identity function $\Xi_{\mathtt{ID}}$ simply defines the output as the input:
\begin{eqnarray}
\Xi_{\mathtt{ID}} &\equiv& \Xi_{\mathtt{PRE}} \quad \text{where:} \\
g_r &=& 15 + 3\Big\lceil \dfrac{|I_\mathbf{d}|}{32} \Big\rceil\\
\mathbf{o} &=& I_\mathbf{d}
\end{eqnarray}

The fifth contract performs arbitrary-precision addition on non-negative integers.
The first word in the input specifies the number of bytes that the first non-negative integer $X$ occupies.
The result is represented in the smallest possible number of bytes. The first byte in the output is never zero.
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...the first byte in the output, if it exists...


\begin{eqnarray}
\Xi_{\mathtt{ADD}} &\equiv& \Xi_{\mathtt{PRE}} \quad \text{except:}\\
\Xi_{\mathtt{ADD}}(\boldsymbol{\sigma}, g, I) &\equiv& (\varnothing, 0, A^0, ()) \quad \text{if} \quad |I_\mathbf{d}| - 32 < I_\mathbf{d}[0..31] \\
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formalise conversion bytes -> integer

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  • Maybe I_\mathbf{d}[0..31] can be assigned to a variable.
  • Or the common parsing logic should be factored out.

g_r &=& G_{addsubbase} + G_{arithword} \Big\lceil \dfrac{|I_\mathbf{d}|}{32} \Big\rceil \\
\mathbf{o} &=& (X + Y) \in \mathbb{B}_\ell \\
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use BE or something similar to define integer -> bytes.

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  • the symbol \ell is already taken
  • the number of bits should be 8 \ell not \ell.

X &=& I_\mathbf{d}[32..(32 + I_\mathbf{d}[0..31] - 1)] \\
Y &=& I_\mathbf{d}[(32 + I_\mathbf{d}[0..31])..(|I_\mathbf{d}| - 1)] \\
\ell &=&
\begin{cases}
0 &\text{if}\, X = Y = 0 \\
\lfloor \log_8(X + Y) \rfloor + 1 & \text{otherwise}
\end{cases}
\end{eqnarray}

The sixth contract performs arbitrary-precision subtraction on non-negative integers. This is similar to the addition. The subtraction contract halts exceptionally if the result would be negative.

\begin{eqnarray}
\Xi_{\mathtt{SUB}} &\equiv& \Xi_{\mathtt{PRE}} \quad \text{except:}\\
\Xi_{\mathtt{SUB}}(\boldsymbol{\sigma}, g, I) &\equiv& (\varnothing, 0, A^0, ()) \quad \text{if} \quad |I_\mathbf{d}| - 32 < I_\mathbf{d}[0..31] \\
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  • ... if ...; ... if ... should go into begin{cases} \end{cases}

\Xi_{\mathtt{SUB}}(\boldsymbol{\sigma}, g, I) &\equiv& (\varnothing, 0, A^0, ()) \quad \text{if} \quad X < Y \\
g_r &=& G_{addsubbase} + G_{arithword} \Big\lceil \dfrac{|I_\mathbf{d}|}{32} \Big\rceil \\
\mathbf{o} &=& (X - Y) \in \mathbb{B}_\ell \\
X &=& I_\mathbf{d}[32..(32 + I_\mathbf{d}[0..31] - 1)] \\
Y &=& I_\mathbf{d}[(32 + I_\mathbf{d}[0..31])..(|I_\mathbf{d}| - 1)] \\
\ell &=&
\begin{cases}
0 &\text{if}\ X = Y \\
\lfloor \log_8(X - Y) \rfloor + 1 & \text{otherwise}
\end{cases}
\end{eqnarray}

The seventh contract performs arbitrary-precision multiplication on non-negative integers. The input format is the same as that of the addition contract.

\begin{eqnarray}
\Xi_{\mathtt{MUL}} &\equiv& \Xi_{\mathtt{PRE}} \quad \text{except:}\\
\Xi_{\mathtt{MUL}}(\boldsymbol{\sigma}, g, I) &\equiv& (\varnothing, 0, A^0, ()) \quad \text{if} \quad |I_\mathbf{d}| - 32 < I_\mathbf{d}[0..31] \\
g_r &=& G_{muldivbase} + G_{arithword} \Big\lceil \dfrac{|I_\mathbf{d}|}{32} \Big\rceil + \ell_X \ell_Y / G_{quaddivisor} \\
\mathbf{o} &=& (X \times Y) \in \mathbb{B}_\ell \\
X &=& I_\mathbf{d}[32..(32 + I_\mathbf{d}[0..31] - 1)] \\
\ell_X &=& I_\mathbf{d}[0..31] \\
Y &=& I_\mathbf{d}[(32 + I_\mathbf{d}[0..31])..(|I_\mathbf{d}| - 1)] \\
\ell_Y &=& |I_\mathbf{d}| - 32 - \ell_X \\
\ell &=&
\begin{cases}
0 &\text{if}\ X = 0 \ \vee\ Y = 0 \\
\lfloor \log_8(X \times Y) \rfloor + 1 & \text{otherwise}
\end{cases}
\end{eqnarray}

The eigth contract performs arbitrary-precision division on non-negative integers. The definition is similar to that of the multiplication contract.

\begin{eqnarray}
\Xi_{\mathtt{DIV}} &\equiv& \Xi_{\mathtt{PRE}} \quad \text{except:}\\
\Xi_{\mathtt{DIV}}(\boldsymbol{\sigma}, g, I) &\equiv& (\varnothing, 0, A^0, ()) \quad \text{if} \quad |I_\mathbf{d}| - 32 < I_\mathbf{d}[0..31] \\
g_r &=& G_{muldivbase} + G_{arithword} \Big\lceil \dfrac{|I_\mathbf{d}|}{32} \Big\rceil + \ell_X \ell_Y / G_{quaddivisor} \\
\mathbf{o} &=&
\begin{cases}
() & \text{if} \ Y = 0 \\
\Big\lfloor \dfrac X Y \Big\rfloor \in \mathbb{B}_\ell & \text{otherwise}
\end{cases} \\
X &=& I_\mathbf{d}[32..(32 + I_\mathbf{d}[0..31] - 1)] \\
\ell_X &=& I_\mathbf{d}[0..31] \\
Y &=& I_\mathbf{d}[(32 + I_\mathbf{d}[0..31])..(|I_\mathbf{d}| - 1)] \\
\ell_Y &=& |I_\mathbf{d}| - 32 - \ell_X \\
\ell &=&
\begin{cases}
0 &\text{if}\ Y = 0 \ \vee \ \Big\lfloor \dfrac X Y \Big\rfloor = 0 \\
\lfloor \log_8(\Big\lfloor \dfrac X Y \Big\rfloor) \rfloor + 1 & \text{otherwise}
\end{cases}
\end{eqnarray}

The ninth contract performs arbitrary-precision exponentiation under modulo. Here, $0 ^ 0$ is taken to be one.
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mention how the arguments are encoded here.

\begin{eqnarray}
\Xi_{\mathtt{EXPMOD}} &\equiv& \Xi_{\mathtt{PRE}} \quad \text{except:}\\
\ell_B &=& I_\mathbf{d}[0..31] \\
B &=& I_\mathbf{d}[32..(31 + \ell_B)] \\
\ell_E &=& I_\mathbf{d}[(32 + \ell_B)..(63 + \ell_B)] \\
E &=& I_\mathbf{d}[(64 + \ell_B)..(63 + \ell_B + \ell_E)] \\
M &=& I_\mathbf{d}[(64 + \ell_B + \ell_E)..(|I_\mathbf{d}| - 1)] \\
\Xi_{\mathtt{EXPMOD}}(\boldsymbol{\sigma}, g, I) &\equiv& (\varnothing, 0, A^0, ()) \quad \text{if} \quad |I_\mathbf{d}| < 64 + \ell_B + \ell_E \\
g_r &=& G_{modexpbase} + G_{arithword} \Big\lceil \dfrac{|I_\mathbf{d}|}{32} \Big\rceil + |M|^2 |E| / G_{quaddivisor} \\
\mathbf{o} &=&
\begin{cases}
() & \text{if} \ M = 0 \\
B ^ E \bmod M \in \mathbb{B}_\ell & \text{otherwise}
\end{cases} \\
\ell &=&
\begin{cases}
0 &\text{if}\ M = 0 \quad \vee \quad {B ^ E} \bmod M = 0 \\
\lfloor \log_8(B ^ E \bmod M) \rfloor + 1 & \text{otherwise}
\end{cases}
\end{eqnarray}

\section{Signing Transactions}\label{app:signing}

Expand Down Expand Up @@ -1506,6 +1607,11 @@ \section{Fee Schedule}\label{app:fees}
$G_{sha3word}$ & 6 & Paid for each word (rounded up) for input data to a {\small SHA3} operation. \\
$G_{copy}$ & 3 & Partial payment for {\small *COPY} operations, multiplied by words copied, rounded up. \\
$G_{blockhash}$ & 20 & Payment for {\small BLOCKHASH} operation. \\
$G_{addsubbase}$ & 15 & Payment for the precompiled addition or subtraction contract. \\
$G_{muldivbase}$ & 30 & Payment for the precompiled multiplication or division contract. \\
$G_{modexpbase}$ & 45 & Payment for the precompiled exponention under modulo. \\
$G_{arithword}$ & 6 & Paid for each word used in precompiled contracts for arbitrary precision arighmetics.\\
$G_{quaddivisor}$ & 32 & The quadratic coefficient of the input sizes of multiplication and division precompiled contracts. \\

%extern u256 const c_copyGas; ///< Multiplied by the number of 32-byte words that are copied (round up) for any *COPY operation and added.
\bottomrule
Expand Down