Remove add, sub, mul and div precompiles

Because ethereum/EIPs#198 supersedes ethereum/EIPs#101 and the new version does not contain these.

Paper.tex

@@ -762,11 +762,7 @@ \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_{\mathtt{EXPMOD}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad r = 5 \\
 \Xi(\boldsymbol{\sigma}_1, g, I) & \text{otherwise} \end{cases} \\
 I_a & \equiv & r \\
 I_o & \equiv & o \\
@@ -1395,67 +1391,7 @@ \section{Precompiled Contracts}\label{app:precompiled}
 \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 rest of the input is interpreted as $Y$.
-Both $X$ and $Y$ are interpreted as a natural number encoded as byte sequences in the big-endian way.
-The result is represented in the smallest possible number of bytes.  The first byte in the output, if exists, is never zero.
-The input and output formats are the same for $\Xi_{\mathtt{SUB}}$, $\Xi_{\mathtt{MUL}}$ and $\Xi_{\mathtt{DIV}}$ as well.
-
-\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 <  l_X \\
-g_r &=& G_{addsubbase} + G_{arithword} \Big\lceil \dfrac{|I_\mathbf{d}|}{32} \Big\rceil \\
-\mathbf{o} &=& \mathtt{\tiny BE}(X + Y) \\
-X &=& I_\mathbf{d}[32..(31 + l_X)] \\
-l_X &=& I_\mathbf{d}[0..31] \in \mathbb{P}_{256} \\
-Y &=& I_\mathbf{d}[(32 + l_X)..(|I_\mathbf{d}| - 1)] \\
-l_Y &=& |I_\mathbf{d}| - 32 - l_X
-\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 <  l_X \, \vee \, X < Y \\
-g_r &=& G_{addsubbase} + G_{arithword} \Big\lceil \dfrac{|I_\mathbf{d}|}{32} \Big\rceil \\
-\mathbf{o} &=& \mathtt{\tiny BE}(X - Y) \\
-X &=& I_\mathbf{d}[32..(31 + l_X)] \\
-l_X &=& I_\mathbf{d}[0..31] \in \mathbb{P}_{256} \\
-Y &=& I_\mathbf{d}[(32 + l_X)..(|I_\mathbf{d}| - 1)] \\
-l_Y &=& |I_\mathbf{d}| - 32 - l_X
-\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 < l_X \\
-g_r &=& G_{muldivbase} + G_{arithword} \Big\lceil \dfrac{|I_\mathbf{d}|}{32} \Big\rceil + l_X l_Y / G_{quaddivisor} \\
-\mathbf{o} &=& \mathtt{\tiny BE}(X \times Y) \\
-X &=& I_\mathbf{d}[32..(32 + I_\mathbf{d}[0..31] - 1)] \\
-l_X &=& l_X \in \mathbb{P}_{256} \\
-Y &=& I_\mathbf{d}[(32 + l_X)..(|I_\mathbf{d}| - 1)] \\
-l_Y &=& |I_\mathbf{d}| - 32 - l_X
-\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 < l_X \\
-g_r &=& G_{muldivbase} + G_{arithword} \Big\lceil \dfrac{|I_\mathbf{d}|}{32} \Big\rceil + l_X l_Y / G_{quaddivisor} \\
-\mathbf{o} &=&
-\begin{cases}
- () & \text{if} \  Y = 0 \\
- \mathtt{\tiny BE}\Big(\big\lfloor \dfrac X Y \big\rfloor\Big) & \text{otherwise}
-\end{cases} \\
-X &=& I_\mathbf{d}[32..(31 + l_X)] \\
-l_X &=& I_\mathbf{d}[0..31] \in \mathbb{P}_{256} \\
-Y &=& I_\mathbf{d}[(32 + l_X..(|I_\mathbf{d}| - 1)] \\
-l_Y &=& |I_\mathbf{d}| - 32 - l_X
-\end{eqnarray}
-
-The ninth contract performs arbitrary-precision exponentiation under modulo.  Here, $0 ^ 0$ is taken to be one.
+The fifth contract performs arbitrary-precision exponentiation under modulo.  Here, $0 ^ 0$ is taken to be one.
 The first word in the input specifies the number of bytes that the first non-negative integer $B$ occupies.
 The second word in the input specifies the number of bytes that the second non-negative integer $E$ occupies.
 These two words are followed by $B$ and $E$; and the rest of the input is interpreted as the third non-negative integer $M$.