\begin{algorithm}[!ht]
\caption{Integer Linear Programming for 3d Subroutine of 4d Sieve} 
\begin{algorithmic}
	\Require
		\State \hspace{5mm}Boundary $[-B,B[\times[-B,B[\times[-B,B[\times[-B,B[$,
		\State Plane $P$ defined by $u=(u_1,u_2,u_3,u_4),v=(v_1,v_2,v_3,v_4),R=(x,y,z,t)$
		\State \hspace{5mm} such that $R \in P$, but $R$ not necessarily in boundary.
	\Ensure
		\State \hspace{5mm}$(a,b)$ such that $p_0 = R + a\cdot u + b\cdot v$ contained in
		both $P$ and boundary.
\Procedure{}{}
    \vspace{1mm}
	\State $U \gets \{ u_1,u_2,u_3,u_4,-u_1,-u_2,-u_3,-u_4 \}^T$
	\State $V \gets \{ v_1,v_2,v_3,v_4,-v_1,-v_2,-v_3,-v_3 \}^T$
	\State $C \gets \{ B-x-1,B-y-1,B-z-1,B-t-1,B+x,B+y,B+z,B_t \}^T$
	\State $L \gets \lvert \text{LCM}\left(u_1,u_2,u_3,u_4\right) \rvert$
	\State Normalize system - multiply all entries by $L$, divide $U_i,V_i,C_i$ by original $\lvert U_i \rvert$
	\State $a \gets B, b \gets B$
	\For {all pairs of rows $(i,j)$}
		\If {$U_i = 0$ and $V_i > 0$}
			\State $b_{_{TRIAL}} = \left\lfloor C_i / V_i \right\rfloor$
			\State if $b_{_{TRIAL}} < b$ then $b \gets b_{_{TRIAL}}$
		\Else
			\If {$U_i < 0$ and $U_j > 0$ and $\lvert i - j \rvert \neq 4$ }
				\State $D \gets V_i + V_j$
				\If {$D > 0$}
					\State $b_{_{TRIAL}} = \left\lfloor \left(C_i + C_j\right)/D \right\rfloor$
					\State if $b_{_{TRIAL}} < b$ then $b \gets b_{_{TRIAL}}$
				\EndIf
			\EndIf
		\EndIf
	\EndFor
	\State for the best value of $b$, compute $a$ by back substitution.
	\State Return $(a,b)$
    \State\EndProcedure
\end{algorithmic}\label{alg2}
\end{algorithm}

\begin{algorithm}
[!ht]
\caption{Integer Linear Programming for 3d Subroutine of 4d Sieve}
\begin{algorithmic}
\Require
\State \hspace{5mm}Boundary $[-B,B[\times[-B,B[\times[-B,B[\times[-B,B[$,
\State Plane $P$ defined by $u=(u_1,u_2,u_3,u_4),v=(v_1,v_2,v_3,v_4),R=(x,y,z,t)$
\State \hspace{5mm} such that $R \in P$, but $R$ not necessarily in boundary.
\Ensure
\State \hspace{5mm}$(a,b)$ such that $p_0 = R + a\cdot u + b\cdot v$ contained in
both $P$ and boundary.
\Procedure{}{}
\vspace{1mm}
\State $U \gets \{ u_1,u_2,u_3,u_4,-u_1,-u_2,-u_3,-u_4 \}^T$
\State $V \gets \{ v_1,v_2,v_3,v_4,-v_1,-v_2,-v_3,-v_3 \}^T$
\State $C \gets \{ B-x-1,B-y-1,B-z-1,B-t-1,B+x,B+y,B+z,B_t \}^T$
\State $L \gets \lvert \text{LCM}\left(u_1,u_2,u_3,u_4\right) \rvert$
\State Normalize system - multiply all entries by $L$, divide $U_i,V_i,C_i$ by original $\lvert U_i \rvert$
\State $a \gets B, b \gets B$
\For {all pairs of rows $(i,j)$}
\If {$U_i = 0$ and $V_i > 0$}
\State $b_{_{TRIAL}} = \left\lfloor C_i / V_i \right\rfloor$
\State if $b_{_{TRIAL}} < b$ then $b \gets b_{_{TRIAL}}$
\Else
\If {$U_i < 0$ and $U_j > 0$ and $\lvert i - j \rvert \neq 4$ }
\State $D \gets V_i + V_j$
\If {$D > 0$}
\State $b_{_{TRIAL}} = \left\lfloor \left(C_i + C_j\right)/D \right\rfloor$
\State if $b_{_{TRIAL}} < b$ then $b \gets b_{_{TRIAL}}$
\EndIf
\EndIf
\EndIf
\EndFor
\State for the best value of $b$, compute $a$ by back substitution.
\State Return $(a,b)$
\State\EndProcedure
\end{algorithmic}\end{algorithm}
{
    "title": "an implementation of the extended tower number field sieve using 4d   sieving in a box and a record computation in fp4",
    "id": "2212.04999",
    "abstract": "we report on an implementation of the extended tower number field sieve (extnfs) and record computation in a medium characteristic finite field $\\mathbb{f}_{p^4}$ of 512 bits size. empirically, we show that sieving in a 4-dimensional box (orthotope) for collecting relations for extnfs in $\\mathbb{f}_{p^4}$ is faster than sieving in a 4-dimensional hypersphere. we also give a new intermediate descent method, `descent using random vectors', without which the descent stage in our extnfs computation would have been difficult/impossible, and analyze its complexity.",
    "categories": "cs.cr",
    "doi": "",
    "created": "2022-12-09",
    "updated": "2024-02-06",
    "authors": [
        "oisin robinson"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2212.04999"
}
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