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I am currently going through Pollard's article on Lattice Sieving and have a few confusions. Firstly,how to figure that $C$ and $D$ in the two  -dimensional array so that every $(c,d)$ pair corresponds to a lattice element (a,b) because if I consider$ c \in [-C,C]$ and $d \in [1,D]$ where $C$ is to be chosen greater than $D$; there are many combinations of type $cV_1 + dV_2$ which go outside the lattice. Here, $V_1$ and $V_2$ are the reduced basis of the lattice.

Pollard, J.M., The lattice sieve, Lenstra, A. K. (ed.) et al., The development of the number field sieve. Berlin: Springer-Verlag. Lect. Notes Math. 1554, 43-49 (1993). ZBL0806.11066, doi: 10.1007/BFb0091538.

Secondly, this 2-D array can't cover entire $L(q)$, so aren't we missing a lot of smooth pairs?

I am currently going through Pollard's article on Lattice Sieving and have a few confusions. Firstly, how to figure that $C$ and $D$ in the two-dimensional array so that every $(c,d)$ pair corresponds to a lattice element $(a,b)$ because if I consider  $ c \in [-C,C]$ and $d \in [1,D]$ where $C$ is to be chosen greater than $D$; there are many combinations of type $cV_1 + dV_2$ which go outside the lattice. Here, $V_1$ and $V_2$ are the reduced basis of the lattice.

Pollard, J.M., The lattice sieve, Lenstra, A. K. (ed.) et al., The development of the number field sieve. Berlin: Springer-Verlag. Lect. Notes Math. 1554, 43-49 (1993). ZBL0806.11066, doi: 10.1007/BFb0091538.

Secondly, this 2-D array can't cover entire $L(q)$, so aren't we missing a lot of smooth pairs?

I am currently going through Pollard's article on Lattice Sieving and have a few confusions. Firstly,how to figure that C and D in the two -dimensional array so that every (c,d) pair corresponds to a lattice element (a,b) because if I consider c \in [-C,C] and d \in [1,D] where C is to be chosen greater than D ; there are many combinations of type cV_1 + dV_2 which go outside the lattice  . Here, V_1 and V_2 are the reduced basis of the lattice.

Pollard, J.M., The lattice sieve, Lenstra, A. K. (ed.) et al., The development of the number field sieve. Berlin: Springer-Verlag. Lect. Notes Math. 1554, 43-49 (1993). ZBL0806.11066.Secondly, this 2-D array can't cover entire L(q),so aren't we missing a lot of smooth pairs?

I am currently going through Pollard's article on Lattice Sieving and have a few confusions. Firstly,how to figure that $C$ and $D$ in the two -dimensional array so that every $(c,d)$ pair corresponds to a lattice element (a,b) because if I consider$ c \in [-C,C]$ and $d \in [1,D]$ where $C$ is to be chosen greater than $D$; there are many combinations of type $cV_1 + dV_2$ which go outside the lattice. Here, $V_1$ and $V_2$ are the reduced basis of the lattice.

Pollard, J.M., The lattice sieve, Lenstra, A. K. (ed.) et al., The development of the number field sieve. Berlin: Springer-Verlag. Lect. Notes Math. 1554, 43-49 (1993). ZBL0806.11066, doi: 10.1007/BFb0091538.

Secondly, this 2-D array can't cover entire $L(q)$, so aren't we missing a lot of smooth pairs?