  Given $(a_1, \dots, a_n, t) \in \mathbb{Z}_{\geq 0}^{n + 1}$, the Subset Sum
problem ($\mathsf{SSUM}$) is to decide whether there exists $S \subseteq [n]$
such that $\sum_{i \in S} a_i = t$. There is a close variant of the
$\mathsf{SSUM}$, called $\mathsf{Subset~Product}$. Given positive integers
$a_1, ..., a_n$ and a target integer $t$, the $\mathsf{Subset~Product}$ problem
asks to determine whether there exists a subset $S \subseteq [n]$ such that
$\prod_{i \in S} a_i=t$. There is a pseudopolynomial time dynamic programming
algorithm, due to Bellman (1957) which solves the $\mathsf{SSUM}$ and
$\mathsf{Subset~Product}$ in $O(nt)$ time and $O(t)$ space.
  In the first part, we present {\em search} algorithms for variants of the
Subset Sum problem. Our algorithms are parameterized by $k$, which is a given
upper bound on the number of realisable sets (i.e.,~number of solutions,
summing exactly $t$). We show that $\mathsf{SSUM}$ with a unique solution is
already NP-hard, under randomized reduction. This makes the regime of
parametrized algorithms, in terms of $k$, very interesting.
  Subsequently, we present an $\tilde{O}(k\cdot (n+t))$ time deterministic
algorithm, which finds the hamming weight of all the realisable sets for a
subset sum instance. We also give a poly$(knt)$-time and $O(\log(knt))$-space
deterministic algorithm that finds all the realisable sets for a subset sum
instance.
  In the latter part, we present a simple and elegant randomized $\tilde{O}(n +
t)$ time algorithm for $\mathsf{Subset~Product}$. Moreover, we also present a
poly$(nt)$ time and $O(\log^2 (nt))$ space deterministic algorithm for the
same. We study these problems in the unbounded setting as well. Our algorithms
use multivariate FFT, power series and number-theoretic techniques, introduced
by Jin and Wu (SOSA'19) and Kane (2010).
