Subexponential Algorithms for Total Search Problems in the Dense Pigeonhole Regime

Overview & Synopsis

The class $\mathsf{PPP}$ captures fundamental total search problems, for which a solution is guaranteed to exist by the pigeonhole principle and the main challenge is to find it. We ask whether problems in this class admit significantly faster algorithms in the \emph{dense regime}. More concretly, we study \emph{binary collision Short Integer Solution} (SIS): given a matrix $A\in\mathbb{Z}_q^{n\times m}$ with $m>n\log q$, the goal is to find distinct $\underline{\mathbf{x}},\underline{\mathbf{y}}\in\{0,1\}^m$ with $A\underline{\mathbf{x}}=A\underline{\mathbf{y}}\pmod q$. Brute-force enumeration can be used to find a collision in $q^{\mathcal{O}(n)} \operatorname{poly}(|I|)$ time, and no significantly faster algorithm for that problem is known. We present an algorithm running in time $q^{o(n)}\cdot\operatorname{poly}(|I|)$ whenever the density parameter $δ:= m/(n\log q)$ is $ω(1)$.\footnote{All logarithms are in base $2$, and $|I|$ is the binary input length.} Building on reductions of Sotiraki et al.~(FOCS~2018), we obtain corresponding algorithms for dense variants of weak constrained SIS, Equal Vector Sums, Minkowski, and Blichfeldt. Our algorithm adapts techniques of Bhati et al.~(FOCS~2026), who gave a subexponential algorithm for Equal Sums in the dense regime and proved a tight conditional lower bound. Their conditional lower bound easily transfers to SIS and shows that the running time of our algorithms is essentially optimal.