We consider fundamental algorithmic number theoretic problems and their relation to a class of block structured Integer Linear Programs (ILPs) called 2-stage stochastic. A 2-stage stochastic ILP is an integer program of the form min{cTx∣Ax=b,ℓ≤x≤u,x∈Zr+ns}\documentclass[12pt]{minimal}
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\begin{document}$$\min \{c^T x \mid {\mathcal {A}} x = b, \ell \le x \le u, x \in {\mathbb {Z}}^{r + ns} \}$$\end{document} where the constraint matrix A∈Znt×r+ns\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {A}} \in {\mathbb {Z}}^{nt \times r +ns}$$\end{document} consists of n matrices Ai∈Zt×r\documentclass[12pt]{minimal}
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\begin{document}$$A_i \in {\mathbb {Z}}^{t \times r}$$\end{document} on the vertical line and n matrices Bi∈Zt×s\documentclass[12pt]{minimal}
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\begin{document}$$B_i \in {\mathbb {Z}}^{t \times s}$$\end{document} on the diagonal line aside. We show a stronger hardness result for a number theoretic problem called Quadratic Congruences where the objective is to compute a number z≤γ\documentclass[12pt]{minimal}
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\begin{document}$$z \le \gamma $$\end{document} satisfying z2≡αmodβ\documentclass[12pt]{minimal}
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\begin{document}$$z^2 \equiv \alpha \bmod \beta $$\end{document} for given α,β,γ∈Z\documentclass[12pt]{minimal}
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\begin{document}$$\alpha , \beta , \gamma \in {\mathbb {Z}}$$\end{document}. This problem was proven to be NP-hard already in 1978 by Manders and Adleman. However, this hardness only applies for instances where the prime factorization of β\documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document} admits large multiplicities of each prime number. We circumvent this necessity proving that the problem remains NP-hard, even if each prime number only occurs constantly often. Using this new hardness result for the QUADRATICCONGRUENCES\documentclass[12pt]{minimal}
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\begin{document}$$\textsc {Quadratic Congruences}$$\end{document} problem, we prove a lower bound of 22δ(s+t)|I|O(1)\documentclass[12pt]{minimal}
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\begin{document}$$2^{2^{\delta (s+t)}} |I|^{O(1)}$$\end{document} for some δ>0\documentclass[12pt]{minimal}
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\begin{document}$$\delta > 0$$\end{document} for the running time of any algorithm solving 2-stage stochastic ILPs assuming the Exponential Time Hypothesis (ETH). Here, |I| is the encoding length of the instance. This result even holds if r, ||b||∞\documentclass[12pt]{minimal}
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\begin{document}$$||b||_{\infty }$$\end{document}, ||c||∞,||ℓ||∞\documentclass[12pt]{minimal}
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\begin{document}$$||c||_{\infty }, ||\ell ||_{\infty }$$\end{document} and the largest absolute value Δ\documentclass[12pt]{minimal}
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\begin{document}$$\varDelta $$\end{document} in the constraint matrix A\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {A}}$$\end{document} are constant. This shows that the state-of-the-art algorithms are nearly tight. Further, it proves the suspicion that these ILPs are indeed harder to solve than the closely related n\documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}-fold ILPs where the constraint matrix is the transpose of A\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {A}}$$\end{document}.
