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**2024/713**

**Title:**

Analyzing Pump and jump BKZ algorithm using dynamical systems

**Authors:**
Leizhang Wang

**Abstract:**

The analysis of the reduction effort of the lattice reduction algorithm is important in estimating the hardness of lattice-based cryptography schemes. Recently many lattice challenge records have been cracked by using the Pnj-BKZ algorithm which is the default lattice reduction algorithm used in G6K, such as the TU Darmstadt LWE and SVP Challenges. However, the previous estimations of the Pnj-BKZ algorithm are simulator algorithms rather than theoretical upper bound analyses. In this work, we present the first dynamic analysis of Pnj-BKZ algorithm. More precisely, our analysis results show that let L is the lattice spanned by (\mathbf{a}_i)_{i\leq d}. The shortest vector \mathbf{b}_1 output by running \Omega \left ( \frac{2Jd^2}{\beta(\beta-J)}\left ( \ln_{}{d} +\ln_{} \ln_{}{\max_{i}\frac{\left \| \mathbf{a}_i^{*} \right \| }{(\mathrm{det}L )^{1/d} } } \right ) \right ) tours reduction of pnj-BKZ$(\beta,J), \mathbf{b}_1$ satisfied that \memo{\left \| \mathbf{b}_1 \right \| \le {\gamma}_{\beta}^{\frac{d-1}{2(\beta-J)}+2 } \cdot \left ( \mathrm{det}L \right ) ^{\frac{1}{d} } }.

**ePrint:**
https://eprint.iacr.org/2024/713

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