https://cryptanalysis.info/blog/tags/%E5%90%8C%E8%B0%83%E4%BB%A3%E6%95%B0/ Recent content in 同调代数 on 密码分析游记 Hugo zh-CN Copyright © 2025, Tim Beyne. Sun, 01 Nov 2026 00:00:00 +0000 https://cryptanalysis.info/blog/posts/2026/monsky-washnitzer/ Sun, 01 Nov 2026 00:00:00 +0000 https://cryptanalysis.info/blog/posts/2026/monsky-washnitzer/ <p>1968年蒙斯基和瓦什尼策发现了一种p进上同调理论。虽然这个理论有几个局限性，但是它与当今的研究仍然相关。</p> <p>令X为任意一条光滑仿射平面曲线，由<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(x,y) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>截出。 如果<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>∈</mo><msub><mi mathvariant="bold">F</mi><mi>p</mi></msub><mo stretchy="false">[</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f \in \mathbf{F}_p[x, y]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathbf">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">]</span></span></span></span>，那么我们可以选一个p进提升<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>f</mi><mo stretchy="true">~</mo></mover><mo>∈</mo><msub><mi mathvariant="bold">Z</mi><mi>p</mi></msub><mo stretchy="false">[</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\widetilde{f} \in \mathbf{Z}_p[x, y]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.1489em;vertical-align:-0.1944em;"></span><span class="mord katex-accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9544em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span><span class="svg-align" style="width:calc(100% - 0.3333em);margin-left:0.3333em;top:-3.6944em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.26em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.26em" viewBox="0 0 600 260" preserveAspectRatio="none"><path d="M200 55.538c-77 0-168 73.953-177 73.953-3 0-7 -2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128 -68.267.847-113-73.952-191-73.952z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathbf">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">]</span></span></span></span>。 咱们考虑一下坐标环</p>