https://i.cauchy.top/tags/Crypto%5CMath/Recent content in Crypto\Math on ZoeeHugozh-CNZoeeWed, 21 Dec 2022 12:00:00 +0000https://i.cauchy.top/2022/12/RSA%E8%A7%A3%E5%AF%86%E6%8A%80%E5%B7%A7/Wed, 21 Dec 2022 12:00:00 +0000https://i.cauchy.top/2022/12/RSA%E8%A7%A3%E5%AF%86%E6%8A%80%E5%B7%A7/<h3 id="保存一些推导好的结论">保存一些推导好的结论</h3> <h2 id="已知n-且pq相邻">已知n 且p、q相邻</h2> <h5 id="线索是在加密脚本中可以找到如下类似代码">线索是在加密脚本中可以找到如下类似代码</h5> <div class="highlight"><div class="chroma"> <table class="lntable"><tr><td class="lntd"> <pre tabindex="0" class="chroma"><code><span class="lnt">1 </span><span class="lnt">2 </span><span class="lnt">3 </span></code></pre></td> <td class="lntd"> <pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="cl"><span class="n">p</span> <span class="o">=</span> <span class="n">getPrime</span><span class="p">(</span><span class="mi">512</span><span class="p">)</span> </span></span><span class="line"><span class="cl"><span class="n">q</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">gmpy2</span><span class="o">.</span><span class="n">next_prime</span><span class="p">(</span><span class="n">p</span><span class="p">))</span> </span></span><span class="line"><span class="cl"><span class="n">n</span> <span class="o">=</span> <span class="n">p</span><span class="o">*</span><span class="n">q</span> </span></span></code></pre></td></tr></table> </div> </div><h5 id="如下代码解出pq">如下代码解出p、q</h5> <div class="highlight"><div class="chroma"> <table class="lntable"><tr><td class="lntd"> <pre tabindex="0" class="chroma"><code><span class="lnt">1 </span><span class="lnt">2 </span><span class="lnt">3 </span><span class="lnt">4 </span></code></pre></td> <td class="lntd"> <pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="cl"><span class="c1"># pq相邻 为n开方</span> </span></span><span class="line"><span class="cl"><span class="n">n2</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">gmpy2</span><span class="o">.</span><span class="n">iroot</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span> </span></span><span class="line"><span class="cl"><span class="n">p</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">nextprime</span><span class="p">(</span><span class="n">n2</span><span class="p">)</span> </span></span><span class="line"><span class="cl"><span class="n">q</span> <span class="o">=</span> <span class="n">n</span><span class="o">//</span><span class="n">p</span> </span></span></code></pre></td></tr></table> </div> </div><h6 id="加以验证">加以验证</h6> <div class="highlight"><div class="chroma"> <table class="lntable"><tr><td class="lntd"> <pre tabindex="0" class="chroma"><code><span class="lnt">1 </span><span class="lnt">2 </span></code></pre></td> <td class="lntd"> <pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="cl"><span class="nb">print</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="p">)</span> </span></span><span class="line"><span class="cl"><span class="nb">print</span><span class="p">(</span><span class="n">n</span> <span class="o">==</span> <span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="p">)</span> <span class="c1">#返回True</span> </span></span></code></pre></td></tr></table> </div> </div><h6 id="q--sympynextprime意味着-p-和-q-是相邻的素数非常接近于是可以将-n-开平方根这个数的-nextprime-就为-q">q = sympy.nextprime()，意味着 p 和 q 是相邻的素数，非常接近，于是可以将 N 开平方根，这个数的 nextprime 就为 q</h6> <hr> <h2 id="与--关系">$q$ 与 $phi(q^2)$ 关系</h2> <h5 id="题目给出">题目给出</h5> <div class="highlight"><div class="chroma"> <table class="lntable"><tr><td class="lntd"> <pre tabindex="0" class="chroma"><code><span class="lnt"> 1 </span><span class="lnt"> 2 </span><span class="lnt"> 3 </span><span class="lnt"> 4 </span><span class="lnt"> 5 </span><span class="lnt"> 6 </span><span class="lnt"> 7 </span><span class="lnt"> 8 </span><span class="lnt"> 9 </span><span class="lnt">10 </span><span class="lnt">11 </span><span class="lnt">12 </span><span class="lnt">13 </span><span class="lnt">14 </span><span class="lnt">15 </span><span class="lnt">16 </span><span class="lnt">17 </span><span class="lnt">18 </span><span class="lnt">19 </span><span class="lnt">20 </span><span class="lnt">21 </span><span class="lnt">22 </span><span class="lnt">23 </span><span class="lnt">24 </span><span class="lnt">25 </span><span class="lnt">26 </span><span class="lnt">27 </span><span class="lnt">28 </span><span class="lnt">29 </span><span class="lnt">30 </span></code></pre></td> <td class="lntd"> <pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="cl"><span class="kn">from</span> <span class="nn">Crypto.Util.number</span> <span class="kn">import</span> <span class="o">*</span> </span></span><span class="line"><span class="cl"><span class="kn">import</span> <span class="nn">gmpy2</span> </span></span><span class="line"><span class="cl"><span class="kn">from</span> <span class="nn">flag</span> <span class="kn">import</span> <span class="n">flag</span> </span></span><span class="line"><span class="cl"> </span></span><span class="line"><span class="cl"><span class="n">p</span><span class="o">=</span><span class="n">getPrime</span><span class="p">(</span><span class="mi">512</span><span class="p">)</span> </span></span><span class="line"><span class="cl"><span class="n">q</span><span class="o">=</span><span class="n">getPrime</span><span class="p">(</span><span class="mi">512</span><span class="p">)</span> </span></span><span class="line"><span class="cl"><span class="n">r</span><span class="o">=</span><span class="n">getPrime</span><span class="p">(</span><span class="mi">512</span><span class="p">)</span> </span></span><span class="line"><span class="cl"><span class="n">n</span><span class="o">=</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="o">*</span><span class="n">r</span> </span></span><span class="line"><span class="cl"><span class="n">e</span><span class="o">=</span><span class="mi">65537</span> </span></span><span class="line"><span class="cl"><span class="n">m</span><span class="o">=</span><span class="n">bytes_to_long</span><span class="p">(</span><span class="n">flag</span><span class="p">)</span> </span></span><span class="line"><span class="cl"> </span></span><span class="line"><span class="cl"><span class="k">def</span> <span class="nf">Euler</span><span class="p">(</span><span class="n">x</span><span class="p">):</span> </span></span><span class="line"><span class="cl"> <span class="n">res</span><span class="o">=</span><span class="mi">0</span> </span></span><span class="line"><span class="cl"> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">):</span> </span></span><span class="line"><span class="cl"> <span class="k">if</span> <span class="n">gmpy2</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">==</span><span class="mi">1</span><span class="p">:</span> </span></span><span class="line"><span class="cl"> <span class="n">res</span><span class="o">+=</span><span class="mi">1</span> </span></span><span class="line"><span class="cl"> <span class="k">return</span> <span class="n">res</span> </span></span><span class="line"><span class="cl"> </span></span><span class="line"><span class="cl"><span class="n">P</span><span class="o">=</span><span class="n">p</span><span class="o">**</span><span class="mi">3</span> </span></span><span class="line"><span class="cl"><span class="n">Q</span><span class="o">=</span><span class="n">q</span><span class="o">**</span><span class="mi">2</span> </span></span><span class="line"><span class="cl"><span class="n">c</span><span class="o">=</span><span class="nb">pow</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">e</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> </span></span><span class="line"><span class="cl"> </span></span><span class="line"><span class="cl"><span class="nb">print</span><span class="p">(</span><span class="n">P</span><span class="p">)</span> </span></span><span class="line"><span class="cl"><span class="nb">print</span><span class="p">(</span><span class="n">Euler</span><span class="p">(</span><span class="n">Q</span><span class="p">))</span> </span></span><span class="line"><span class="cl"><span class="nb">print</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> </span></span><span class="line"><span class="cl"><span class="nb">print</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> </span></span><span class="line"><span class="cl"><span class="c1">#P = 1686761823519516525084824311416810253107853832929411677237594989001281261421956188747941222367576127569696216513071075733130132251383529469095077597202999362675041210639065389821237728348981344440193122126487447235175127680730304754656661704596111547454161716607787386914764780833658069534913186485846587027674567133467341836048413431174183101579802349498153899249182793495245916757355079598668221097821452488627067390724198617676379698358212167618567704428433303</span> </span></span><span class="line"><span class="cl"><span class="c1">#EQ = 54800501457630149544580145188029519076092032026436445384163914536965196942938808746487258773679836358732387355329080483568564046906919385574994390974732491368590525875801103056613954297623835159311237599961507385582029709732950222118171961946571285930711702624160354541459438994349318149872111029043942485620</span> </span></span><span class="line"><span class="cl"><span class="c1">#c = 568846080701555049788706647255668980211679838950729382006912035332305772256748203239331545262283165739670330060735508231578298253855583985677482008855909565463834639005910652510802915373310537390293061001384655286359323437737989289787972131460392977341024828530868508329336263146882773903176326250063921456707975853839017504122823304303509269793133132036479219404842827556015566627129747816769486873563843578029479179692030808518925753268233301452280242586076493</span> </span></span><span class="line"><span class="cl"><span class="c1">#n = 1069981867450019752454430625015273180922733107799929958042241890002915414684562764186875387471850290817321430141222917656674447229697676236077201897275059270515637506529666384968535578683380559782336910645306992981172862940944536463561840412558764760962107958365575095435157363812028759723055357681895134974760386884254380189603418912937553755099672511307377054933171384741715642510754214768859689909974996095149155241791151425031489280537907842378844226410097051</span> </span></span></code></pre></td></tr></table> </div> </div><h5 id="解出q">解出q</h5> <blockquote> <p><code>Euler(q^2) = q * (q-1)</code>，那么 <code>q-1 &lt; sqrt(Euler(q^2)) &lt; q</code>.(已翻译)</p>