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| 連結 | Text |
|---|---|
| https://github.com/HeroCTF/HeroCTF_v6/tree/main/Crypto/Interpolation/interpolation | Source files |
| https://en.wikipedia.org/wiki/Modular_arithmetic | modular arithmetic |
| https://cryptohack.org/courses/modular/ma0/ | cryptohack |
| https://en.wikipedia.org/wiki/Lagrange_polynomial | lagrange interpolating polynomial |
| https://en.wikipedia.org/wiki/Shamir%27s_secret_sharing | https://en.wikipedia.org/wiki/Shamir%27s_secret_sharing |
| https://x.com/cryptanalyse | cryptanalyse |
| http://exo7.emath.fr/cours/ch_polynomes.pdf | http://exo7.emath.fr/cours/ch_polynomes.pdf |
| https://www.youtube.com/watch?v=nvkX1Bd90Gk | https://www.youtube.com/watch?v=nvkX1Bd90Gk |
| https://doc.sagemath.org/html/en/reference/polynomial_rings/sage/rings/polynomial/polynomial_ring.html | https://doc.sagemath.org/html/en/reference/polynomial_rings/sage/rings/polynomial/polynomial_ring.html |
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HeroCTF v6 - Interpolation [crypto]
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<time datetime="2024-10-29 22:28:51 +0100 CET">
2024-10-29
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<h1>HeroCTF v6 - Interpolation [crypto]</h1>
<aside>
<nav id="TableOfContents">
<ul>
<li><a href="#tldr">TL;DR</a></li>
<li><a href="#introduction">Introduction</a></li>
<li><a href="#solving">Solving</a>
<ul>
<li><a href="#reading-the-challenge-source-code">Reading the challenge source code</a></li>
<li><a href="#vulnerabilities">Vulnerabilities</a>
<ul>
<li><a href="#polynomial-recovery">Polynomial recovery</a></li>
<li><a href="#flag-recovery">Flag recovery</a></li>
<li><a href="#solve-final">Solve final</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#conclusion">Conclusion</a></li>
<li><a href="#links">Links</a></li>
</ul>
</nav>
</aside>
<p>HeroCTF v6 writeup for interpolation challenge</p>
<blockquote>
<p>Difficulty : Easy</p>
</blockquote>
<blockquote>
<p>Team : Langskip</p>
</blockquote>
<blockquote>
<p><a href="https://github.com/HeroCTF/HeroCTF_v6/tree/main/Crypto/Interpolation/interpolation"> Source files </a></p>
</blockquote>
<blockquote>
<p>Has missing data really stopped anyone ?</p>
</blockquote>
<blockquote>
<p>Author : alol</p>
</blockquote>
<h1 id="tldr">TL;DR</h1>
<ul>
<li>Get the computed points that the server send us</li>
<li>Compute f(0) with the by knowing that <code>Hero</code> is used to generate the first coefficient a0</li>
<li>Interpolate with the points we have</li>
<li>Compute every flag part by using a rainbow table</li>
</ul>
<h1 id="introduction">Introduction</h1>
<p>We have a server that creates a polynomial from the flag which is cut into several parts and then hashed.</p>
<p>It calculates points with this polynomial and sends them to us.</p>
<h1 id="solving">Solving</h1>
<h2 id="reading-the-challenge-source-code">Reading the challenge source code</h2>
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<td style="vertical-align:top;padding:0;margin:0;border:0;;width:100%">
<pre tabindex="0" style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code class="language-python" data-lang="python"><span style="display:flex;"><span><span style="color:#8b949e;font-style:italic">#!/usr/bin/sage</span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">import</span> <span style="color:#ff7b72">hashlib</span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">import</span> <span style="color:#ff7b72">re</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">with</span> open(<span style="color:#a5d6ff">"flag.txt"</span>, <span style="color:#a5d6ff">"rb"</span>) <span style="color:#ff7b72">as</span> f:
</span></span><span style="display:flex;"><span> FLAG <span style="color:#ff7b72;font-weight:bold">=</span> f<span style="color:#ff7b72;font-weight:bold">.</span>read()
</span></span><span style="display:flex;"><span> <span style="color:#ff7b72">assert</span> re<span style="color:#ff7b72;font-weight:bold">.</span><span style="color:#ff7b72">match</span>(<span style="color:#79c0ff">rb</span><span style="color:#a5d6ff">"Hero{[0-9a-zA-Z_]</span><span style="color:#a5d6ff">{90}</span><span style="color:#a5d6ff">}"</span>, FLAG)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>F <span style="color:#ff7b72;font-weight:bold">=</span> FiniteField(<span style="color:#a5d6ff">2</span><span style="color:#ff7b72;font-weight:bold">**</span><span style="color:#a5d6ff">256</span> <span style="color:#ff7b72;font-weight:bold">-</span> <span style="color:#a5d6ff">189</span>)
</span></span><span style="display:flex;"><span>R <span style="color:#ff7b72;font-weight:bold">=</span> PolynomialRing(F, <span style="color:#a5d6ff">"x"</span>)
</span></span><span style="display:flex;"><span>H <span style="color:#ff7b72;font-weight:bold">=</span> <span style="color:#ff7b72">lambda</span> n: int(hashlib<span style="color:#ff7b72;font-weight:bold">.</span>sha256(n)<span style="color:#ff7b72;font-weight:bold">.</span>hexdigest(), <span style="color:#a5d6ff">16</span>)
</span></span><span style="display:flex;"><span>C <span style="color:#ff7b72;font-weight:bold">=</span> <span style="color:#ff7b72">lambda</span> x: [H(x[i : i <span style="color:#ff7b72;font-weight:bold">+</span> <span style="color:#a5d6ff">4</span>]) <span style="color:#ff7b72">for</span> i <span style="color:#ff7b72;font-weight:bold">in</span> range(<span style="color:#a5d6ff">0</span>, len(FLAG), <span style="color:#a5d6ff">4</span>)]
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>f <span style="color:#ff7b72;font-weight:bold">=</span> R(C(FLAG))
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>points <span style="color:#ff7b72;font-weight:bold">=</span> []
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">for</span> _ <span style="color:#ff7b72;font-weight:bold">in</span> range(f<span style="color:#ff7b72;font-weight:bold">.</span>degree()):
</span></span><span style="display:flex;"><span> r <span style="color:#ff7b72;font-weight:bold">=</span> F<span style="color:#ff7b72;font-weight:bold">.</span>random_element()
</span></span><span style="display:flex;"><span> points<span style="color:#ff7b72;font-weight:bold">.</span>append([r, f(r)])
</span></span><span style="display:flex;"><span>print(points)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>flag <span style="color:#ff7b72;font-weight:bold">=</span> input(<span style="color:#a5d6ff">">"</span>)<span style="color:#ff7b72;font-weight:bold">.</span>encode()<span style="color:#ff7b72;font-weight:bold">.</span>ljust(len(FLAG))
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>g <span style="color:#ff7b72;font-weight:bold">=</span> R(C(flag))
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">for</span> p <span style="color:#ff7b72;font-weight:bold">in</span> points:
</span></span><span style="display:flex;"><span> <span style="color:#ff7b72">if</span> g(p[<span style="color:#a5d6ff">0</span>]) <span style="color:#ff7b72;font-weight:bold">!=</span> p[<span style="color:#a5d6ff">1</span>]:
</span></span><span style="display:flex;"><span> print(<span style="color:#a5d6ff">"Wrong flag!"</span>)
</span></span><span style="display:flex;"><span> <span style="color:#ff7b72">break</span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">else</span>:
</span></span><span style="display:flex;"><span> print(<span style="color:#a5d6ff">"Congrats!"</span>)
</span></span></code></pre></td></tr></tbody></table>
</div>
</div><ul>
<li>
<p>Here : <code>assert re.match(rb "Hero{[0-9a-zA-Z_]{90}}", FLAG)</code>, we know the characters used for the flag, their number of 90, and added to this <code>Hero{}</code>, the flag has a length of <code>96</code> characters.</p>
</li>
<li>
<p>A finite field is defined: <code>F = FiniteField(2**256 - 189)</code> also noted <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">Z</mi><mi mathvariant="normal">/</mi><mi>n</mi><mi mathvariant="double-struck">Z</mi></mrow><annotation encoding="application/x-tex"> \mathbb{Z}/n\mathbb{Z} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathbb">Z</span><span class="mord">/</span><span class="mord mathnormal">n</span><span class="mord mathbb">Z</span></span></span></span></span> with <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><msup><mn>2</mn><mn>256</mn></msup><mo>−</mo><mn>189</mn></mrow><annotation encoding="application/x-tex"> n = 2^{256} -189 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal">n</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="base"><span class="strut" style="height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8141em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">256</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">189</span></span></span></span></span>. This is a notion of <a href="https://en.wikipedia.org/wiki/Modular_arithmetic">modular arithmetic</a> which implies that all the numbers we work with are modulo <code>n</code>. There’s a <a href="https://cryptohack.org/courses/modular/ma0/">cryptohack</a> module that covers these notions.</p>
</li>
<li>
<p>Next, we initialize a <code>polynomial</code> object in the finite field <code>F</code>: <code>R = PolynomialRing(F, "x")</code>. It is polynom with one variable <code>x</code>, so we call it a <code>univariate polynomial</code>.</p>
</li>
<li>
<p>A <code>H</code> function that only hashes given bytes and the <code>C</code> function slices a string into 4-byte chunks, hash each of them with <code>H</code> and add them to a list.</p>
</li>
</ul>
<div class="highlight"><div style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;">
<table style="border-spacing:0;padding:0;margin:0;border:0;"><tbody><tr><td style="vertical-align:top;padding:0;margin:0;border:0;">
<pre tabindex="0" style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">1
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">2
</span></code></pre></td>
<td style="vertical-align:top;padding:0;margin:0;border:0;;width:100%">
<pre tabindex="0" style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code class="language-python" data-lang="python"><span style="display:flex;"><span>H <span style="color:#ff7b72;font-weight:bold">=</span> <span style="color:#ff7b72">lambda</span> n: int(hashlib<span style="color:#ff7b72;font-weight:bold">.</span>sha256(n)<span style="color:#ff7b72;font-weight:bold">.</span>hexdigest(), <span style="color:#a5d6ff">16</span>)
</span></span><span style="display:flex;"><span>C <span style="color:#ff7b72;font-weight:bold">=</span> <span style="color:#ff7b72">lambda</span> x: [H(x[i : i <span style="color:#ff7b72;font-weight:bold">+</span> <span style="color:#a5d6ff">4</span>]) <span style="color:#ff7b72">for</span> i <span style="color:#ff7b72;font-weight:bold">in</span> range(<span style="color:#a5d6ff">0</span>, len(FLAG), <span style="color:#a5d6ff">4</span>)]
</span></span></code></pre></td></tr></tbody></table>
</div>
</div><ul>
<li>Then the polynomial is generated with: <code>f = R(C(FLAG))</code>.</li>
</ul>
<p>By the way, a polynomial <code>P</code> is written:</p>
<br>
<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>a</mi><mi>n</mi></msub><msup><mi>X</mi><mi>n</mi></msup><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mi>X</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mtext> </mtext><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><msup><mi>X</mi><mn>2</mn></msup><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><msup><mi>X</mi><mn>1</mn></msup><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> P(X) = a_{n}X^{n} + a_{n-1}X^{n-1} + ... \ + a_{2}X^{2} + a_{1}X^{1} + a_{0} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</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="base"><span class="strut" style="height: 0.8644em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.7144em;"><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 1.0724em; vertical-align: -0.2083em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.6667em; vertical-align: -0.0833em;"></span><span class="mord">...</span><span class="mspace"> </span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 1.0141em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 1.0141em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.5806em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span>
<br>
<p>Basically, what we saw in high school was a polynomial:</p>
<br>
<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow><annotation encoding="application/x-tex"> ax^{2} + bx +c </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.9474em; vertical-align: -0.0833em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.7778em; vertical-align: -0.0833em;"></span><span class="mord mathnormal">b</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span></span></span>
<br>
<p>It’s called a polynomial of <strong>second degree</strong>, because the degree of a polynomial is the largest integer <code>i</code> such that <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub><mo mathvariant="normal">≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> a_{i} \neq 0 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8889em; vertical-align: -0.1944em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3117em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height: 0.8889em; vertical-align: -0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">0</span></span></span></span></span>.</p>
<p>And the coefficients are the values <code>a</code> of the polynomial P.</p>
<ul>
<li>There are <code>n</code> random numbers <code>x</code> generated, their <code>y</code> image is calculated through the polynomial and the [x,y] pair is added to the <code>points</code> list.
And this list of points is sent to us.</li>
</ul>
<div class="highlight"><div style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;">
<table style="border-spacing:0;padding:0;margin:0;border:0;"><tbody><tr><td style="vertical-align:top;padding:0;margin:0;border:0;">
<pre tabindex="0" style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">1
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">2
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">3
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">4
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">5
</span></code></pre></td>
<td style="vertical-align:top;padding:0;margin:0;border:0;;width:100%">
<pre tabindex="0" style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code class="language-python" data-lang="python"><span style="display:flex;"><span>points <span style="color:#ff7b72;font-weight:bold">=</span> []
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">for</span> _ <span style="color:#ff7b72;font-weight:bold">in</span> range(f<span style="color:#ff7b72;font-weight:bold">.</span>degree()):
</span></span><span style="display:flex;"><span> r <span style="color:#ff7b72;font-weight:bold">=</span> F<span style="color:#ff7b72;font-weight:bold">.</span>random_element()
</span></span><span style="display:flex;"><span> points<span style="color:#ff7b72;font-weight:bold">.</span>append([r, f(r)])
</span></span><span style="display:flex;"><span>print(points)
</span></span></code></pre></td></tr></tbody></table>
</div>
</div><ul>
<li>Then a user input is requested, and a polynomial will be generated as it was with the <code>FLAG</code> and the results obtained with our input will be compared with the results obtained with the polynomial generated with the <code>FLAG</code>.</li>
</ul>
<div class="highlight"><div style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;">
<table style="border-spacing:0;padding:0;margin:0;border:0;"><tbody><tr><td style="vertical-align:top;padding:0;margin:0;border:0;">
<pre tabindex="0" style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">1
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">2
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">3
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">4
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">5
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">6
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">7
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">8
</span></code></pre></td>
<td style="vertical-align:top;padding:0;margin:0;border:0;;width:100%">
<pre tabindex="0" style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code class="language-python" data-lang="python"><span style="display:flex;"><span>g <span style="color:#ff7b72;font-weight:bold">=</span> R(C(flag))
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">for</span> p <span style="color:#ff7b72;font-weight:bold">in</span> points:
</span></span><span style="display:flex;"><span> <span style="color:#ff7b72">if</span> g(p[<span style="color:#a5d6ff">0</span>]) <span style="color:#ff7b72;font-weight:bold">!=</span> p[<span style="color:#a5d6ff">1</span>]:
</span></span><span style="display:flex;"><span> print(<span style="color:#a5d6ff">"Wrong flag!"</span>)
</span></span><span style="display:flex;"><span> <span style="color:#ff7b72">break</span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">else</span>:
</span></span><span style="display:flex;"><span> print(<span style="color:#a5d6ff">"Congrats!"</span>)
</span></span></code></pre></td></tr></tbody></table>
</div>
</div><p>The FLAG will not be displayed at the end of our user input, so we should be able to find the flag on our side.</p>
<h2 id="vulnerabilities">Vulnerabilities</h2>
<p>The logical path for me is that, from the points and their images sent by the server, reconstruct the original polynomial.</p>
<p>Recover the <code>coefficients</code> of the polynomial and then trace back to the value of each coefficients before being hashed.</p>
<p>It’s a kind of bruteforce and I’m going to use a rainbow table for this.</p>
<h3 id="polynomial-recovery">Polynomial recovery</h3>
<p>A polynomial can be reconstructed if we have enough of its points and images with the <a href="https://en.wikipedia.org/wiki/Lagrange_polynomial">lagrange interpolating polynomial</a>. This is the concept behind <a href="https://en.wikipedia.org/wiki/Shamir%27s_secret_sharing">Shamir’s secret key sharing</a>.</p>
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Lagrange_polynomial.svg/640px-Lagrange_polynomial.svg.png" style="filter: invert(100%)" alt="lagrange polynomial">
<p>For the theory, we’ll draw a curve that passes through a point thanks to an equation, reproduce this for all the submitted points and multiply all the equations, here’s what the final equation looks like:</p>
<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>l</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mi>j</mi><mo mathvariant="normal">≠</mo><mi>i</mi></mrow><mi>n</mi></munderover><mfrac><mrow><mi>X</mi><mo>−</mo><msub><mi>x</mi><mi>j</mi></msub></mrow><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>x</mi><mi>j</mi></msub></mrow></mfrac><mtext> </mtext><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mtext> </mtext><mfrac><mrow><mi>X</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub></mrow><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub></mrow></mfrac><mfrac><mrow><mi>X</mi><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac><mtext> </mtext><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mtext> </mtext><mfrac><mrow><mi>X</mi><mo>−</mo><msub><mi>x</mi><mi>n</mi></msub></mrow><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>x</mi><mi>n</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex"> l_{i}(X) = \prod_{j=0,j\neq i}^{n} \frac{X - x_{j}}{x_{i} - x_{j}} \ ... \ \frac{X - x_{0}}{x_{i} - x_{0}} \frac{X - x_{1}}{x_{i} - x_{1}} \ ... \ \frac{X - x_{n}}{x_{i} - x_{n}} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0197em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3117em;"><span class="" style="top: -2.55em; margin-left: -0.0197em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</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="base"><span class="strut" style="height: 3.0896em; vertical-align: -1.4382em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.6514em;"><span class="" style="top: -1.8479em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0572em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">0</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right: 0.0572em;">j</span><span class="mrel mtight"><span class="mrel mtight"><span class="mord vbox mtight"><span class="thinbox mtight"><span class="rlap mtight"><span class="strut" style="height: 0.8889em; vertical-align: -0.1944em;"></span><span class="inner"><span class="mord mtight"><span class="mrel mtight"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel mtight">=</span></span><span class="mord mathnormal mtight">i</span></span></span></span><span class="" style="top: -3.05em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∏</span></span></span><span class="" style="top: -4.3em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.4382em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.3603em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3117em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3117em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0572em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3117em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0572em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.9721em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace"> </span><span class="mord">...</span><span class="mspace"> </span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.3603em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3117em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.3603em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3117em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace"> </span><span class="mord">...</span><span class="mspace"> </span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.3603em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3117em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span><p>The server sends us 23 points and we can try to reconstruct the polynomial with them:</p>
<div class="highlight"><div style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;">
<table style="border-spacing:0;padding:0;margin:0;border:0;"><tbody><tr><td style="vertical-align:top;padding:0;margin:0;border:0;">
<pre tabindex="0" style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 1
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 2
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 3
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 4
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 5
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 6
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 7
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 8
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 9
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">10
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">11
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">12
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">13
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">14
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">15
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">16
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">17
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">18
</span></code></pre></td>
<td style="vertical-align:top;padding:0;margin:0;border:0;;width:100%">
<pre tabindex="0" style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code class="language-python" data-lang="python"><span style="display:flex;"><span><span style="color:#ff7b72">from</span> <span style="color:#ff7b72">pwn</span> <span style="color:#ff7b72">import</span> remote
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>host <span style="color:#ff7b72;font-weight:bold">=</span> <span style="color:#a5d6ff">'crypto.heroctf.fr'</span>
</span></span><span style="display:flex;"><span>port <span style="color:#ff7b72;font-weight:bold">=</span> <span style="color:#a5d6ff">9000</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>conn <span style="color:#ff7b72;font-weight:bold">=</span> remote(host, port)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>data <span style="color:#ff7b72;font-weight:bold">=</span> conn<span style="color:#ff7b72;font-weight:bold">.</span>recvuntilS(<span style="color:#a5d6ff">'>'</span>)<span style="color:#ff7b72;font-weight:bold">.</span>decode()
</span></span><span style="display:flex;"><span>given_points <span style="color:#ff7b72;font-weight:bold">=</span> eval(data<span style="color:#ff7b72;font-weight:bold">.</span>split(<span style="color:#a5d6ff">'</span><span style="color:#79c0ff">\n</span><span style="color:#a5d6ff">'</span>)[<span style="color:#a5d6ff">0</span>])
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>F <span style="color:#ff7b72;font-weight:bold">=</span> FiniteField(<span style="color:#a5d6ff">2</span><span style="color:#ff7b72;font-weight:bold">**</span><span style="color:#a5d6ff">256</span> <span style="color:#ff7b72;font-weight:bold">-</span> <span style="color:#a5d6ff">189</span>)
</span></span><span style="display:flex;"><span>R <span style="color:#ff7b72;font-weight:bold">=</span> PolynomialRing(F, <span style="color:#a5d6ff">"x"</span>)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>interp <span style="color:#ff7b72;font-weight:bold">=</span> R<span style="color:#ff7b72;font-weight:bold">.</span>lagrange_polynomial(given_points)
</span></span><span style="display:flex;"><span>coefs <span style="color:#ff7b72;font-weight:bold">=</span> interp<span style="color:#ff7b72;font-weight:bold">.</span>coefficients()
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>print(coefs)
</span></span><span style="display:flex;"><span>print(interp)
</span></span></code></pre></td></tr></tbody></table>
</div>
</div><p>spoiler: we get the wrong polynomial.</p>
<p>I tested locally with a flag of the same format as the flag used in the challenge. I generated the polynomial from the flag and another polynomial by interpolation and they don’t match.</p>
<p>The explanation is that I’m not generating the polynomial with enough points. I’m using 23, which generates 23 coefficients, whereas the original polynomial has 24 (to calculate this, just calculate the length of the list generated by <code>C(FLAG)</code>). However, I can calculate one more.</p>
<p>When the polynomial is generated with <code>f = R(C(FLAG))</code>, a list of int is generated with <code>C(FLAG)</code>. The polynomial is then generated from this list.
Except that the first element of the list will be converted to a coefficient with <code>x</code> to the power 0: <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>0</mn></msub><msup><mi>X</mi><mn>0</mn></msup><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> a_{0}X^{0} = a_{0} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.9641em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8141em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></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="base"><span class="strut" style="height: 0.5806em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>. In other words, <code>points[0] becomes $ a_{0}X $ and </code>points[1]` becomes <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><msup><mi>X</mi><mn>1</mn></msup></mrow><annotation encoding="application/x-tex"> a_{1}X^{1} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.9641em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8141em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span> …</p>
<p>For example, with a list of points: <code>[7, 4 ,3]</code> we’ll have this polynomial: <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msup><mi>X</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>X</mi><mn>1</mn></msup><mo>+</mo><mn>7</mn></mrow><annotation encoding="application/x-tex"> 3X^{2} + 4X^{1} + 7 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8141em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord">4</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8141em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">7</span></span></span></span></span>. And so, calculating the polynomial for x = 0 is like <code>3*0 + 4*0 + 7 = 7</code>.</p>
<p>It turns out that the first coefficient is generated with the first four characters of the flag : <code>Hero</code> and will represent the variable <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> a_{0} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.5806em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>. So calculating <span><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><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f(0) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="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">0</span><span class="mclose">)</span></span></span></span></span> gives <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> a_{0} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.5806em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>.</p>
<p>So we computes a0 :</p>
<div class="highlight"><div style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;">
<table style="border-spacing:0;padding:0;margin:0;border:0;"><tbody><tr><td style="vertical-align:top;padding:0;margin:0;border:0;">
<pre tabindex="0" style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">1
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">2
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">3
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">4
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">5
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">6
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">7
</span></code></pre></td>
<td style="vertical-align:top;padding:0;margin:0;border:0;;width:100%">
<pre tabindex="0" style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code class="language-python" data-lang="python"><span style="display:flex;"><span><span style="color:#ff7b72">from</span> <span style="color:#ff7b72">sage.all</span> <span style="color:#ff7b72">import</span> <span style="color:#ff7b72;font-weight:bold">*</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>F <span style="color:#ff7b72;font-weight:bold">=</span> FiniteField(<span style="color:#a5d6ff">2</span><span style="color:#ff7b72;font-weight:bold">**</span><span style="color:#a5d6ff">256</span> <span style="color:#ff7b72;font-weight:bold">-</span> <span style="color:#a5d6ff">189</span>)
</span></span><span style="display:flex;"><span>R <span style="color:#ff7b72;font-weight:bold">=</span> PolynomialRing(F, <span style="color:#a5d6ff">"x"</span>)
</span></span><span style="display:flex;"><span>H <span style="color:#ff7b72;font-weight:bold">=</span> <span style="color:#ff7b72">lambda</span> n: int(hashlib<span style="color:#ff7b72;font-weight:bold">.</span>sha256(n)<span style="color:#ff7b72;font-weight:bold">.</span>hexdigest(), <span style="color:#a5d6ff">16</span>)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>known_coefficient <span style="color:#ff7b72;font-weight:bold">=</span> H(<span style="color:#a5d6ff">'Hero'</span>) <span style="color:#8b949e;font-style:italic">#51862623363251592162508517414206794722184767070638202339849823866691337237984</span>
</span></span></code></pre></td></tr></tbody></table>
</div>
</div><p>We can then add this point to the point list <code>given_points.append([0, known_coefficient])</code>.</p>
<p>We then regenerate the list as before and obtain the correct polynomial:</p>
<pre tabindex="0"><code>91356407137791927144958613770622174607926961061379368852376771002781151613901*x^23 + 58688474918974956495962699109478986243962548972465028067725936901754910032197*x^22 + 71177914266346294875020009514904614231152252028035180341047573071890295627281*x^21 + [...] + 51862623363251592162508517414206794722184767070638202339849823866691337237984
</code></pre><p>Now that we have the polynomial, we’ll try to find their value in text, i.e. before being hashed avec sha256.</p>
<p>The vulnerability lay in the fact that we could calculate an extra point.</p>
<p>Another solution proposed by <a href="https://x.com/cryptanalyse">cryptanalyse</a> is to query the server many times until we have enough points to reconstruct the polynomial.</p>
<h3 id="flag-recovery">Flag recovery</h3>
<p>Here the vulnerability is that we have pieces of 4 characters and a small character set, so solutions are reduced.</p>
<p>We know that the coefficients are generated with chunks of four characters from this character set: “abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789_{}” (I’ve added the braces). This gives us a total of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn><msup><mn>5</mn><mn>4</mn></msup><mo>=</mo><mn>17850625</mn></mrow><annotation encoding="application/x-tex"> 65 ^{4} = 17850625 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8141em;"></span><span class="mord">6</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8141em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></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="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">17850625</span></span></span></span></span> possibilities (bruteforcable).</p>
<p>We’ll generate a dictionary of matches <code>value in int:plaintext</code> and see if it matches any of the characters in our list of coefficients:</p>
<div class="highlight"><div style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;">
<table style="border-spacing:0;padding:0;margin:0;border:0;"><tbody><tr><td style="vertical-align:top;padding:0;margin:0;border:0;">
<pre tabindex="0" style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 1
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 2
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 3
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 4
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 5
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 6
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 7
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 8
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679"> 9
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">10
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">11
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">12
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">13
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">14
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">15
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">16
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">17
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">18
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">19
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">20
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">21
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">22
</span><span style="white-space:pre;-webkit-user-select:none;user-select:none;margin-right:0.4em;padding:0 0.4em 0 0.4em;color:#737679">23
</span></code></pre></td>
<td style="vertical-align:top;padding:0;margin:0;border:0;;width:100%">
<pre tabindex="0" style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code class="language-python" data-lang="python"><span style="display:flex;"><span>charset <span style="color:#ff7b72;font-weight:bold">=</span> bytes(string<span style="color:#ff7b72;font-weight:bold">.</span>ascii_letters <span style="color:#ff7b72;font-weight:bold">+</span> string<span style="color:#ff7b72;font-weight:bold">.</span>digits <span style="color:#ff7b72;font-weight:bold">+</span> <span style="color:#a5d6ff">'_</span><span style="color:#a5d6ff">{}</span><span style="color:#a5d6ff">'</span>)
</span></span><span style="display:flex;"><span>candidate_list <span style="color:#ff7b72;font-weight:bold">=</span> list(itertools<span style="color:#ff7b72;font-weight:bold">.</span>product(charset, repeat<span style="color:#ff7b72;font-weight:bold">=</span><span style="color:#a5d6ff">4</span>))
</span></span><span style="display:flex;"><span>candidate_list <span style="color:#ff7b72;font-weight:bold">=</span> [<span style="color:#a5d6ff">""</span><span style="color:#ff7b72;font-weight:bold">.</span>join(v)<span style="color:#ff7b72;font-weight:bold">.</span>encode() <span style="color:#ff7b72">for</span> v <span style="color:#ff7b72;font-weight:bold">in</span> candidate_list]
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>rainbow_table <span style="color:#ff7b72;font-weight:bold">=</span> {}
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>print(<span style="color:#79c0ff">f</span><span style="color:#a5d6ff">'Building the rainbow table with </span><span style="color:#a5d6ff">{</span>len(candidate_list} entries <span style="color:#f85149">!</span><span style="color:#a5d6ff">')</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">for</span> candidate <span style="color:#ff7b72;font-weight:bold">in</span> candidate_list:
</span></span><span style="display:flex;"><span> int_hash <span style="color:#ff7b72;font-weight:bold">=</span> H(candidate)
</span></span><span style="display:flex;"><span> rainbow_table[int_hash] <span style="color:#ff7b72;font-weight:bold">=</span> candidate
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#8b949e;font-style:italic"># it tooks 15 seconds to build the table</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">assert</span> rainbow_table[H(<span style="color:#79c0ff">b</span><span style="color:#a5d6ff">'test'</span>)] <span style="color:#ff7b72;font-weight:bold">==</span> <span style="color:#79c0ff">b</span><span style="color:#a5d6ff">'test'</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>flag_parts <span style="color:#ff7b72;font-weight:bold">=</span> [<span style="color:#a5d6ff">0</span>] <span style="color:#ff7b72;font-weight:bold">*</span> len(flag_coefficients)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>print(<span style="color:#a5d6ff">"Looking up values in the rainbow table"</span>)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">for</span> coef <span style="color:#ff7b72;font-weight:bold">in</span> rainbow_table:
</span></span><span style="display:flex;"><span> <span style="color:#ff7b72">if</span> coef <span style="color:#ff7b72;font-weight:bold">in</span> flag_coefficients:
</span></span><span style="display:flex;"><span> flag_parts[flag_coefficients<span style="color:#ff7b72;font-weight:bold">.</span>index(H(<span style="color:#79c0ff">b</span><span style="color:#a5d6ff">'Hero'</span>))] <span style="color:#ff7b72;font-weight:bold">=</span> rainbow_table[coef]
</span></span></code></pre></td></tr></tbody></table>
</div>
</div><h3 id="solve-final">Solve final</h3>
<div class="highlight"><div style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;">
<table style="border-spacing:0;padding:0;margin:0;border:0;"><tbody><tr><td style="vertical-align:top;padding:0;margin:0;border:0;">
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<td style="vertical-align:top;padding:0;margin:0;border:0;;width:100%">
<pre tabindex="0" style="color:#e6edf3;background-color:#0d1117;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code class="language-python" data-lang="python"><span style="display:flex;"><span><span style="color:#ff7b72">import</span> <span style="color:#ff7b72">hashlib</span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">import</span> <span style="color:#ff7b72">re</span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">import</span> <span style="color:#ff7b72">string</span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">import</span> <span style="color:#ff7b72">requests</span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">import</span> <span style="color:#ff7b72">itertools</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">from</span> <span style="color:#ff7b72">pwn</span> <span style="color:#ff7b72">import</span> remote, context
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">from</span> <span style="color:#ff7b72">sage.all</span> <span style="color:#ff7b72">import</span> <span style="color:#ff7b72;font-weight:bold">*</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>host <span style="color:#ff7b72;font-weight:bold">=</span> <span style="color:#a5d6ff">'crypto.heroctf.fr'</span>
</span></span><span style="display:flex;"><span>port <span style="color:#ff7b72;font-weight:bold">=</span> <span style="color:#a5d6ff">9000</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>conn <span style="color:#ff7b72;font-weight:bold">=</span> remote(host, port)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>data <span style="color:#ff7b72;font-weight:bold">=</span> conn<span style="color:#ff7b72;font-weight:bold">.</span>recvuntilS(<span style="color:#a5d6ff">'>'</span>)<span style="color:#ff7b72;font-weight:bold">.</span>decode()
</span></span><span style="display:flex;"><span>given_points <span style="color:#ff7b72;font-weight:bold">=</span> eval(data<span style="color:#ff7b72;font-weight:bold">.</span>split(<span style="color:#a5d6ff">'</span><span style="color:#79c0ff">\n</span><span style="color:#a5d6ff">'</span>)[<span style="color:#a5d6ff">0</span>])
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>F <span style="color:#ff7b72;font-weight:bold">=</span> FiniteField(<span style="color:#a5d6ff">2</span><span style="color:#ff7b72;font-weight:bold">**</span><span style="color:#a5d6ff">256</span> <span style="color:#ff7b72;font-weight:bold">-</span> <span style="color:#a5d6ff">189</span>)
</span></span><span style="display:flex;"><span>R <span style="color:#ff7b72;font-weight:bold">=</span> PolynomialRing(F, <span style="color:#a5d6ff">"x"</span>)
</span></span><span style="display:flex;"><span>H <span style="color:#ff7b72;font-weight:bold">=</span> <span style="color:#ff7b72">lambda</span> n: int(hashlib<span style="color:#ff7b72;font-weight:bold">.</span>sha256(n)<span style="color:#ff7b72;font-weight:bold">.</span>hexdigest(), <span style="color:#a5d6ff">16</span>)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>known_coefficient <span style="color:#ff7b72;font-weight:bold">=</span> H(<span style="color:#a5d6ff">'Hero'</span>) <span style="color:#8b949e;font-style:italic"># 51862623363251592162508517414206794722184767070638202339849823866691337237984</span>
</span></span><span style="display:flex;"><span>given_points<span style="color:#ff7b72;font-weight:bold">.</span>append([<span style="color:#a5d6ff">0</span>, known_coefficient])
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>interp <span style="color:#ff7b72;font-weight:bold">=</span> R<span style="color:#ff7b72;font-weight:bold">.</span>lagrange_polynomial(given_points)
</span></span><span style="display:flex;"><span>flag_coefficients <span style="color:#ff7b72;font-weight:bold">=</span> interp<span style="color:#ff7b72;font-weight:bold">.</span>coefficients()
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>charset <span style="color:#ff7b72;font-weight:bold">=</span> bytes(string<span style="color:#ff7b72;font-weight:bold">.</span>ascii_letters <span style="color:#ff7b72;font-weight:bold">+</span> string<span style="color:#ff7b72;font-weight:bold">.</span>digits <span style="color:#ff7b72;font-weight:bold">+</span> <span style="color:#a5d6ff">'_</span><span style="color:#a5d6ff">{}</span><span style="color:#a5d6ff">'</span>)
</span></span><span style="display:flex;"><span>candidate_list <span style="color:#ff7b72;font-weight:bold">=</span> list(itertools<span style="color:#ff7b72;font-weight:bold">.</span>product(charset, repeat<span style="color:#ff7b72;font-weight:bold">=</span><span style="color:#a5d6ff">4</span>))
</span></span><span style="display:flex;"><span>candidate_list <span style="color:#ff7b72;font-weight:bold">=</span> [<span style="color:#a5d6ff">""</span><span style="color:#ff7b72;font-weight:bold">.</span>join(v)<span style="color:#ff7b72;font-weight:bold">.</span>encode() <span style="color:#ff7b72">for</span> v <span style="color:#ff7b72;font-weight:bold">in</span> candidate_list]
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>rainbow_table <span style="color:#ff7b72;font-weight:bold">=</span> {}
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>print(<span style="color:#79c0ff">f</span><span style="color:#a5d6ff">'Building the rainbow table with </span><span style="color:#a5d6ff">{</span>len(candidate_list} entries <span style="color:#f85149">!</span><span style="color:#a5d6ff">')</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">for</span> candidate <span style="color:#ff7b72;font-weight:bold">in</span> candidate_list:
</span></span><span style="display:flex;"><span> int_hash <span style="color:#ff7b72;font-weight:bold">=</span> H(candidate)
</span></span><span style="display:flex;"><span> rainbow_table[int_hash] <span style="color:#ff7b72;font-weight:bold">=</span> candidate
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">assert</span> rainbow_table[H(<span style="color:#79c0ff">b</span><span style="color:#a5d6ff">'test'</span>)] <span style="color:#ff7b72;font-weight:bold">==</span> <span style="color:#79c0ff">b</span><span style="color:#a5d6ff">'test'</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>flag_parts <span style="color:#ff7b72;font-weight:bold">=</span> [<span style="color:#a5d6ff">0</span>] <span style="color:#ff7b72;font-weight:bold">*</span> len(flag_coefficients)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>print(<span style="color:#a5d6ff">"Looking up values in the rainbow table"</span>)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#ff7b72">for</span> coef <span style="color:#ff7b72;font-weight:bold">in</span> rainbow_table:
</span></span><span style="display:flex;"><span> <span style="color:#ff7b72">if</span> coef <span style="color:#ff7b72;font-weight:bold">in</span> flag_coefficients:
</span></span><span style="display:flex;"><span> flag_parts[flag_coefficients<span style="color:#ff7b72;font-weight:bold">.</span>index(H(<span style="color:#79c0ff">b</span><span style="color:#a5d6ff">'Hero'</span>))] <span style="color:#ff7b72;font-weight:bold">=</span> rainbow_table[coef]
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>flag <span style="color:#ff7b72;font-weight:bold">=</span> <span style="color:#a5d6ff">""</span><span style="color:#ff7b72;font-weight:bold">.</span>join(flag_parts)
</span></span><span style="display:flex;"><span>print(<span style="color:#a5d6ff">"Flag : </span><span style="color:#a5d6ff">{flag}</span><span style="color:#a5d6ff">"</span>)
</span></span></code></pre></td></tr></tbody></table>
</div>
</div><p>It tooks 20 to 30 seconds overall.</p>
<p>Flag : <code>Hero{th3r3_4r3_tw0_typ35_0f_p30pl3_1n_th15_w0rld_th053_wh0_c4n_3xtr4p0l4t3_fr0m_1nc0mpl3t3_d474}</code></p>
<h1 id="conclusion">Conclusion</h1>
<p>The challenge was easy to understand after a quick read of the doc, it’s easy to understand where to look.</p>
<p>I was able to better understand Lagrange interpolation and polynomials.</p>
<h1 id="links">Links</h1>
<ul>
<li><a href="http://exo7.emath.fr/cours/ch_polynomes.pdf">http://exo7.emath.fr/cours/ch_polynomes.pdf</a></li>
<li><a href="https://www.youtube.com/watch?v=nvkX1Bd90Gk">https://www.youtube.com/watch?v=nvkX1Bd90Gk</a></li>
<li><a href="https://en.wikipedia.org/wiki/Lagrange_polynomial">https://en.wikipedia.org/wiki/Lagrange_polynomial</a></li>
<li><a href="https://en.wikipedia.org/wiki/Modular_arithmetic">https://en.wikipedia.org/wiki/Modular_arithmetic</a></li>
<li><a href="https://en.wikipedia.org/wiki/Shamir%27s_secret_sharing">https://en.wikipedia.org/wiki/Shamir%27s_secret_sharing</a></li>
<li><a href="https://doc.sagemath.org/html/en/reference/polynomial_rings/sage/rings/polynomial/polynomial_ring.html">https://doc.sagemath.org/html/en/reference/polynomial_rings/sage/rings/polynomial/polynomial_ring.html</a></li>
</ul>
</article>
</div>
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