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<p><strong>Warning</strong>: I want to state clearly that I&rsquo;m not a mathematician, I&rsquo;m just an amateur on math studying in my free time, and this article is just an attempt to try to sort the notes that I took while reading about the KZG Commitments.</p>
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<p>Few weeks ago I started reading about <a href="https://www.iacr.org/archive/asiacrypt2010/6477178/6477178.pdf">KZG Commitments</a> from the articles written by <a href="https://dankradfeist.de/ethereum/2020/06/16/kate-polynomial-commitments.html">Dankrad Feist</a>, by <a href="https://hackmd.io/@tompocock/Hk2A7BD6U">Tom Walton-Pocock</a> and by <a href="https://alinush.github.io/2020/05/06/kzg-polynomial-commitments.html">Alin Tomescu</a>. I want to thank them, because their articles helped me to understand a bit the concepts. I recommend spending the time reading their articles instead of this current notes.</p>
<p>Few weeks ago I started reading about <a href="https://www.iacr.org/archive/asiacrypt2010/6477178/6477178.pdf">KZG Commitments</a> from the articles written by <a href="https://dankradfeist.de/ethereum/2020/06/16/kate-polynomial-commitments.html">Dankrad Feist</a>, by <a href="https://hackmd.io/@tompocock/Hk2A7BD6U">Tom Walton-Pocock</a> and by <a href="https://alinush.github.io/2020/05/06/kzg-polynomial-commitments.html">Alin Tomescu</a>. I want to thank them, because their articles helped me to understand a bit the concepts. I recommend spending the time reading their articles (<a href="https://dankradfeist.de/ethereum/2020/06/16/kate-polynomial-commitments.html">1</a>, <a href="https://hackmd.io/@tompocock/Hk2A7BD6U">2</a>, <a href="https://alinush.github.io/2020/05/06/kzg-polynomial-commitments.html">3</a>) instead of this current notes.</p>
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<h3>Conclusions</h3>
<p>The content covered in this notes is just a quick overview, but allows us to see the potential of the scheme. One next iteration from what we&rsquo;ve seen is the approach to do batch proofs, which allows us to evaluate at multiple points with a single evaluation proof. This scheme can be used as a <em>vector commitment</em>, using a polynomial where the <span class="math inline">\(p(i) = x_i\)</span> for all values of <span class="math inline">\(x_i\)</span> of the vector, which can be obtained from the <span class="math inline">\(x_i\)</span> values and computing the <a href="https://en.wikipedia.org/wiki/Lagrange_polynomial">Lagrange interpolation</a>. This is quite useful combined with the mentioned batch proofs. The <em>batch proofs</em> logic can be found at the <a href="https://arnaucube.com/blog/kzg-batch-proof.html">blog/kzg-batch-proof</a> notes (kind of the continuation of the current notes).</p>
<p>The content covered in this notes is just a quick overview, but allows us to see the potential of the scheme. One next iteration from what we&rsquo;ve seen is the approach to do batch proofs, which allows us to evaluate at multiple points with a single evaluation proof. This scheme can be used as a <em>vector commitment</em>, using a polynomial where the <span class="math inline">\(p(i) = x_i\)</span> for all values of <span class="math inline">\(x_i\)</span> of the vector, which can be obtained from the <span class="math inline">\(x_i\)</span> values and computing the <a href="shamir-secret-sharing.html#lagrange-polynomial%20interpolation">Lagrange interpolation</a>. This is quite useful combined with the mentioned batch proofs. The <em>batch proofs</em> logic can be found at the <a href="https://arnaucube.com/blog/kzg-batch-proof.html">blog/kzg-batch-proof</a> notes (kind of the continuation of the current notes).</p>
<p>As a final note, in order to try to digest the notes, I&rsquo;ve did a <em>toy implementation</em> of this scheme at <a href="https://github.com/arnaucube/kzg-commitments-study">https://github.com/arnaucube/kzg-commitments-study</a>. It&rsquo;s quite simple, but contains the logic overviewed in this notes.</p>
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