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<p>Let <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">A</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> be a polynomial over <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.974998em;vertical-align:-0.286108em;"></span><spanclass="mord"><spanclass="mord mathbb">F</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.15139200000000003em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">p</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> with formal indeterminate <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span></span></span></span>. As an example,</p>
<p>defines a degree-<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">3</span></span></span></span> polynomial. <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.58056em;vertical-align:-0.15em;"></span><spanclass="mord"><spanclass="mord mathnormal">a</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.30110799999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">0</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is referred to as the constant term. Polynomials of
degree <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.66666em;vertical-align:-0.08333em;"></span><spanclass="mord mathnormal">n</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1</span></span></span></span> have <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span> coefficients. We will often want to compute the result of replacing
the formal indeterminate <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span></span></span></span> with some concrete value <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">x</span></span></span></span>, which we denote by <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">A</span><spanclass="mopen">(</span><spanclass="mord mathnormal">x</span><spanclass="mclose">)</span></span></span></span>.</p>
<blockquote>
<p>In mathematics this is commonly referred to as "evaluating <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">A</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> at a point <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">x</span></span></span></span>".
The word "point" here stems from the geometrical usage of polynomials in the form
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical-align:-0.19444em;"></span><spanclass="mord mathnormal"style="margin-right:0.03588em;">y</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">A</span><spanclass="mopen">(</span><spanclass="mord mathnormal">x</span><spanclass="mclose">)</span></span></span></span>, where <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mopen">(</span><spanclass="mord mathnormal">x</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord mathnormal"style="margin-right:0.03588em;">y</span><spanclass="mclose">)</span></span></span></span> is the coordinate of a point in two-dimensional space.
However, the polynomials we deal with are almost always constrained to equal zero, and
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">x</span></span></span></span> will be an <ahref="fields.html">element of some field</a>. This should not be confused
with points on an <ahref="curves.html">elliptic curve</a>, which we also make use of, but never in
<li>Given a polynomial <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">A</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> of degree <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.66666em;vertical-align:-0.08333em;"></span><spanclass="mord mathnormal">n</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1</span></span></span></span>, if we obtain <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span> evaluations of the
polynomial at distinct points then these evaluations perfectly define the polynomial. In
other words, given these evaluations we can obtain a unique polynomial <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">A</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> of degree
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.66666em;vertical-align:-0.08333em;"></span><spanclass="mord mathnormal">n</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1</span></span></span></span> via polynomial interpolation.</li>
<li><spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mopen">[</span><spanclass="mord"><spanclass="mord mathnormal">a</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.30110799999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">0</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord"><spanclass="mord mathnormal">a</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.30110799999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">1</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="minner">⋯</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord"><spanclass="mord mathnormal">a</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.301108em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mathnormal mtight">n</span><spanclass="mbin mtight">−</span><spanclass="mord mtight">1</span></span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.208331em;"><span></span></span></span></span></span></span><spanclass="mclose">]</span></span></span></span> is the <strong>coefficient representation</strong> of the polynomial
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">A</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span>. Equivalently, we could use its <strong>evaluation representation</strong>
at <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span> distinct points. Either representation uniquely specifies the same polynomial.</li>
<p>Horner's rule allows for efficient evaluation of a polynomial of degree <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.66666em;vertical-align:-0.08333em;"></span><spanclass="mord mathnormal">n</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1</span></span></span></span>, using
only <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.66666em;vertical-align:-0.08333em;"></span><spanclass="mord mathnormal">n</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1</span></span></span></span> multiplications and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.66666em;vertical-align:-0.08333em;"></span><spanclass="mord mathnormal">n</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1</span></span></span></span> additions. It is the following identity:
<p>The FFT is an efficient way of converting between the coefficient and evaluation
representations of a polynomial. It evaluates the polynomial at the <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span>th roots of unity
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.064108em;vertical-align:-0.25em;"></span><spanclass="mopen">{</span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.8141079999999999em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">0</span></span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.8141079999999999em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">1</span></span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="minner">⋯</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.8141079999999999em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mathnormal mtight">n</span><spanclass="mbin mtight">−</span><spanclass="mord mtight">1</span></span></span></span></span></span></span></span></span><spanclass="mclose">}</span><spanclass="mpunct">,</span></span></span></span> where <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span></span></span></span> is a primitive <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span>th root
of unity. By exploiting symmetries in the roots of unity, each round of the FFT reduces
the evaluation into a problem only half the size. Most commonly we use polynomials of
length some power of two, <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.849108em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord">2</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.849108em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight"style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span>, and apply the halving reduction recursively.</p>
<h3id="motivation-fast-polynomial-multiplication"><aclass="header"href="#motivation-fast-polynomial-multiplication">Motivation: Fast polynomial multiplication</a></h3>
<p>In the coefficient representation, it takes <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.064108em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal"style="margin-right:0.02778em;">O</span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mord mathnormal">n</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.8141079999999999em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">2</span></span></span></span></span></span></span></span><spanclass="mclose">)</span></span></span></span> operations to multiply two
<p>where each of the <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span> terms in the first polynomial has to be multiplied by the <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span> terms
of the second polynomial.</p>
<p>In the evaluation representation, however, polynomial multiplication only requires <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal"style="margin-right:0.02778em;">O</span><spanclass="mopen">(</span><spanclass="mord mathnormal">n</span><spanclass="mclose">)</span></span></span></span>
<p>where each evaluation is multiplied pointwise.</p>
<p>This suggests the following strategy for fast polynomial multiplication:</p>
<ol>
<li>Evaluate polynomials at all <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span> points;</li>
<li>Perform fast pointwise multiplication in the evaluation representation (<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal"style="margin-right:0.02778em;">O</span><spanclass="mopen">(</span><spanclass="mord mathnormal">n</span><spanclass="mclose">)</span></span></span></span>);</li>
<li>Convert back to the coefficient representation.</li>
</ol>
<p>The challenge now is how to <strong>evaluate</strong> and <strong>interpolate</strong> the polynomials efficiently.
Naively, evaluating a polynomial at <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span> points would require <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.064108em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal"style="margin-right:0.02778em;">O</span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mord mathnormal">n</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.8141079999999999em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">2</span></span></span></span></span></span></span></span><spanclass="mclose">)</span></span></span></span> operations (we use
the <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal"style="margin-right:0.02778em;">O</span><spanclass="mopen">(</span><spanclass="mord mathnormal">n</span><spanclass="mclose">)</span></span></span></span> Horner's rule at each point):</p>
<p>(<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.9495499999999999em;vertical-align:0em;"></span><spanclass="mord accent"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.9495499999999999em;"><spanstyle="top:-3em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord mathbf">A</span></span><spanstyle="top:-3.25511em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="accent-body"style="left:-0.11110999999999999em;"><spanclass="mord">^</span></span></span></span></span></span></span></span></span></span> is known as the <em>Discrete Fourier Transform</em> of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68611em;vertical-align:0em;"></span><spanclass="mord mathbf">A</span></span></span></span>;
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.83611em;vertical-align:-0.15em;"></span><spanclass="mord"><spanclass="mord mathbf"style="margin-right:0.01597em;">V</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.151392em;"><spanstyle="top:-2.5500000000000003em;margin-left:-0.01597em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight"style="margin-right:0.03588em;">ω</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is also called the <em>Vandermonde matrix</em>.)</p>
<p>Our strategy is to divide a DFT of size <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span> into two interleaved DFTs of size <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">n</span><spanclass="mord">/2</span></span></span></span>. Given
<p>Notice that we are only evaluating <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.151392em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord text mtight"><spanclass="mord mtight">even</span></span></span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.33610799999999996em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord text mtight"><spanclass="mord mtight">odd</span></span></span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> over half
This gives us all the terms we need to reconstruct <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">A</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> over the full domain
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.064108em;vertical-align:-0.25em;"></span><spanclass="mopen">{</span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.8141079999999999em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">0</span></span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="minner">⋯</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.8141079999999999em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mathnormal mtight">n</span><spanclass="mbin mtight">−</span><spanclass="mord mtight">1</span></span></span></span></span></span></span></span></span><spanclass="mclose">}</span></span></span></span>: which means we have transformed a
<p>We choose <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.849108em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord">2</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.849108em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight"style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span> to be a power of two (by zero-padding if needed), and apply this
divide-and-conquer strategy recursively. By the Master Theorem<supclass="footnote-reference"><ahref="#master-thm">1</a></sup>, this gives us
an evaluation algorithm with <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal"style="margin-right:0.02778em;">O</span><spanclass="mopen">(</span><spanclass="mord mathnormal">n</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin-right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.20696799999999996em;"><spanstyle="top:-2.4558600000000004em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">2</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.24414em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord mathnormal">n</span><spanclass="mclose">)</span></span></span></span> operations, also known as the Fast Fourier
<li>multiply our final result by a factor of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord">1/</span><spanclass="mord mathnormal">n</span></span></span></span>.</li>
<p>(To understand why the inverse FFT has a similar form to the FFT, refer to Slide 13-1 of
<supclass="footnote-reference"><ahref="#ifft">2</a></sup>. The below image was also taken from <supclass="footnote-reference"><ahref="#ifft">2</a></sup>.)</p>
<p>The Schwartz-Zippel lemma informally states that "different polynomials are different at
most points." Formally, it can be written as follows:</p>
<blockquote>
<p>Let <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">p</span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mord mathnormal">x</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.30110799999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">1</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord"><spanclass="mord mathnormal">x</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.30110799999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">2</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="minner">⋯</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord"><spanclass="mord mathnormal">x</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.151392em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">n</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mclose">)</span></span></span></span> be a nonzero polynomial of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span> variables with degree <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathnormal">d</span></span></span></span>.
Let <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span> be a finite set of numbers with at least <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathnormal">d</span></span></span></span> elements in it. If we choose random
<p>In the familiar univariate case <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">p</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span>, this reduces to saying that a nonzero polynomial
of degree <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathnormal">d</span></span></span></span> has at most <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathnormal">d</span></span></span></span> roots.</p>
<p>The Schwartz-Zippel lemma is used in polynomial equality testing. Given two multi-variate
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical-align:-0.19444em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.0037em;">α</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.30110799999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:-0.0037em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">1</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="minner">⋯</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.0037em;">α</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.151392em;"><spanstyle="top:-2.5500000000000003em;margin-left:-0.0037em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">n</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">←</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="mpunct">,</span></span></span></span> where the size of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span> is at least
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord">∣</span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="mord">∣</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">≥</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mord mathnormal">d</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.30110799999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">1</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord"><spanclass="mord mathnormal">d</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.30110799999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">2</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mclose">)</span><spanclass="mord">.</span></span></span></span> If the two polynomials are identical, this will always be true,
whereas if the two polynomials are different then the equality holds with probability at
<p>Consider the order-<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span> multiplicative subgroup <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathcal"style="margin-right:0.00965em;">H</span></span></span></span> with primitive root of unity
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span></span></span></span>. For all <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.863764em;vertical-align:-0.0391em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.824664em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">∈</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><spanclass="mord mathcal"style="margin-right:0.00965em;">H</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord mathnormal">i</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">∈</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mopen">[</span><spanclass="mord mathnormal">n</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord">1</span><spanclass="mclose">]</span><spanclass="mpunct">,</span></span></span></span> we have
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.0746639999999998em;vertical-align:-0.25em;"></span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.824664em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.664392em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.0746639999999998em;vertical-align:-0.25em;"></span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.664392em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.824664em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.0746639999999998em;vertical-align:-0.25em;"></span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.8141079999999999em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">0</span></span></span></span></span></span></span></span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.824664em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1.</span></span></span></span> In other words, every element of
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathcal"style="margin-right:0.00965em;">H</span></span></span></span> fulfils the equation </p>
<p>meaning every element is a root of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.07153em;">Z</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.32833099999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight"style="margin-right:0.08125em;">H</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span><spanclass="mord">.</span></span></span></span> We call <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.07153em;">Z</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.32833099999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight"style="margin-right:0.08125em;">H</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> the <strong>vanishing polynomial</strong>
over <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathcal"style="margin-right:0.00965em;">H</span></span></span></span> because it evaluates to zero on all elements of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathcal"style="margin-right:0.00965em;">H</span><spanclass="mord">.</span></span></span></span></p>
check that <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">A</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal"style="margin-right:0.05017em;">B</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal"style="margin-right:0.07153em;">C</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> over <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><spanclass="mord mathcal"style="margin-right:0.00965em;">H</span><spanclass="mpunct">,</span></span></span></span> we simply have to check that
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">A</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal"style="margin-right:0.05017em;">B</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal"style="margin-right:0.07153em;">C</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> is some multiple of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.07153em;">Z</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.32833099999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight"style="margin-right:0.08125em;">H</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span>. In other words, if dividing our
constraint by the vanishing polynomial still yields some polynomial
over <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathcal"style="margin-right:0.00965em;">H</span><spanclass="mord">.</span></span></span></span></p>
<p>Polynomials are commonly written in the monomial basis (e.g. <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.008548em;vertical-align:-0.19444em;"></span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.8141079999999999em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">2</span></span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord">...</span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.664392em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span>). However,
when working over a multiplicative subgroup of order <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span>, we find a more natural expression
<p>Consider the order-<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span> multiplicative subgroup <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathcal"style="margin-right:0.00965em;">H</span></span></span></span> with primitive root of unity
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span></span></span></span>. The Lagrange basis corresponding to this subgroup is a set of functions
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.03785em;">δ</span></span></span></span> is the Kronecker delta function. </p>
<p>Now, we can write our polynomial as a linear combination of Lagrange basis functions,</p>
<p>which is equivalent to saying that <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">p</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> evaluates to <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.58056em;vertical-align:-0.15em;"></span><spanclass="mord"><spanclass="mord mathnormal">a</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.30110799999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">0</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> at <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8141079999999999em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.8141079999999999em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span>,
<p>where <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.58056em;vertical-align:-0.15em;"></span><spanclass="mord"><spanclass="mord mathnormal">c</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.31166399999999994em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">i</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the barycentric weight. (To understand how this form was derived, refer to
<supclass="footnote-reference"><ahref="#barycentric">3</a></sup>.) For <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.65952em;vertical-align:0em;"></span><spanclass="mord mathnormal">i</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><spanclass="mord">0</span><spanclass="mpunct">,</span></span></span></span> we have
<p>Suppose we are given a set of evaluation points <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mopen">{</span><spanclass="mord"><spanclass="mord mathnormal">x</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.30110799999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">0</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord"><spanclass="mord mathnormal">x</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.30110799999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">1</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="minner">⋯</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord"><spanclass="mord mathnormal">x</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.301108em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mathnormal mtight">n</span><spanclass="mbin mtight">−</span><spanclass="mord mtight">1</span></span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.208331em;"><span></span></span></span></span></span></span><spanclass="mclose">}</span></span></span></span>.
Since we cannot assume that the <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.58056em;vertical-align:-0.15em;"></span><spanclass="mord"><spanclass="mord mathnormal">x</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.31166399999999994em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">i</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>'s form a multiplicative subgroup, we consider also
the Lagrange polynomials <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.83333em;vertical-align:-0.15em;"></span><spanclass="mord"><spanclass="mord mathcal">L</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.31166399999999994em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">i</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>'s in the general case. Then we can construct:</p>
the whole product to evaluate to zero. On the other hand, <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.58056em;vertical-align:-0.15em;"></span><spanclass="mord"><spanclass="mord mathnormal">x</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.31166399999999994em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">i</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> will evaluate to
<p><ahref="https://people.eecs.berkeley.edu/%7Evazirani/algorithms/chap2.pdf">Dasgupta, S., Papadimitriou, C. H., & Vazirani, U. V. (2008). "Algorithms" (ch. 2). New York: McGraw-Hill Higher Education.</a></p>
<p><ahref="http://www.cs.ust.hk/mjg_lib/Classes/COMP3711H_Fall16/lectures/FFT_Slides.pdf">Golin, M. (2016). "The Fast Fourier Transform and Polynomial Multiplication" [lecture notes], COMP 3711H Design and Analysis of Algorithms, Hong Kong University of Science and Technology.</a></p>
<p><ahref="https://people.maths.ox.ac.uk/trefethen/barycentric.pdf">Berrut, J. and Trefethen, L. (2004). "Barycentric Lagrange Interpolation."</a></p>