I was going through a presentation titled "SIKE in Hardware" by professor Reza Azarderakhsh. On the page $10$ of the presentation he introduces a variable $w$. Could you please explain what the variable is in layman's terms? I am a master student with a background in software engineering and little to none background in algebra.
1 Answer
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Coordinates of points on the curves in SIDH are in a finite field $\mathbf F_{p^2}$.
First, $\mathbf F_p$ are integers mod $p$. Some polynomials on this field does not have roots, and what we can do about it is extend the field so it has a root. The same way $X^2 + 1$ does not have a root in real numbers, so we introduce $i$.
On this case, let's say we have an irreducible (it has no roots) polynomial of degree $2$ in $\mathbf F_p$, so we introduce a root $w$ and that's how we can create $\mathbf F_{p^2}$. Then, any element $z$ in this new field can be written $z = a + bw$ for $a$ and $b$ in $\mathbf F_p$.
Again, think of the complex numbers: anyone of them can be written $a + bi$ for $a$ and $b$ two real numbers.
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$\begingroup$ Thank you very much for your reply. That certainly clears things up. $\endgroup$– samarCommented Jul 10, 2019 at 7:53