Math and stuff

Exercise from P.62 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Show (by example) that the congruence $ax \equiv ay \pmod n$ does not necessarily imply that $x \equiv y \pmod n$. $1 \not\equiv 0 \pmod 2$ but $2 \cdot 1 \equiv 2 \cdot 0 \pmod 2$. On...

Exercise from P.47 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Let $a, b \in \mathbb{Z}$, $b = 0$. Then there exist unique integers $q$ and $r$ with $a = bq + r$ and $0 \leq r < \abs{b}$, where $\abs{b}$ is the absolute value of $\abs{b}$. Let...

Exercise from P.44 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Now consider $y^2 = g(x)$ where $g$ is a cubic, that is the zero set of $f(x, y) = y^2 − g(x)$. We want to see that a nonvertical tangent has multiplicity at least 2 at the...

Exercise from P.44 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Show that $h$ has a zero of order $k$ at $x = a$ if and only if $h(a) = h’(a) = \cdots = h^{(k−1)}(a) = 0$ and $h^{(k)}(a) = 0$, where $h^{(i)}$ is the $i$th derivative of...

Exercise from P.44 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Show that the curve $y = (x − a)^k$ intersects the $x$-axis with multiplicity $k$ at $x = a$ and with multiplicity 0 at all other points $x = b$. The curve is the zero set of...