Math and stuff

Exercise from P.44 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Let $h(x)$ be a polynomial of degree $n \geq 2$ with coefficients in a field $F$, and let $a \in F$. (1) Prove $h(x) = (x − a)q(x) + h(a)$ for some polynomial $q$ having coefficients in...

Exercise from P.42 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. In how many points can two (distinct) conics intersect? Based on drawings of my own and on P.43, it seems that there can be up to 4 intersections. In how many points can a conic and a...

Exercise from P.40 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. In how many points can two arbitrary lines (in the plane) intersect? For each $i = 1, 2$, let $f_i(x, y) = a_ix + b_iy + c_i$. At least one of $a_1$ or $b_1$ must be nonzero...

Exercise from P.40 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Is every point on a rational line a rational point? No, $\sqrt{2}$ is famously irrational, and the rational line $f(x, y) = x - y$ passes $(\sqrt{2}, \sqrt{2})$. If a line passes through at least two rational...

Exercise from P.38 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Given an elliptic curve $y^2 = x^3 + k$ and a point $(a, b)$, we’ll find the intersection between the line tangent at $(a, b)$ and the curve. $\frac{dy}{dx} = \frac{3x^2}{2y}$, so the slope of the tangent...