Math and stuff

Exercise from P.185 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Calculate the elliptic curve group generated by $y^2 = x^3 + 3x + 6$ over $\mathbb{F}_7$. By substituting all possible points, it is easy to determine that $[0, 1, 0], [3, 0, 1], [6, 3, 1]$ and...

Exercise from P.180 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Transform $p(z) = z^3+ Az^2+ Bz + C$ into the form of $q(x) = x^3 + ax + b$ without changing the discriminant. As discussed earlier in the chapter, setting $z = x - A / 3$...

Exercise from P.170 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Show that in contrast to the uniqueness result which Lagrange interpolation provides, there are an infinite number of polynomials of degree $n + 1$ which pass through the $n + 1$ given points. Let $p(x)$ be the...

Exercise from P.164 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Show that $x^2 + y^2 = 3$ has no rational points, even in $\mathbb{P}^2(\mathbb{Q})$. Suppose there is. Let $(a / b, c / d)$ with $\gcd(a, b) = \gcd(c, d) = 1$ be a point on it....

Exercise from P.159 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Given affine curves $x = y^2$ and $y = −3$, find the points of intersection of the corresponding projective curves. After homogenization, we obtain: $zx = y^2$ $y = -3z$. We split this into two cases: $z...