Math and stuff

Let $R = \{ b_0, b_1, \cdots, b_n \}$ be such a ring where $b_0 = 0$ and $b_1 = 1$. If $n = 1$, we are done. Suppose otherwise. All we need to show that each $b_i$ has a multiplicative inverse. Let $i = 2, \cdots, n$ be given....

Let $y^2 = x^3 + ax + b$ be an elliptic curve over a field $F$. We claim that all points are one of the following: $[0, 1, 0]$. $[x, y, 1]$ for some $x, y \in F$ First, we will show that none of the following points are on...

Important property in Modern Cryptography and Elliptic Curves - A Beginner’s Guide. For any prime $p \geq 3$, $U_p$ has exactly $(p - 1) / 2$ squares. More formally, we are trying to show: \[\begin{align*} \abs{ \{ x \in U_p \mid \exists y, y \cdot y = x \} }...

Important property used in Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Given a finite field $F$, $F^{\times}$ is cyclic. We will expand the proof given in Dummit and Foote in Proposition 18 in Section 9.5. First, we prove a lemma: Given a cyclic group $G$ and $d \in...

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...