Math 502 Daily Assignment

Exercises from the text

262: 6, 7, 12 – 14, 16, 17, 20, 21, 24, 25

Further Exercises

1. Find a point $A$ of largest possible order on the cubic curve $y^2+y=x^3-x$ over the finite field $\mathbf{Z}⁄59\mathbf{Z}$; then express the affine point $\left(0,0\right)$ as a multiple of $A$.

2. It is agreed between George and Karla to construct jointly a private value using the Diffie-Hellman algorithm with powers of the primitive root $6$ modulo $761$. Karla randomly picks the power $221$ and, accordingly, sends George the number $481$. George sends Karla the number $22$. What private value have they jointly constructed?

3. George and Karla decide to use the Diffie-Hellman algorithm with powers (i.e., additive multiples) of the point $\left(0,0\right)$ in the arithmetic of points on the cubic curve $E$ with equation $y^2+y=x^3-x$ over the finite field $\mathbf{Z}⁄761\mathbf{Z}$.

   1. How many points does $E$ contain in the finite field $\mathbf{Z}⁄761\mathbf{Z}$.

   2. What is the order of $\left(0,0\right)$ in the arithmetic of points on $E$ over this field?

   3. If Karla sends George the $221$st power of the point $\left(0,0\right)$ and George sends Karla the point $\left(265,321\right)$, what private point have they jointly constructed on the curve $E$?