Tagged Questions
3
votes
1answer
76 views
Finding number of solutions to an equation in $\mathbb F_p$
$p=3 \pmod 4$ is a prime. Let $b\in \mathbb F_p^*$.
Show that the equation $v^2=u^4-4b$ has $p-1$ solutions $(u,v)$ with $u, v \in \mathbb F_p$.
If we write the given equation as $v+u^2=x$ and ...
2
votes
1answer
32 views
Representing an element mod $n$ as a product of two primes
Given a positive integer $n$ and $x \in (\mathbb{Z}/n\mathbb{Z})^*$ what is the most efficient way to find primes $q_1,q_2$ st
$$q_1q_2 \equiv x \bmod n$$
when $n$ is large?
One option is just to ...
-1
votes
1answer
87 views
Prove that in $\mathbb{Z}_p$, $[a]^{-1} = [a]$ if and only if $[a]=[1]$ or $[a]=[p-1]$
Let $p$ be a prime and a an integer. Prove that in $\mathbb{Z}_p$, $[a]^{-1} = [a]$ if and only if $[a]=[1]$ or $[a]=[p-1]$.
I greatly appreciate your help on this question!
2
votes
1answer
73 views
Is the modular multiplicative inverse of $a$ equal to that of $-a$?
In javascript, I am implementing Lagrange interpolation over a finite field $GF_p$ for some prime $p$. I only need to compute the value of the $y$-intercept of the Lagrange interpolation polynomial ...
0
votes
1answer
51 views
Prove that there are $p+1$ points on the elliptic curve $y^2 = x^3 + 1$ over $\mathbb{F}_p$, where $p > 3$ is a prime such that $p \equiv 2 \pmod 3$.
Let $p > 3$ be a prime such that $p \equiv 2 \pmod 3$. Define the elliptic curve $E$ over $\mathbb{F}_p$ by $y^2 = x^3 + 1$. Prove that $E(\mathbb{F}_p)$ consists of $p+1$ points.
Using Fermat's ...
0
votes
2answers
53 views
Find the inverse of $\alpha^{38}$ in $\mathbb F = \mathbb Z_2[x]/\left<x^4+x+1\right>$
Let $\alpha$ be a root of $x^4+x+1$ and we are given some powers of $\alpha$ as linear combinations of $1,\alpha,\alpha^2$ and $\alpha^3$
$\alpha^4=\alpha+1$
$\alpha^5=\alpha^2+\alpha$
... (the rest ...
3
votes
4answers
71 views
$Y^3$ congruent to $1 \pmod {p}$
How to get the condition on $p$ for which $y^3$ congruent to $1$ modulo $p$ has $3$ solutions ( $1$ solution $x= 1$ is always possible, right ?).
0
votes
1answer
87 views
$\mathbb{Z}/n\mathbb{Z}[T]$ and zero divisors
I read the following in wiki, but I can't understand what is meant by "divisor" there.
Notice that $(\mathbb{Z}/2\mathbb{Z})[T]/(T^{2}+1)$ is not a field since it admits a zero divisor ...
1
vote
1answer
103 views
Mathematical names of the sets and elements of standard computer numbers
In standard computer arithmetic, there are two sets of numbers.
N-bit unsigned numbers. The elements are natural numbers in $(0, 2^N]$. Arithmetic operations is defined as for the natural numbers ...
