# Tagged Questions

21 views

### Elements with order 3 in group $F_{16}/\{0\}$

If you have the finite field $GF(16)$ and you define the group $GF(16)/\{0\},*$ this group is cyclic. I need to determine how many elements in this group have order 3. Of course you could just try out ...
45 views

### Elements of subfield $F_{8}$ of $F_{2}[x]/(x^{6}+x+1)$

I need to find the elements of the subfield $F_{8}$ of $F_{2}[x]/(x^{6}+x+1)$ in their standard representation. I know that $F_{2}[x]/(x^{6}+x+1)$ represents the residu classes of polynomials modulo ...
41 views

### Discrete Logarithm Problem in $GF(p^m)$

I have question regarding DLP in $GF(p^m)$ I know the algorithms for solving the DLP in $GF(p)$ like Baby Step-Giant Step, Pohlig-Hellman etc... But what if we move into the $GF(p^m)$ and are ...
52 views

### Solving the discrete logarithm using index calculus, finite fields and factor bases.

(a) Let $p$ be the prime 1073741827, with $\Bbb{F}_p$ the corresponding finite field. A primitive root in $\Bbb{F}_p$ is equal to $g=2$. Use a factor base of primes up to 13 to find the discrete ...
68 views

### How to find multiplicative orders of all elements in field $\Bbb F$ (say $\Bbb F_{13}$)?

I am working on some finite fields and I was referring to some online class material. Is there any way to find the multiplicative orders of all elements in a field $\Bbb F$?
38 views

### Prove that for two vectors x,y over GF(q), the number of vectors that are closer to x is the same as the number of vectors that closer to y.

Let $x,y\in\mathbb F_q^n$ be vectors. We'll define: $X= \{ u\in\mathbb F_q^n \mid d(x,u)<d(y,u)\}$ $Y= \{ u\in\mathbb F_q^n \mid d(y,u)<d(x,u)\}$ Prove that $|X|=|Y|$. Well. ...
86 views

### polynomials over finite field with irreducible factors of odd degrees

It is well-known that the number of monic $n$-degree polynomials over a finite field of size $q$ is $q^n$. How many such degree-$n$ polynomials can be completely factored into only irreducible ...
127 views

### Definition of a primitive polynomial

I understand there are already some questions (A, B) on primitive polynomials. But none of these clears my confusion. In page 84 of Handbook of Applied Cryptography, primitive polynomial has been ...
141 views

### Construction of the projective plane over $\mathbb{F}_3$

I have a question about constructing projective plane over $\mathbb{F}_3$. We first establish seven equivalence classes $P= \{ [0,0,1], [0,1,0], [1,0,0], [0,1,1], [1,1,0], [1,0,1], [1,1,1] \}$. ...
157 views

### The linearity of the extended code

If we add a last digit to the code $C$ of length $n$, we obtain a new code called extended code. My question is: If the code $C$ is linear, can we prove that the extended code $C'$ is linear too? ...
Suppose we extend the $[n,k]$ linear code $C$ over the field $\Bbb F_q$ to the code $C'$, where $$C' = \{(x_1,\ldots ,x_n,x_{n+1})\in \Bbb F_q^{n+1} : (x_1,\ldots,x_n) \in C \text{ and } ... 0answers 184 views ### Error correction code handling deletions and insertions I have data which is expressed in form of fixed-length sequence of decimal digits, typically 10 digits in length. I'm looking for error correction code that would allow me to append one or more ... 1answer 111 views ### Field construction Explain how to construct a field of order 343 not using addition and multiplication tables. I understand that every finite field has order p^n for some prime p. Since 343 is 7^3, let ... 6answers 279 views ### How to construct minimal polynomial? This is an exam question from last semester. We have the finite field$$ \mathbb F_{81} = \mathbb Z_3 [x]/(x^4+x^2+x+1)$$(a) Prove that the polynomial$$ x^4+x^2+x+1  is irreducible (b) ...
A finite subring $R$ of a field $V$ contains $1$ (so $1$ is an element of $R$). The question is: True or False: The ring $R$ must be a field. I thought that if $R$ was a field it had to be a finite ...