All Questions
Tagged with cryptography finite-fields
109
questions
1
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1
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32
views
AES S-box as simple algebraic transformation
The next matrix represents the Rijndael S-box according to wikipedia and other sources
$$\begin{bmatrix}s_0\\s_1\\s_2\\s_3\\s_4\\s_5\\s_6\\s_7\end{bmatrix} =
\begin{bmatrix}
1 & 0 & 0 &...
- 473
1
vote
0
answers
45
views
Conway Polynomial for p=2, n=3?
Im doing an exercise on Conway polynomials. As far as im concerned, for p=2, n=3 both
$f(x)=x^3 + x^2 + 1$
and
$g(x)=x^3 + x + 1$
satisfy every condition. According to every source i found, the latter ...
- 11
0
votes
0
answers
120
views
Unexpected Result from Finite Field Calculations in GF(2^8)
I'm performing calculations within the finite field $GF(2^8)$ and I can't seem to get the expected results. This is my first time working with finite fields, so my understanding is quite basic. I ...
1
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1
answer
30
views
Distinct derivations of polynomial over finite field
I am a student studying algebra and cryptography.
I wonder below question is possible.
Can I make some polynomials $f(x)$ over finite field that all derivations $f^{(k)}(x)$ are distinct when x is ...
- 13
0
votes
1
answer
229
views
Binary multiplication in Galois Field GF(2^8)
I am working on a project (high school), and I need to explain the process of AES MixColumns for one of the parts.
I am trying to show an example of the matrix multiplication in MixColumns that uses ...
- 3
0
votes
1
answer
80
views
Clarification on Multiplication in $GF(2^3)$ vs. Boolean Algebra
While experimenting with finite fields, specifically $GF(2^3)$, I stumbled upon a puzzling situation when comparing multiplication operations to those in Boolean algebra.
Let's take two elements $A$ ...
- 15
1
vote
1
answer
88
views
Show that $f(x)=x^2+2x-1 \in \mathbb{Z}_3[x]$ is irreducible over $\mathbb{Z}_3$. And find the elements of a finite field with 9 elements.
Show that $f(x)=x^2+2x-1 \in \mathbb{Z}_3[x]$ is irreducible over $\mathbb{Z}_3$. Using this fact construct a finite field $\mathbb{F}_9$ of $9$ elements. If $\alpha$ is a root of $f(x)$, then find ...
- 761
4
votes
2
answers
323
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Endomorphisms of a supersingular elliptic curve defined over the prime field
Let $E$ be a supersingular elliptic curve defined over a prime field $K=\Bbb{F}_p$. It is well known (see for example chapter V of J. Silverman, The Arithmetic of Elliptic Curves, my copy is the 1986 ...
- 135k
0
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0
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23
views
$\frac{g(x)}{f^*(x)}=\sum_{n\geq 0}s_nx^n$ generates recursion
I need help proving that in $\mathbb{F}_q$, if $g$ is a polynomial of degree less than $k$ and $f^*(x)=1-(a_{k-1}x+\dots+a_0x^k)$ a polynomial of degree $k$, then $$\frac{g(x)}{f^*(x)}=\sum_{n\geq 0}...
- 67
0
votes
0
answers
24
views
Given a support set calculating Walsh transform.
I have support set of length 120 of a bent function over $GF(256)$.
$GF(256)=<\beta>$ is generated by the polynomial $x^8+ x^4 + x^3 + x^2 + 1$. The support set is of the form
{$\alpha 0 1 0$, $\...
- 77
2
votes
1
answer
214
views
Understanding dimension of Goppa code
My concern is regarding the understanding of the dimension of Goppa Codes and the corresponding dimension of its parity check matrix. The classical definition often referred to as classical view of ...
- 35
1
vote
1
answer
117
views
Point on elliptic curve that is torsion over algebraic closure
Say I have an elliptic curve $E: y^2 = x^3+4$ over $\mathbb{F}_{7}$. I want to find an $7$-torsion point in $\overline{\mathbb{F}}_7$ which is not in $\mathbb{F}_7$. How do I do that?
The $n$-torsion ...
- 640
0
votes
0
answers
145
views
Evaluation of a polynomial over the finite field $\mathrm{GF}\left(2^{8}\right).$
I am trying to make a program that, among other things, considers a polynomial $p$ whose coefficients are elements of $\mathrm{GF}\left(2^{8}\right)$ and shows the user the graph of that polynomial. ...
- 152
1
vote
1
answer
109
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Construction of a recurrence sequence with given period
I want to construct a binary recurrence sequence which has period 1023. Moreover, it shouldn't have pre-period.
Can anyone help me with the procedure? I truly have no ideas where to start.
Also, I don'...
- 175
3
votes
1
answer
47
views
Can polynomial in Z[x]/(x^n-1) have non-integer coefficients?
I am trying to compute an inverse of some polynomial $f$ in $\mathbb{Z}[x]/(x^5-1)$. Is it possible that $f^{-1}$ has coefficients that are non-integer like 0.33?
Thanks in advance.
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