All Questions
Tagged with cryptography finite-fields
13 questions
23
votes
1
answer
1k
views
Is hyperelliptic cryptography "practical"?
Previosly my impression on this subject was that hyperelliptic cryptography systems (as well as other possible cryptosystems based on abelian varieties of dimension $>1$) have no advantages over ...
- 16.9k
10
votes
1
answer
637
views
Discrete logarithm for polynomials
Let $p$ be a fixed small prime (I'm particularly interested in $p = 2$), and let $Q, R \in \mathbb{F}_p[X]$ be polynomials.
Consider the problem of determining the set of $n \in \mathbb{N}$ such that $...
- 12.4k
9
votes
4
answers
1k
views
The "interplay" between additive and multiplicative structure in a field
A field is an ordered triple $(F, +,\cdot)$ of a set $F$ and binary operations $+,\times$ on $F$ such that $(F,+)$ and $(F\backslash 0,\times)$ are abelian groups satisfying the distributive laws
$\...
- 2,075
7
votes
0
answers
199
views
Polynomial representation of modular arithmetic in finite fields
Let $n \in \mathbb{N}$ be a predefined integer. Consider the following bijection (between the ring of integers modulo $2^n$ and finite field with $2^n$ elements:
$$ \phi: \mathbb{Z}_{2^n} \to \mathbb{...
- 81
5
votes
1
answer
667
views
A silly question: is the number of points on a Jacobian (of a curve, over a finite field) known?
In a cryptography book I read that people does not known how to compute the number of points on a Jacobian of a hyperelliptic curve $C$ over a finite field $F_q$? Is this true? It seems easy to ...
- 16.9k
4
votes
1
answer
309
views
Discrete logarithms and primitive elements in finite fields
The recent papers:
R. Granger, T. Kleinjung, J. Zumbragel, "On the Discrete Logarithm
Problem in Finite Fields of Fixed Characteristic," Trans. Amer. Math.
Soc., 370(5) (2018), 3129–3145.
T....
- 503
3
votes
1
answer
266
views
p-adic logarithms with fixed precision
Probably this is easy, but we would like to see it on paper.
Let $p$ be prime and $D,g,n$ positive integers.
Let $A=g^n \bmod p^D$.
Let $\log(p,a,D)$ be the p-adic logarithm with precision $D$.
In ...
- 25.4k
3
votes
0
answers
215
views
Why we are interested in p>3 Schoof's algorithm
In the Schoof's algorithm we are particularly interested in $char(K)>3$, where $K$ is the field. I know Schoof's algorithm is mostly used over large prime fields. Also, when we are transforming ...
- 149
2
votes
0
answers
132
views
Solving efficiently a quadratic equation in a large finite field of characteristic two
I'm trying to solve efficiently a quadratic equation in the finite field $\text{GF}(2^{128})$ represented as $(\mathbb{Z}/2\mathbb{Z})[x] / (x^{128} + x^7 + x^2 + x + 1)$.
Until now, I came across ...
- 21
0
votes
1
answer
1k
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Elliptic curve over finite field: scalar multiplication
I'm implementing arithmetics for elliptic curves over secp256r1 as a homework assignment.
For scalar multiplication, the assignment specifically specifies that $k$ is "any hexidecimal encoded integer"...
- 3
0
votes
0
answers
123
views
Is it in theory possible to create a subexponential algorithm for solving discrete logarithms in multiplicative subgroups or within an Integer range?
As far I understand, when it comes to finite fields, Pollard rho and Pollard’s lambda are still the best algorithm for solving discrete logarithms in a multiplicative subgroup/suborder…
Index calculus ...
- 87
-1
votes
2
answers
164
views
Is there any way to solve this equation without knowing the inverse modulo? [closed]
Suppose I have an arbitrary 256 bit number $m$ another secret number $k$ of the same bit length, and then I multiply them both modulo a 256 bit prime number $p$ to get $c$ as follows:
$$
c = (m\cdot k)...
- 117
-1
votes
0
answers
74
views
Why is there in theory no morphism/isogenies when enlarging a prime field sharing a common suborder/subgroup? [closed]
Simple question : I have a prime field having modulus $p$ where $p−1$ contains $O$ as prime factor, and I have a larger prime field $q$ also having $O$ as its suborder/subgroup. Why are there no ...
- 87