Number theory is the study of the properties and construction of numbers, particularly integers. Prime numbers are of particular interest to number theorists and consequently cryptographers as they are considered the "building blocks" of numbers and produce many interesting results which are useful ...
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3answers
464 views
Would the ability to efficiently find Discrete Logs have any impact on the security of RSA?
This answer makes the claim that the Discrete Log problem and RSA are independent from a security perspective.
RSA labs makes a similar statement:
The discrete logarithm problem bears the same ...
21
votes
4answers
6k views
How can I generate large prime numbers for RSA?
What is the currently industry-standard algorithm used to generate large prime numbers to be used in RSA encryption?
I'm aware that I can find any number of articles on the Internet that explain how ...
9
votes
3answers
2k views
What is the relation between RSA & Fermat's little theorem?
I came across this while refreshing my cryptography brain cells.
From the RSA algorithm I understand that it somehow depends on the fact that, given a large number (A) it is computationally ...
11
votes
3answers
1k views
For Diffie-Hellman, must g be a generator?
Due to a number of recently asked questions about Diffie-Hellman, I was thinking this morning: must $g$ in Diffie-Hellman be a generator?
Recall the mathematics of Diffie-Hellman:
Given public ...
7
votes
3answers
536 views
Modern integer factorization software
What are the modern software packages that can be used to factoring large numbers into primes. By modern I mean developed and made public within the last 5 years. I'm interested in things that are ...
8
votes
1answer
188 views
Do recent announcements about solving the DLP in $GF(2^{6120})$ apply to schemes proposed for cryptographic use?
A recent paper by Göloğlu, Granger, McGuire, and Zumbrägel: Solving a 6120-bit DLP on a Desktop Computer seems to "demonstrate a practical DLP break in the finite field of $2^{6120}$ elements, using ...
13
votes
3answers
814 views
How robust is discrete logarithm in $GF(2^n)$?
"Normal" discrete logarithm based cryptosystems (DSA, Diffie-Hellman, ElGamal) work in the finite field of integers modulo a big prime p. However, there exist other finite fields out there, in ...
7
votes
2answers
502 views
Selecting a large random prime
Say I want a random 1024-bit prime $p$. The obviously-correct way to do this is select a random 1024-bit number and test its primality with the usual well-known tests.
But suppose instead that I do ...
4
votes
5answers
867 views
Galois fields in cryptography
I don't really understand Galois fields, but I've noticed they're used a lot in crypto. I tried to read into them, but quickly got lost in the mess of heiroglyphs and alien terms. I understand they're ...
8
votes
3answers
378 views
Is it reasonable to assure that p-1 and q-1 aren't smooth?
I came across the requirement that, in RSA, $p-1$ and $q-1$ shouldn't be smooth, shouldn't consist of lots of small factors. Therefore my question:
How complicated is it to check whether $p-1$ is ...
3
votes
1answer
180 views
A discrete-log-like problem, with matrices: given $A^k x$, find $k$
Let $p$ be a large prime; we will work in $GF(p)$. Let $A$ be a $n\times n$ matrix. Also, let $x$ be a $n$-vector and $k$ a positive integer.
Suppose we are given $p$, $A$, $x$, and $y$. The goal ...
3
votes
0answers
127 views
Finding where I am in a linear recurrence relation
Suppose I have a linear recurrence relation
$$a(n) = c_1 a(n-1) + \dots + c_k a(n-k) + d,$$
where the constants $c_1,\dots,c_k,d$ are given and the initial values $a(0),\dots,a(k-1)$ are given as ...
