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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Why RSA encryption key is based on modulo $\varphi(n)$ rather than modulo $n$

While calculating RSA encryption key we take modulo $\varphi(n)$ rather that modulo $n$. I couldn't understand why its so?
9
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5answers
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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 ...
7
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3answers
1k 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 ...
8
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3answers
363 views

Building a hard to factor number without knowing its factorization

It is possible to find an efficient algorithm for constructing a provably hard to factor number $N$, together with a witness that shows that it is indeed hard to factor. EDIT, since it was not clear: ...
28
votes
4answers
18k 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 ...
10
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3answers
4k 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 ...
10
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1answer
6k views

Calculating RSA private exponent when given public exponent and the modulus factors using extended euclid

When given $p = 5, q = 11, N = 55$ and $e = 17$, I'm trying to compute the RSA private key $d$. I can calculate $\varphi(N) = 40$, but my lecturer then says to use the extended Euclidean algorithm to ...
16
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3answers
2k 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 ...
10
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2answers
847 views

Can I select a large random prime using this procedure?

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
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1answer
758 views

AES mixcolumn stage

I'm studying AES, and am having problems with the "mixcolumn" stage. I read about finite fields, but I still don't know. How do I construct $GF(2^8)$? ...
19
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4answers
12k views

Basic explanation of Elliptic Curve Cryptography?

I have been studying Elliptic Curve Cryptography as part of a course based on the book Cryptography and Network Security. The text for provides an excellent theoretical definition of the algorithm but ...
15
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3answers
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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 ...
12
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1answer
457 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 ...
8
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3answers
1k 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 ...
4
votes
4answers
416 views

Where does the $\varphi(n)$ part of RSA come from?

$e d \equiv 1 \pmod{\varphi(n)}$ Where does the $\varphi(n)$ part come from? How did the inventors of RSA arrive at $\varphi(n)$?
7
votes
3answers
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Understanding CRC

There are zillions of articles describing CRC. What can I read to (more deeply) understand what's really going on? Both from an algebraic perspective and a bit-manipulation perspective, I'd like to ...
7
votes
2answers
415 views

Given a message and signature, find a public key that makes the signature valid

Given a message $M$ and a signature $S$, is it feasible to find a RSA public key $(n,e)$ such that $S$ verifies as a valid signature on $M$ (using this public key)? What if we're given one public key ...
6
votes
1answer
228 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 ...
4
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1answer
279 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 ...
2
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1answer
1k views

How can I use eulers totient and the chinese remainder theorem for modular exponentiation?

I'm trying to implement modular exponentiation in Java using Lagrange and the Chinese remainder theorem. The example we've been given is: Let $N = 55 = 5 · 11$ and suppose we want to compute ...
1
vote
1answer
323 views

What does $(\mathbb{Z}_n^*)^2$ mean?

In a paper they write once, $(\mathbb{Z}_n^*)^2$. Is this the group of quadratic residues or is it something else? Here the theorem: Under the strong RSA assumption, given a modulus $n$, along with ...
9
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3answers
461 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 ...
5
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5answers
478 views

Is there a technique to confirm that a given large integer value is a product of two primes?

Given a list of 2048 bit integer values in which one or few 2048 bit integer values may be product of two prime numbers and other values may be just 2048 bit odd integers numbers. My question is - ...
2
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1answer
269 views

Is it possible to determine the group order by knowing the “public” and “private” key exponents in an RSA group?

I have an RSA group with modulus $n = p \cdot q$, two safe primes $p=2p'+1$ and $q=2q'+1$ and the "public" and "private" key exponents $d$ and $e$. $\phi(n) = 4p'q'$ is the order of the RSA group. If ...
1
vote
1answer
136 views

Equal length of primes in paillier cryptosystem

In the key generation step of paillier cryptosystem , In order to satisfy $\gcd(pq,(p-1)(q-1))=1$ , we can take equal length primes. Instead of taking(length as parameter to generate $p,q$) equal ...
1
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1answer
69 views

Non-commutitive and nonassociative algebraic structures in cryptography

Are there any cryptographic algorithms or primitives that have been developed and studied that make use of non-commutative or non-associative algebraic structures such as quaternion integers or ...