# Tagged Questions

Factorization is the problem of taking an integer and finding the set of prime numbers that produce the integer when multiplied together. For large integers with large factors, factorization is hard and is the basis for cryptosystems like RSA.

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### RSA: revealing the modulus factorization by choosing a bad message

I started reading the book Cryptanalysis of RSA and its variants by M. Jason Hinek and I stumbled upon a phrase that intrigued me: plaintext messages that are relatively prime to the modulus (i....
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### Factors of the group order to secure against Pohlig-Hellman

I am looking into the security of Diffie-Hellman and the discrete log in general. To make sure an attacker can not use Pohlig-Hellman to solve the discrete log quickly we need to make sure that the ...
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### Algorithm for factoring a number $n$ of a specific form given $n$ and $\varphi(n)$

Given the natural number $n$, which is in the form $p^2 \cdot q^2$ with $p$,$q$ prime numbers. Also $\varphi(n)$ is given. Describe a fast algorithm (polynomial time) that calculates $p$ and $q$. ...
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### Blinding to mask private key operations

Blinding is often used to mask private key operations when the underlying problem is integer factorization. For example, its used in both RSA and Rabin-Williams signature schemes. This presumes ...
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### Why is Rabin encryption equivalent to factoring?

I don't understand the proof of equivalence I've read everywhere (e.g., in Rabin's paper or on Wikipedia). Here's my objection: let's say you have a Rabin decryption oracle that takes ...
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### Why should the primes used in RSA be distinct?

The two primes $p$ and $q$ part of the public key need to be distinct. What's the reason for them to be distinct? Is it because factorization of $p^2$ where $p$ is a prime is relatively easier, or is ...
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### Are there UFDs where the factorization problem is difficult but finding irreducibles is cheap?

Factorization of integers is hard, but finding irreducibles is expensive. Is there a ring where factorization is assumed hard but finding irreducibles is much cheaper than over $\Bbb Z$? It could ...
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### Logjam-style attack on Factoring?

We're all aware of the Logjam attack, which is known as "FREAK on discrete logarithms". The attack works by doing a large pre-computation step, which needs only to be done once per field and then ...
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### Attack for RSA 1024 bit with Low Public Exponent

I am facing a challenge at university. Our teacher give us the challenge to try to break an RSA 1024 bit. We have public modulus N and public exponent e (0x03), we don't know the padding. We have a ...
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### Can spatial filters be used to factor composite numbers?

$Z=(N-XY)^2$ is a surface with absolute minima ($0s$) anywere $Y=N/X$. I know this question is naiive, but shouldn't it be possible to apply a lossy compression filter to this function which ...
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### If someone had a list of all primes, would it be possible for them to factor any integer in polynomial time? [duplicate]

For example, if they somehow got a function that would churn out any arbitrary amount of primes in a row. Could they break the RSA problem then?
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### Prime factorization

What is the largest integer that can be factored by modern algorithm like Msieve and GGNFS in a time less than 5 hours with normal computers? For example, can an integer like ...
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### Factoring two RSA moduli $N_i=p_i\cdot q_i$ knowing that $p_2=p_1+2$?

It is given two RSA moduli $N_1$ and $N_2$, known to be of the form $N_i=p_i\cdot q_i$, with $p_i$ and $q_i$ unknown primes, and such that $p_2=p_1+2$. Can we make use of that relation to factor the ...
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### Zero knowledge of two factor

Here I overconfident in myself state that I can show, that n has two factors. This is not completely true, can possibly show $n$ is composite - prover generates RSA key with modulo $n$, and gives $e$ ...
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### Are factorization algorithms parallelizable?

I was reading about the Blum-Blum-Shub random number generator, and its security depends on the hardness of factoring very large numbers (like many things in crypto do). I'm just wondering, if I have ...
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### Is the half-homomorphic property of RSA a problem for blind RSA signatures?

For blind RSA signatures, is it problematic that RSA is half-homomorph? Take a scenario where blind RSA signatures are used for something like a voting procedure or this proposal: Lots of people, ...
Given an integer you want to factor $N$, GNFS starts by selecting a monic irreducible polynomial $f \in \mathbb{Z}[X]$ and an integer $m$ such that $f(m) \equiv 0 \text{ mod } N$. In practice, if $m$ ...
### Why is factoring $p-1$ easy when $p$ is a safe prime?
A paper states: [...] $(p,g,y)$ is a correct ElGamal public key if $g^x=y\pmod p$. To verify this the order of $g$, and thus the factorization of $p-1$, is needed. This is easy for safe primes (i....