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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Factorization of the semi-palprime $N = pq$ [migrated]

I define semi-palprime be a prime number that remains the prime when its digits are reversed, like $p = 13$, and its mate is $q = 31$. I know that number $N$, $ N ...
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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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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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287 views

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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263 views

Factoring large numbers

I am trying to factor few integers that are each between 115 and 135 digits long. I was wondering if anyone knew of any efficient methods or any programs that I could use to find the two primes $p$ ...
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130 views

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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114 views

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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219 views

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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How to choose the integer m in the general number field sieve (GNFS)?

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 $m$ ...
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67 views

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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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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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, ...
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133 views

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 ...
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How big an RSA key is considered secure today?

I think 1024 bit RSA keys were considered secure ~5 years ago, but I assume that's not true anymore. Can 2048 or 4096 keys still be relied upon, or have we gained too much computing power in the ...
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139 views

Programming language for modular arithmetic over large numbers [closed]

I'am trying to implement algorithms on integer factorization.This involves dealing with integers of 200-500 digits and doing modular arithmetic over them.Which programming language has inbuilt support ...
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184 views

Factors of RSA modulus

In the article A Method for Obtaining Digital Signatures and Public-Key Cryptosystems, the original RSA article, it is mentioned that Miller has shown that n (the modulus) can be factored using any ...
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How to find the factors of the modulus? [duplicate]

In the article "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems", the original RSA article, it is mentioned that Miller has shown that n (the modulus) can be factored using any ...
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162 views

Discrete logarithm modulo a smooth number

I am solving the discrete logarithm problem modulo $N$. $N$ is a composite number, I found its factors — lots of small primes and two big primes ($> 2^{50}$). Does the factorization of $N$ somehow ...
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Relations between RSA and DLOG, factoring and DLOG

Definition: (The generalized Diffie-Hellman problem) Let $n=pq$ for two large primes $p,q$. Given $x, x^a, x^b,n$, find $x^{ab}\pmod{n}$. (1) Is there a known reduction from the GDH problem ...
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OTP from Sony BIOS password recover [closed]

From Dogbert's blog: Sony has a line of laptops ("Vaio") which compete mainly in the high value market segments. They implemented a master password bypass which is rather sane in comparison to the ...
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What is the value of Q such that Q|P-1 where P is a prime number?

For my crypto assignment, I'm asked to enter a prime P and generate Q such that Q|P-1 Can anyone guide me what is Q|P-1?
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253 views

Square roots, prime factorization

I´m stuck in my homework and since its homework, I would rather get some hints than full solution. The problem goes: factorize n = 88416763 in case that you know that the square roots of 51733469 ...
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216 views

Compare two approaches for cracking RSA key

I came across these questions while studying for a crypto course, does anyone have any ideas on how to answer these? (a) Random prime numbers of size 1536 bits are chosen to generate an RSA modulus ...
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Hard problems in composite order group even when factorization is known

Composite discrete log problem has been proved to be reducible to hardness of factorization and discrete log on the prime factor groups. Are there any problems apart from that in composite order ...
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604 views

Attacks on the RSA Cryptosystem

I was reading some articles about attacks on RSA system and I wonder about some generalization of the following theorem. Theorem (Coppersmith). Let $N=pq$ be an $n$-bit RSA modulus, where ...
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Adi Shamir's secret database of all primes

I was going through these presentation slides (PDF) on Crypto 2013. It summarizes the paper, Factoring RSA keys from certified smart cards: Coppersmith in the wild. In the last slide, it was ...
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133 views

Brute force RSA cracking

Suppose one had a complete list of primes up to $2^{n+1}-1$. Then wouldn't one be able to crack an $n$-bit RSA public key in time $O(\pi(2^{n+1}-1))$, making RSA insecure? Thanks, René
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Security strength of RSA in relation with the modulus size

NIST SP 800-57 §5.6.1 p.62–64 specifies a correspondence between RSA modulus size $n$ and expected security strength $s$ in bits: ...
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Computational Diffie-Hellman problem over the group of quadratic residues

Suppose that $N=pq$ where $p$ and $q$ are safe primes. $\mathbb{QR}_N$ is the group of quadratic residues which is a cyclic group with order $\frac{\phi(N)}{4}$. Let $g$ be the generator of ...
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Is knowing the private key of RSA equivalent to the factorization of $N$?

Given the RSA modulus $N$ the fastest method to factor it is of sub-exponent order. But, now if I know the private key $d$ of RSA, does that mean I can factor $N$ efficiently?. It intuitively seems ...
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From Factorisation of semiprimes to breaching confidentiality

If someone or some group found an efficient way to factor large composites with two distinct prime divisors, would this make it easier to decode any messages?
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Which algorithms are used to factorize large integers?

Even if RSA decided to cancel the Factoring Challenge, it seems that some teams keep working on it. According to Wikipedia, RSA-768 has been factored in late 2009. What are the current large integer ...
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RSA: Letting $p$ and $q$ have different bit-size

I am aware that there are concerns if $p$ and $q$ are close i.e. $\Delta=|p-q|$ can't be too small. But I would like to know if there are any known attacks for cases where $p$ and $q$ take on ...
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Quadratic Sieve: what's the next step after it fails?

Factoring some 20-45 digit values n with a (simple) quadratic sieve, the quadratic sieve may end up with pairs of x and y s.t. $x^2 \equiv y^2 \pmod n$, but neither x+y nor x-y has a nontrivial gcd ...
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How can I create an RSA modulus for which no one knows the factors?

It's easy to create an RSA modulus where almost no one knows the factors: for example, I can generate two 1024-bit primes $p$ and $q$ and set $n=pq$. If I publish $n$, I will be the only person in ...
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Determine the iteration times using Pollard's rho Method for factoring

Let's say, we have a large number $n=181937053$ and $f(x)=x^2+1$. And also we know that $n=12391 \times 14683$. The problem is that ,using Pollard rho method, can we find the algorithm iteration ...
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How to calculate the time it'll take to crack RSA or DH?

Sometimes the easiest way to describe security of a type of cryptography is to say that "the time it takes to solve for an x-bit key would be y years". How would one go about doing such a calculation ...
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Why can ECC key sizes be smaller than RSA keys for similar security?

I understand how ECC is based on the discrete log problem and RSA on integer factorization. I've read several references that show how a solution to either of these problems can typically be adapted ...
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What does “Inverting the RSA function is as hard as factoring” mean (a rigorous explanation or intuitive will do)?

I was reading that a current open problem is if inverting the RSA function is as hard as factoring. Does this mean that, its an open problem whether, if given a subroutine that computes in ...
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Distributed integer factorization?

I'm looking around for publicly published work on factorization of large numbers using distributed systems of any kind. So far I've come across the PDF "Mapreduce for integer factorization" by Javier ...
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Is it possible to attack RSA with a WalkSat derivative?

We consider a large $n$-bit number $N$. We want to find a factor, if it admits any. For $m$ taking values from $1$ to $n$, perform the following three steps (actually, for each $m$, perform many ...
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Elliptic Curve Factorization: Why are elliptic curves suited for this kind of task?

Currently I'm working on a presentation of a paper that talks about the factorization of large numbers. In the paper elliptic curves are presented as a way to factorize large numbers. After hearing a ...
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What is the difference between exhaustive search and factorization in relationship to determining a key?

It seems to me an exhaustive search would simply try to use all the possible bit combinations of a key, while factorization is some mathematical formula for determining the key? When discussing the ...
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Are there asymmetric cryptographic algorithms that are not based on integer factorization and discrete logarithm?

In the computer security class (in which cryptography is a big chapter) that I took, I remembered the professor said about current asymmetric cryptography algorithms are based on integer factorization ...