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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Optimising Pollard's Rho algorithm for large semi-primes

I have programmed an implementation of Pollard's Rho factoring algorithm using C++ and the GMP library. It is reasonably fast with large numbers, however I haven't implemented any form of cycle ...
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Discrete log analog of ECM factoring algorithm?

Anecdotally, most factoring algorithms have a corresponding variant algorithm that can be used to attack the discrete log problem using similar ideas. Is there an analog of the elliptic curve (ECM) ...
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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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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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What role plays Quantum Fourier Transformation in Shor's integer factorization algorithm?

I cannot seem to understand the role or goal of Quantum Fourier Transformation in Shor's integer factorization algorithm. Is it used to collapse all quantum states into one, in which it has a factor ...
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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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Is there an algorithm for factoring N, which is just as simple as this one, but faster?

I found a simple algorithm for factoring semiprime numbers, you can read about it in Factoring Semiprimes and Possible Implications for RSA. It basically works like this: You reverse the digits in ...
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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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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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45 views

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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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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218 views

Relationship between Elliptic Curve Discrete Log, Integer Discrete Log, and Integer Factorization

I am trying to look into a relation between the following three problems which are widely used to build public crypto systems: Integer Discrete log Elliptic Curve Discrete log Integer Factorization ...
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Does it necessarily mean that an RSA moduli generated with poor randomness is not random?

In 2012 a group of researchers collected a large amount of RSA moduli and calculated their greatest common divisor in order to find common factors between them. By finding a common factor they could ...
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Reduction of Integer factorization to Discrete logarithm problem

I was reading Eric Bach paper entitles "Discrete logarithms and factoring", in which he states the following reductions: solving the integer factorization problem suffices to solve the discrete ...
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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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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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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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Discrete log problem with modulus prime

I am a bit confused on the hardness of the discrete logarithm problem. Does it become intractale only when it is mod n, where n is a large composite number (Like RSA key). What about if it is mod a ...
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130 views

Security relevance of random factor in Paillier

In the Paillier cryptosystem [1] the encryption of $m \in \mathbb{Z}_N$ with randomness $r \in \mathbb{Z}_n^*$ is $c = g^m r^n \bmod{n^2}$. The additive-homomorphic property of the system shows that ...
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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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86 views

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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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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Why is it impractical to generate a semiprime dictionary? [duplicate]

This might be a very simple question. However, I am just learning the concept, so just excuse me. I am wondering why there is not any attempt to generate all semiprime numbers? (as an dict. attack to ...
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Generating Polynomials for the MPQS

I'm going to try and eventually factor RSA-100, but my current QS needs a lot of improvement, so I'm going to try and switch over to the MPQS. I'm a bit confused as to how the MPQS works, which is ...
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Quadratic Sieve Bottleneck, Multiple Polynomials an option?

After my failed attempt at trying to implement the ECM, I started working on the quadratic sieve. It works, but the bottleneck is finding smooth values over the factor base. The way I implemented it ...
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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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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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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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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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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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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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39 views

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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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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Integer factorization via geometric mean problem

presume: we have A and B integers, and C - product of multiplication of A and B, A and B are prime numbers (strong or usual) G - is geometric mean of A and B (square root of C) also, we have: if B > ...
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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 $p$ ...