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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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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Is it feasible to build an index of prime factors?

Would it be possible to break an RSA key, in for example 1 week of time, if the cracker have already spent X number of years building an index of primes by performing every permutation of existing ...
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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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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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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 ...
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RSA leak bits to factor N

Suppose you randomly generate large primes p and q as in RSA, and then tell me N=pq but not p or q. Then, you would like to actually let me factor N, except you should tell me as few bits of ...
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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 ...
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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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RSA and prime difference

It is known that the two prime factors $p$ and $q$ of an RSA modulus $n$ should not be too close to each other, otherwise an attacker may factor the modulus. In other words, $\Delta = \left| p - q ...
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Why are elliptic curve variants of RSA “chiefly of academic interest”?

Yesterday I was thinking about elliptic curve variants of popular protocols/algorithms (ECDH, ECES[1], etc) and the thought occured that I had never seen an elliptic curve variant of RSA. My ...
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Why has the RSA factoring challenge been withdrawn?

Wikipedia states that RSA challenge has been withdrawn. Does it mean that an efficient factoring algorithm is "just around the corner"? or are there some other reasons? If the challenge was still ...
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What is the progress on the MIT LCS35 Time Capsule Crypto-Puzzle?

Ron Rivest posed a puzzle in 1999. MIT LCS35 Time Capsule Crypto-Puzzle. The problem is to compute $2^{2^t} \pmod n$ for specified values of $t$ and $n$. Here $n$ is the product of two large ...
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What is the difference between Shor's algorithm for factoring and Shor's algorithm for logarithm

There is a paper from Peter W. Shor from 1994: http://www.csee.wvu.edu/~xinl/library/papers/comp/shor_focs1994.pdf "Algorithms for Quantum Computation: Discrete Logarithms and Factoring", and I have a ...
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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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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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In RSA, how to make sure that $p-1$ and $q-1$ are still hard to factorize?

See this question. The comment by Brett Hale stated: On the other hand, ensuring $(p - 1)$ has a large prime factor requires very little extra effort. What's actually the 'little extra effort'?
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Why would this method of discrete logarithm finding not work?

Say we do know $b$ but not $k$, and are given $g$ such that $g\equiv b^k\pmod p$. And say there exist factors $E = e + m'p$ ($e \equiv b^i \bmod p$) and $F = f + m''p$ ($f \equiv b^j \bmod p$) of $g$. ...
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Carmichael number factoring

Unsure whether this is the right forum for this question, worth a try. The task im faced with is to implement a poly-time algorithm that finds a nontrivial factor of a carmichael number. Many ...
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How can I take advantage of repeated patterns in non random RSA prime factors?

I am researching vulnerable RSA moduli which are composed of primes generated with poor entropy. Having a list of these primes I searched for variable sized repeated patterns among them and I noticed ...
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uniqueness of the RSA public modulus

What is the probability that two separate RSA public moduli are the same? For example, consider a 2048-bit modulus. The number seems to be huge, but the choice for prime factors p and q is much more ...
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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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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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Questions about William's p+1

First off, if you're doing William's p+1 test, then also doing Pollard's p-1 is redundant, since the p+1 test covers both cases, right? Second, why is the recurrence $V_{n+1} = aV_n - V_{n-1}$ used? ...
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An Elliptic curve cryptography implementation which can be terminated

I'd like to have an implementation of elliptic curve cryptography along the lines of secp256k1 which is secure until some information is published after which it is broken. One idea would be to use ...
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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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RSA security assumptions - does breaking the DLP also break RSA? [duplicate]

Possible Duplicate: Would the ability to efficiently find Discrete Logs have any impact on the security of RSA? I'm wondering if breaking the DLP, that is the basis for ElGamal and DSA, ...
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Is the new preprint “An Algorithm For Factoring Integers” by Yingpu Deng and Yanbin Pan worth reading?

I just discovered on the eprint server of the IACR the paper mentioned in the title. Scanning quickly over the paper I didn't find anything spectacular, so I doubt that their new(?) approach will be ...
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Is it possible that two distinct RSA moduli share both of their prime factors?

I am using an algorithm (can be found here1) that can compute efficiently the GCD of multiple RSA keys. It intended for RSA keys that were generated with low entropy and may have one of their primes ...
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Efficiency of finding sub group order vs factorization

Suppose you got a prime $p = 2\mathbb\Pi_{i=0}^{n-1}q_i+1$, where $2^{k-1} \lt q_i \lt 2^k$ for some $k$ and all $0 \le i \lt n$, and that you also got a generator $g$ of one of the prime order sub ...
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Quadratic residuosity problem reduction to integer factorization

How can one show how to reduce the quadratic residuosity problem to an integer factorization?
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Impact of algorithms for factoring using elliptic curves over $\mathbb{Q}$

Recently a few papers have appeared that describe a new approach to factoring, using elliptic curves over $\mathbb{Q}$. See, e.g., Factoring integers and computing elliptic curve rational points, ...
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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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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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How did they factor RSA-704?

I don't understand the 'Wiedemann algorithm' works. Can someone explain the factoring of RSA-704 in an easy way?
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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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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's the strategy for future directions in cryptography? Bigger numbers/faster searching, or new methods, say, of factoring?

I'm taking a course in cryptography, and I would value any comments. This is not too technical a question, but more about directions or strategy in cryptography. My question is, is public key ...
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Choosing good parameter for Lenstra's elliptic curve factorization

In Wikipedia, there is an article explaining Lenstra's factorization algorithm. As far as I got it, we choose some $e \in \mathbb{N}$ and a point $P$ on the curve and then calculate $eP$. While ...
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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

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 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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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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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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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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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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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 ...