# 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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### 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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### 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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### 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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### 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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### Coppersmith method for factorisation

Is anyone familiar with the Coppersmith method? Does anyone know how is the $3\times3$ basis matrix obtained in this case?
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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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### 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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### 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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### 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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### Factoring large numbers

I am trying to factor few integers that are each between 115 and 135 digits long. I have just, little over a month ago, began my study of Cryptography. I was wondering if anyone knew of any efficient ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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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### 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 ...
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 ...