# Questions tagged [factoring]

The decomposition of an integer number to the product of other integers. Algorithms such as RSA are based on the premise that no practical way has been found was to factorize large integers when they have been produced by multiplying two large primes.

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### Compute discrete logarithm mod $n=p \times q$ knowing factorisation

I read in a document that for a given $n = p\times q$ ($p$, $q$ primes), if you know $p$ and $q$ then you can easily solve the discrete logarithm problem, i.e. for fixed $a,b$, you can find $x$ such ...
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Let $N=pq$ where $p$ and $q$ are safe primes. If the adversary knows the inverse of $p$ mod $q$ and the inverse of $q$ mod $p$, can this help him factor $N$ or break the textbook RSA?
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### Lenstra's ECM Algorithm - field requirement

In Lenstra's ECM algorithm, $\#E(\mathbb{F}_{p})$ is required to have small prime factors. Why is this so? I understand that the p-1 method is efficient for factoring N with small factors. The ECM ...
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### Bounds on failure probability for universal exponent method?

The following definition is from Trappe and Washington, "Introduction to Cryptography with Coding Theory". Given a number $n$ and an integer $r > 0$ such that $a^r \equiv 1 \pmod{n}$ for all ...
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### What is the fastest integer factorization to break RSA?

I read on Wikipedia, the fastest Algorithm for breaking RSA is GNFS. And in one IEEE paper (MVFactor: A method to decrease processing time for factorization algorithm), I read the fastest algorithms ...
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### Is it possible to check if a number is the product of two primes without factorizing it?

I have a large number which I suspect may be a private RSA key (although its size, at 613 bits, seems a bit unorthodox). I have started to run a factorization algorithm on it, and after a few hours ...
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### Time-memory tradeoffs in Shor's algorithm

Can a quantum computer with insufficient qubits to factor an integer of a given size make any progress in factoring it? For example, what if a quantum computer is only one qubit short of what is ...
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### Fermat's factorization method on weak RSA modulus

Given a public key for RSA, I have extracted the modulus which looks like this: ...
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### Factoring RSA weak modulus

Given a public key for RSA, I have extracted the modulus which looks like this : ...
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### Equivalence of cryptographic problems

Are integer factorization, discrete log and ECDH problems equivalent? I know that factorization and discrete log are equivalent but are one of those two problem equivalent with ECDH? Cand someone ...
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### In RSA, what is $P[x \notin \mathbb{Z}_N^*]$

In the RSA problem, picking a message $x \in \mathbb{Z}_N \setminus \mathbb{Z}_N^*$ implies factorizing $N$. Since factorization with respect to the standard RSA generator is hard assuming the RSA ...
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### RSA: security of LSB in The Generic Model of Computation

In this paper Maurer and Aggarwal showed that in generic model of computation breaking RSA is equivalent to factoring. It is also known that the LSB of an encrypted message is as hard as breaking RSA.(...
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### Does choosing N to be product of safe primes avoid William's p+1 factoring attack on RSA?

In this post, I found that choosing RSA modulus $N$ to be product of safe primes avoids Willam's $p + 1$ factoring attack. Suppose $N = p \cdot q$, where $p$, $q$, $(p-1)/2$ and $(q-1)/2$ are primes. ...
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### Can we find the exact number given remainder of the numbers with mod m?

I have around 1500 numbers. The numbers $x_i$ are calculated as $x_i$=($p*t_i$) mod m. $p$ constant and same for all the numbers while $t_i$ are chosen randomly everytime. For example the given ...
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### Upper bounds on difference of RSA primes

I was wondering whether given a concrete $N = p \cdot q$ whether we can find a upper bound on $\Delta = | p - q|$ as function of $N$ e.g, $N^\delta$, and thus test whether a given $N$ is vulnerable ...
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### What happens for factoring algorithms if P=NP?

If someone ever demonstrates that P=NP, will it give us a polynomial factoring algorithm, or will it only tell us that such an algorithm exists, but we still have to find it?
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### RSA finding p and q integer with condition

I'm given $N=p\,q$ and told that $44\,p\approx 17\,q$ (with the value given for $N$ some 49-digit integer 8124642558124642555899928124642555899924479992447). In ...
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### Reduction from integer factoring to computational Diffie Hellman

The computational Diffie Hellman (CDH) problem for ${\mathbb{Z}}^*_p$ is given a prime $p$, a generator $g$ of ${\mathbb{Z}}^*_p$, and a pair $(g^i, g^j)$ to compute $g^{ij}$. The value $g$ is called ...
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### Efficient way of knowing large factors of $\phi(n)$ given small prime factors and $n$

Knowing large prime factor$(r > n^{1/4})$ of $\phi(n)$ can easily factorize n and hence learn $\phi(n)$. If we have knowledge on all small prime factors $(2< r_i << n^{1/4})$ of $\phi(n)$...
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### Algorithm to factorize $N$ given $N$, $e$, $d$

I have an RSA public key (public modulus $N$ and public exponent $e$), and the private exponent $d$ of matching private key. How can I compute $p$ and $q$, the primes factor of $N$ ?
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### Counting the number of binary solutions of quadratic system

I have a quadratic system of equations related to a balanced RSA modulus $n=pq$ (i.e. $\log p\approx\log q$), and I want to give an upper bound on the number of solutions. Indeed, let $p_i,q_i$ be ...
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### RSALib prime generation - derive number of primes

I'm working on factorizing a ~450 bit key that I know has been generated with RSALib and thus is vulnerable to ROCA. Now reading the original paper, I can see that the primes are generated in the ...
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### Why would an efficient integer factorization algorithm render RSA insecure?

I know that RSA relies on the integer factorization problem: given two primes p and q, their product p . q is easy to compute. But not feasible (i.e., polynomial-time) an algorithm is known that could ...
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### Can multiplication of two primes be seen as a strong cipher?

If we were define such a cipher: A reversible function that would accept a message $M$ and an initialization vector $\text{IV}_1$ $\operatorname{map}(\text{IV}_1, M)$ which can map an input $M$ to a ...
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### What are SNFS-safe limits for an RSA moduli optimized for simple modular reduction?

I consider $n$-bit RSA moduli $N$ having high-order bits starting by with $k$ bits at 1, then $k$ bits at 0, then $m-2k$ bits at ...
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### How to find p,q in this problem?

Suppose \begin{align*} g^r &\equiv h \pmod N, \\ h^s &\equiv g \pmod N, \end{align*} for known $g$, $h$, $r$, $s$, and $N$, but not $\phi(N)$. Then $$g^{r\cdot s - 1} \equiv 1 \pmod N,$$...
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### Is the matrix step of GNFS still the hardest part?

When the factorization of RSA-768 was announced in December 2009: the sieving took about 24 months and the matrix step took 119 days (4 months). So sieving took about 6 times as long. This is despite ...
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### Number of bits specified in standards implementation?

Currently deployed RSA and discrete logarithm implementation uses $1024$ to $2048$ bits. Hypothetically speaking if a crypto team produces a faster algorithm that moves current factoring and discrete ...
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### RSA: Is possible get p and q from this d and n?

I have this algorithm and Im searching p and q: $n=p^2 * q$ $l=(p-1)*(q-1) / \gcd (p-1,q-1)$ $d\equiv l^{-1} \pmod n$ And the values are: For $n$: ...
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### Recovering 3 private keys if Eve knows that the keys are shared prime numbers and knows their public keys, How would this be done?

okay so here is the original question: Alice Bob and Carl are generating public keys for RSA, but they are lazy and decide to share some of the work of generating prime numbers. They find 3 large ...
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### RSA factorization knowing N, e=65537 and g=d*(p-17)

Having known the values for $N$ a large number, $e = 65537$ and another large number $g = d \cdot (p-17)$, how can I use that info to find out $p$ and $q$? I guess that have something to do with ...
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### Can we efficiently factor if we are given a Pocklington certificate of one of the prime factors?

I recently read Squeamish Ossifrage's answer on generating RSA keys from (short) randomness where they make the following comment: (You might want to keep the certificate secret too.) As the ...
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### Why does TWINKLE use light instead of current?

TWINKLE is a device devised by Adi Shamir to optimize the sieving step of GNFS. It consists of a cylinder, at the bottom of which are LEDs corresponding to factor base primes which blink with ...
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### Factoring RSA-129 with a personal computer today

I have read the history of the RSA-129 challenge, and now I would like to know if it would be possible to factor RSA-129 with a single "average" personal computer, today. Has someone tried to do this ...
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### Finding the first few digits of p and q

Is there a way to find out the first few digits of the factors of the RSA numbers (RSA-1024 or RSA-2048)? I do not want to get all the digits but only first 4-5 digits. My question is thus more ...
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### Malicious DH parameters without using composite numbers [duplicate]

I know that it's possible to generate DH parameters that lead to it being easy to attack (e.g. trivial composite numbers), but is it possible to create a malicious parameter that is not a composite ...
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### Factoring a 512-bit number?

I want to know how to factor this number only given $n$ and $e$, I have tried to factorize $n$ using Fermat's little theorem and also tried primefac module in python (running for the past 4 days) but ...
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### Integer Factorisation

If I have a set of numbers of the form $\{ {kp+r}:k\geq0\}$ with p a prime or product of primes k large in $\in Z^+$ and r fixed, is it computationally feasible to find a factorisation for any one ...
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### Fewest qubits required for the discrete logarithm problem and integer factorization

According to a paper from 2002, the most efficient circuit to factor an $n$-bit integer requires $2n+3$ qubits and $O(n^{3}\lg(n))$ elementary quantum gates, assuming ideal qubits. Later on, according ...
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### Factorisation for Coprimes of Large Numbers - RSA

What is the current conventional algorithm used to calculate factors of large numbers in order to determine if they are coprimes, or if there is a way to do it without calculating factors, what would ...
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### Factor RSA modulus given many valid encryption decryption exponent pairs?

I have read Is sharing the modulus for multiple RSA key pairs secure?, which explains an algorithm for factoring an RSA modulus n given only one encryption and decryption exponent tuple. Given ...
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### Is RSA vulnerable to possible PRNG + Miller Rabin test weaknesses?

Factoring a 2048 bit number is a difficult topic with a well known complexity. But it seems that p, q, the prime numbers used in RSA (order of magnitude: 10^308) are generated thanks to the ...
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### What is the number RSA 2048 used for?

The RSA factoring challenge lists this RSA 2048 value: ...
We are given an arbitrary number $n$ and a sequence of primes $p_1=2$, $p_2=3$, ..., $p_k$. I am interested in the following question: Are the events "Prime $p_i$ is a factor of $n$" independent for ...
For example, let, $p$ and $q$ be two large prime numbers. We set $n = p \cdot q$. Now, let $a \cdot b = c \pmod n$. Given $c$ and $n$, is finding the factors $a$ and $b$ computationally difficult? I ...