# 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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### Quadratic Sieve: Sieving with prime powers

I am trying to understand the Quadratic Sieve algorithm. Currently I am stuck at the sieving part. Let's say the number to be factored is 9788111. I decide to look for 50-smooth factors. My initial ...
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### Cracking RSA (or other algorithms) manually by a savant

RSA cryptography strength comes from the hardness (or so we believe) of factoring big numbers. For key lengths over 2048 bits, it is infeasible for current or near-future computers to factor those ...
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### Prime factorization (102 digits)

I have a number that consists of 102 digits and I need to factor it. I ran it in alpertrom.com.ar, but it'll take up to 40 hours if I counted all right. Is there any way to make it by hand (stupid ...
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### Why doesn't this factoring to order-finding reduction work?

Scott Aaronson likes to motivate the factoring-to-period-finding algorithm used inside Shor's algorithm as follows. Now, I want you to step back and think about what this means. It means that, if we ...
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### How long to reestablish PKI if Diffie Hellman and Factoring are in classical $P$?

Supposing there is a classical (no need quantum) $O(\log N)$ algorithm to factor integers $N$ and supposing there is a classical (no need quantum) $O(\log p)$ algorithm to find $g^{xy}$ given $g^x$ ...
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### How do I retrieve a number which has been multiplied with a random number?

I have a 1024-bit number $n$ obtained by multiplying two 512-bit randomly generated prime numbers $p$ and $q$. Then there's $\phi = (p-1)(q-1)$, which is another 1024-bit number. I do not have $\phi$ ...
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### Possibility of computing a and b values from the ciphertext?

Using paillier encryption, $N$ is the product of two large prime numbers, $s$ is sampled randomly from $Z_{N^2}$ we get $C \leftarrow g^ms^N \bmod N^2$ where $g=1+N$, By multiplying the cipher $c$ ...
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### How Does Prime Factorization Break ECDSA?

I have heard that ECDSA will be broken in the not-to-distant future (roughly 15-25 years) by Quantum Computers running Shor's Algorithm. However, to my understanding, the only purpose of Shor's ...
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### Is phi-hiding assumption as hard as integer factorization?

Phi-hiding assumption can be simply stated as (wrt hardness) It is difficult to find small factors of $\varphi(m)$ where $m$ is a number whose factorization is unknown and $\varphi$ is Euler's ...
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### Does knowing modular eth roots help in factoring n?

Consider $x^e \equiv a\pmod n$, given $n$, $a$, and $e>2$, with $n$ being a composite integer and unknown $x$. Can a hypothetical function $f(a)=x$, an $eth$ root extractor, be used / adapted to ...
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### Why discrete logarithm modulo composite moduli not popular and not defined in standards?

The classical discrete logarithm problem is to find $x$ such that $g^x\equiv h\bmod p$ where $p$ is a prime and $g$ is generator of multiplicative group modulo $p$. The demerit of this approach seems ...
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### Calculating RSA Public Modulus from Private Exponent and Public Exponent

If I know the private and public exponents ($d$ and $e$) of an RSA key pair, is it possible to (efficiently) calculate the public modulus $n$?
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### Decrypting small integers under RSA

Let $(n,e)$ be an RSA public key. Suppose $c = m^e \pmod n$, where $c>1$ is a very small integer. For concreteness, say $c=2$ or $c=4$. Is it hard to find $m$ under the RSA assumption (or any of ...
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### Algorithm for factoring a 30 decimal digit number

My professor has given me an RSA factoring problem as an assignment. The given modulus is 30 decimal digits long. I have been searching a lot about factoring algorithms. But it has been quite a ...
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### Prime Factorization in RSA always leads to the product of two primes?

Lets prime factorize $30$: $$30 = 3 \cdot 10 = 3 \cdot 2 \cdot 5$$ We see that the number $30$ is a product of $3$ primes. But in RSA, when factorizing huge numbers, we always seem to only get two ...
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### Factorization problem

Say, $X= a\cdot b$, where $(a, b) \in Z_q^*$ and $q$ is a large prime. If $X$ is given, then what is the complexity (or hardness) of finding $a$ and $b$? Note that, either $a$ or $b$ can be reused to ...
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### Brute force integer factorization - back of the envelope calculation

RSA-240, an integer with 240 decimal digits from the original RSA Factoring Challenge, has recently been factorized. According to the researchers, the factorization took a total of 900 core-years on ...
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### How large a product out of 3 close-by factors need to be to avoid factorization?

For encryption a prime $P = 2 \cdot Q \cdot R \cdot S +1$ was used. An adversary want to solve the discrete log problem $m \equiv g^i \bmod P$. For this he want to use the Pholig-Hellmann algorithm. ...
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### Help with next step in the Quadratic Sieve

So I am at the same step as someone from math.stackexchange but he never recieved an answer so I will copy-paste his question here: Say, for N = 90283, I compute bound 𝐵=𝑒(12+𝑜(1))(ln(𝑛)ln(ln𝑛√))...
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### RSA: If the least significant bits of the factors are leaked, what advantage is there in factoring N?

For $N=pq$, if the first $x$ least significant bits of both $p$ and $q$ are leaked. what is the advantage in factoring $N$? Does this give an advantage beyond simply lowering the number of bits we ...