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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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0answers
36 views

Estimating the probability of sucess of Pollard's p-1

I'm trying to estimate the probability that Pollard's p-1 factorization in its two-stages variant finds a factor of an RSA modulus product of $k$ random $b$-bit primes, as a function of the bounds $...
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53 views

How do I interpret the CADO-NFS output for discrete logarithm calculation in GF(p)?

I'm using CADO-NFS to calculate discrete logarithm in a finite field GF(p). The problem is when I type ...
3
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1answer
110 views

Attacking any one in many public keys

The problem of finding private key from public key is typically studied in the one-key setup: what's the expected cost of breaking one key (e.g. by factoring a public modulus, or solving a discrete ...
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1answer
182 views

When is factoring semi-primes thought to be hard?

In Lattice Cryptography, problems like LWE or SIS have relatively easy to specify distributions that are thought to be average case hard. I'm curious what specific distributions on semi-primes $(p,q)$ ...
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340 views

What is the efficiency of the new Crown Sterling semiprime factoring method?

In their press release a company called Crown Sterling describes they are working a paper that includes four different geometric and arithmetic methods for public key (semiprime) factorization and ...
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1answer
46 views

Purpose of using a=2 in Pollard p-1 factorization method

The Pollard p-1 factorization method states if $\gcd(2^{B!}-1,n)=p$ where $p>1$ and $B$ bounds the prime factors of $p$, then $p$ is a prime factor of $n$. Shouldn't it be $\gcd(a^{B!}-1,n)$ for ...
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84 views

Pollard's $(p - 1)$ factorization method runtime

Wondering if anyone knows a good reference for Pollard's $p-1$ algorithm's runtime? I was looking on the Wikipedia page and the runtime cited there is $\mathcal{O}(B\cdot \log B\cdot \log^2 n)$. ...
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57 views

General factoring and one-way functions

Let a function $f$ be one-way, if there exists a probabilistic polynomial time algorithm to find the preimage of $y = f(x)$ for uniformly chosen $x$ with non-negligible probability. Define the ...
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1answer
71 views

Elliptic Curve Discrete Log in a Composite Ring

Elliptic curves are usually defined over prime rings (fields), but what if we chose a ring of composite order? Let $n = pq$ for $p,q$ large primes. Say I have elliptic curve $y^2 = x^3 + ax + b$ over ...
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51 views

Subexponential algorithms that apply only one of factoring and discrete logarithm?

Shor (quantum polynomial), Number Field Sieve (subexponential), Pollard rho (square root) all have both factoring and discrete logarithm over $\mathbb F_p^*$ variants. What are the subexponential ...
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1answer
109 views

What is the best strategy to avoid getting even orders in Shor's algorithm?

I do understand Shor's algorithm wants the order of an element to be even so that it can use the factoring identity and find a non-trivial factor. But is there a relationship between safe primes and ...
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85 views

Variant of Pollard rho using small factors of p - 1

Given an integer $N$ to factor which is divisible by some prime $p$, suppose you know (or guess) that $p - 1$ has a few small factors, e.g. $3, 2^2, 5$. Define $B$ as a product of small prime powers ...
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41 views

Factorization of the public value $N$ from the RSA cryptosystem

It is mentioned here that the public value $N=p*q$ of the RSA cryptosystem can be factorized if one of the factors is reused. Thus, if $N_1=p*q_1$ and $N_2=p*q_2$ and only $N_1$ and $N_2$ are known, ...
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141 views

What does it imply if in RSA pow(m, e, n) is same as the m

So, Let's assume we have n which is made up of 2 strong primes which cannot be factored & e which is textbook value of e ...
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326 views

RSA factorization with special primes

Suppose that primes for RSA modulus are generated using formula: $P_i(x,y) = \operatorname{next\_prime}(x^{z_i}+y^{z_i}) = x^{z_i}+y^{z_i}+d_i$ where $x,y$ are unknown random numbers with size 128 ...
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Finding the period of a function with a single output qubit - impact on RSA

In this paper,May and Schlieper claim that one can find the period of a function $f()$ by embedding $h \circ f = h(f(x))$ for input $x$. This would have the immediate consequence of reducing the ...
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4k views

How long does it take to crack RSA 1024 with a PC?

Using an Intel Core i5 CPU, how long does it take to crack RSA using a key size of 1024 bit (generated using a secure key pair generation function)? Suppose for instance that we have thousands of ...
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400 views

Is this paper's technique for factoring RSA 2048 with noisy qubits realistic?

A paper titled How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits has just come out which proposes a technique to factor RSA keys with moduli up to 2048 bits with a design ...
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76 views

How many iterations for Pollard's $p-1$ with $p = r^k + 1$ for prime $r$?

$p$ and $q$ are large primes. What is the lowest upper bound for the number of iterations for Pollard's $p-1$ algorithm for factoring $N = pq$, provided that $p = r^k + 1$, for a prime $r$, and $r^k + ...
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78 views

Is there a fast way to solve $k = n \cdot g^a \mod P$? (get $a$ for unknown $n$)

Would a factor besides the normal discrete logarithm problem increase or decrease the solving time? $k = n \cdot g^a \mod P$ with given $k,g,P$ and the knowledge $P= 2 \cdot N \cdot f+1$, while $...
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1answer
33 views

Which is the fastest way to find a member of a subgroup with known size modulo prime $P$, with know factorization of $P-1$?->$x$ with $x^s \mod P = 1$

Assuming you know the factorization of used prime $P-1$ $P-1 = s \cdot f_2\cdot f_3...f_i$ Now you want to find a member of a subgroup $\mathbb{Z}_s$. This means any $x$ with $x^s \equiv 1 \mod P $ ...
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352 views

Can Shor's algorithm factor multi-prime numbers?

I know that Shor's algorithm can factor semi-primes ($N = p \times q \space, \{p, \space q \in \Bbb{P} \space \vert \space p, \space q \gt 0 \} $). Assuming that all prime numbers are so large that ...
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119 views

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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About integer factorization

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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44 views

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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68 views

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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3k views

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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3answers
184 views

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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126 views

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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1answer
269 views

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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204 views

Factoring RSA weak modulus

Given a public key for RSA, I have extracted the modulus which looks like this : ...
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79 views

Yukel's Sieve - Factorization of Numbers into a Square Sieve

https://www.youtube.com/watch?v=liTTGeitpGQ https://www.youtube.com/watch?v=2nOwgiweyqc https://www.youtube.com/watch?v=rGwFsOG27DQ I came across these videos explaining a pattern that is found in ...
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1answer
65 views

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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1answer
128 views

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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48 views

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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99 views

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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1answer
39 views

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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1answer
153 views

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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3k views

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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315 views

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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65 views

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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127 views

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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712 views

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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54 views

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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84 views

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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1answer
223 views

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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3answers
2k views

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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27 views

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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1answer
122 views

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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683 views

RSA factorization for special primes $p$ and $q$

I want to factorize the modulus $n = pq$ knowing that $p$ and $q$ are not random, but constructed based on integer numbers $a$ and $b$ as following ($a$ and $b$ are not given): $$p = a^2 + b^2, \...