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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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 ...
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Modification of RSA using two inverses, one for P mod (Q-1) and one for Q mode (P-1), instead of inverse d mod [(p-1)(q-1)], more or less secure?

Lets say I have the following modified RSA scheme We choose two large primes P, Q, with additional restriction that these are relatively prime to (p-1) and (q-1) We choose N = PQ as public key We ...
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88 views

Msieve & Yafu - RSA Exponents and bruteforcing

I am a layman in regards to the math behind RSA (and in general, relatively), and my goal is to bruteforce a large quantity of 512-bit RSA keys. Having searched around, I see that msieve, yafu, and an ...
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26 views

If they exist a relation between decisional Diffie-Hellman assumption and composite decisional residuosity assumption

From the cryptographic hardness assumptions, we have DDH and CDR assumptions. It is known that the composite decisional residuosity assumption is related to a factoring problem, while the DDH is ...
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99 views

Show that $\text{FACTORING} \le_P \text{SQROOT}$

I tried to prove that $\text{FACTORING} \le_P \text{SQROOT}$ in a general setting, so $n = p_1^{\alpha_1} \cdot p_2^{\alpha_2} \cdot \ldots \cdot p_k^{\alpha_k}$. THEOREM:Let $n$ be a composite ...
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56 views

If there is an algorithm A can calculate the modular square root of input n, How to use it to get prime factors?

Suppose you are given an algorithm $A$ which takes $y \in \{0, 1, \ldots , N − 1\}$ as input, and outputs $x \in \{0,1,\ldots,N − 1\}$ such that $x^2 \equiv y \pmod{N}$. Design an efficient, ...
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Given a deterministic oracle that calculates square roots modulo n, factor n

When $n = pq$ where $p$ and $q$ are primes, we can generate random numbers until we get $a$ and $b$ such that $a^2 \equiv b^2 \pmod n$. This implies $n$ has some common factor with $a^2-b^2$, and then ...
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Factoring 2048 bit number is easy?

my PC found a factor for (2^2048)-1 in under a second...so does that make RSA-2048 less secure right? i used prime 95. and actually i am kinda curious how it found a factor so fast? i can even factor ...
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Recursive RSA encryption

I have a ciphertext $C$ encrypted with public key $pub_C$, which contains ciphertext $B$ and $pub_B$, $$C= E_{pub_C}(B\mathbin\|pub_B)$$ Ciphertext $B$ is encrypted with $pub_B$ and contains $pub_A$ ...
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99 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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88 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 ...
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117 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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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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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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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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98 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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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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105 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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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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113 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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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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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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192 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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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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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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522 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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82 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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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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38 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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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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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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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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331 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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255 views

Factoring RSA weak modulus

Given a public key for RSA, I have extracted the modulus which looks like this : ...
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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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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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130 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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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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103 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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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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161 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 ...