Questions tagged [prime-numbers]

A prime number is an integer greater than 1 with no divisors other than itself and 1. Primes and prime products play an important role in public key cryptography.

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How can I convert numbers into prime numbers?

I'm working with one-way accumulators, but I'm not knowledgable in cryptography. Is there an easy peasy way to hash numbers (or whatever) into prime numbers? Obviously I'd like it to be collision ...
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Three different numbers with x³=x mod p

p is a prime greater than 2 and $a \in \mathbb{Z}_p$. Why are there exactly three solutions for a³ = a mod p? Obviously 0 and 1 are both in $\mathbb{Z}$ and valid solutions, but that still means, ...
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If $((p-1)*(q-1) -1)$ divisible by $e$ ($e$ is odd number) , then $\text{gcd}(e,(p-1)*(q-1)) = 1$

If $((p-1)(q-1) -1)$ divisible by $e$ ($e$ is odd number) , then $\text{gcd}(e,(p-1)(q-1)) = 1$. ($p,q$ are prime numbers ) Is this true, if yes why, if not why not ?
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ZKP for product of two primes

I'm struggling to understand the intuition of the zero knowledge-ness of this proof from the following paper. The proof is a 2 round where the verifier asks the prover to extract square roots of ...
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Computational trapdoor where the problem is tractable for both parties but easier for one

Usually the sort of trapdoors which are talked about are designed such as to make the computation intractable for one party and tractable for the other. But what if one party merely has a big ...
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With composite $n_1$ = $p_1q_1$, and a separate $n_2 = p_1q_2$, can the primes be calculated more efficiently than factorization?

Supposing that the (3 total) primes are kept secret? Does the reuse of $p_1$ allow an attacker to compromise $n_1$ and $n_2$ if the attacker guesses that both were generated with a shared prime ...
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Problem with RSA deciphering

I don't quite get the algorithm yet. Sometimes it works and other times it doesn't,so clearly I am overseeing or misunderstanding something. I will just write what I did. My $N=143$ and has factors ...
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How to produce large list of BigInteger prime numbers (RSA) in fast and efficient way

Hello fellow experts here, in order to use RSA to encrypt and decrypt in a controlled environment, we will actually need to have a list of prime numbers to do so instead of using library generated RSA ...
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Prime numbers of the form $(2^k)p+1$, for a given prime $p$

Let $p$ be a prime. (say 256 bit) Does there a exist a prime $q$ such that $q = (2^k)p + 1$, for a large $k$ (something like 256), if it does exist, is there a way to find out for which all $k$ such ...
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How does Python's pycrypto library generate primes?

The pycrypto library in Python can generate random n-bit prime numbers. The syntax I use is as follows: from Crypto.Util import number number.getPrime(2048) The ...
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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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RSA calculate $d$ using Chinese Remainder Theorem with $d_p$, $d_q$ and $e$

Suppose for a RSA system I have the following variables given: modulus $n$, expononent $e$, $d_p$ and $d_q$Where, $d_p = d\bmod(p-1)$ and $d_q = d\bmod(q-1)$, Is it possible to find the private ...
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Trivariate Coppersmith Implementation

Bivariate Coppersmith is standard package in math software with number theory support. Bauer and Antoine Joux introduced trivariate Coppersmith in https://www.iacr.org/archive/eurocrypt2007/45150361/...
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Why is Multiplicative group used in RSA or Euler's theorem but not additive?

I am on the verge to understanding RSA, but suddenly a question popped into mind. When we are calculating $U(N)$ i.e $U(PQ)$, we are taking invertible elements that are co-prime to $N$. For example, $...
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FLT is partly applying to RSA equation, and also relation between ED mod phi and Phi + 1 mod N

After numerous attempts from myself and all of you guys, I finally came to understand RSA. I can now prove it and understand how I got there. But I still have some very few polishing questions. 1) We ...
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Relation between $N = P \times Q$, and $\Phi(N)$

When studying RSA, and proving simple concepts to myself, I went and understood groups and rings, but I failed to understand Lagrange's theorem. I did understand how from invertible finite groups I ...
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RSA with probable primes

I am a bit of a newbie to RSA encryption, so please be patient. I understand that for a 4096 bit RSA, the numbers p and q should be prime. And to have the best security, the p and q should both be ...
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Security of Pohlig-Hellman exponentation cipher?

I am looking into implementing Pohlig-Hellman exponentiation cipher and I would like to know how secure that algorithm is? I am guessing it's security relates greatly to the prime number used in it. ...
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Is it hard to find a big random easy to factor number?

Suppose that I give you the challenge of successfully factoring any very big random number. That is, you pick a big random number (say, 65536 bits) and try to factor it. If you manage to, you win. If ...
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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, \...
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Why is a prime number used in ECDSA?

So I need to write a piece for school about ECDSA and how it is secure. Now I thought I had a simple question, however, I can't seem to find an answer anywhere: Why does the p in the formula need to ...
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Three Pass Protocol Question!

Alice and Bob have agreed to use the Three Pass Protocol. p=1009 Alice chooses the encryption exponent e_A = 101 Bob chooses the encryption exponent e_B = 209 Now Alice and Bob send three ...
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Diffie-Hellman in Https: How are prime numbers picked?

I am trying to understand https, as I understand https uses the Diffie–Hellman method for keys exchange and then AES for encryption. But Diffie–Hellman needs two prime numbers, where do these come ...
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What place do prime numbers have in cryptography?

My understanding of hashing and encryption is rather limited. I certainly do not understand the mathematical formulas at play in these algorithms. With that said, what part do prime numbers play in ...
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About the irreducible test for a polynomial over $GF(2^m)$ (its coefficients belong to $GF(2^m)$)

For the Miller-Rabin probabilistic primality test, we can apply it to test a big number whether a probably prime number or not. Does there exist a method to test a higher-degree polynomial over $GF(2^...
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Why do we need Euler's totient function $\varphi(N)$ in RSA?

After we calculated $N = p * q$, we calculate $\varphi(N)$ and use it later to determine $e$ (PR) and $d$ (PU). But why? For decryption and encryption we only use $N$ and don't need $\varphi(N)$. So ...
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Is it feasible to build an index of prime factors?

Would it be possible to break an RSA key, in for example 1 week of time, if the cracker have already spent X number of years building an index of primes by performing every permutation of existing ...
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Distance between consecutive primes distribution

In several prime generation schemes I saw we pick a random number uniformly at random from a wide range and find the next prime after it. Obviously with such a scheme some primes are more likely than ...
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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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Trial division + Miller-Rabin complexity

I was reading a paper that shows time complexity of Trial division + Miller-rabin test for creating prime number. T = N(T_rnd + T_td + T_mr) T_rnd denotes time for making rnd number, T_td denotes ...
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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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Generalized Benaloh cryptosystem with $r=2$

Benaloh cryptosystem requires $\gcd(r, (q-1))=1$ which is impossible if $q>2$ (since it needs to be a large prime) and $r=2$. This confuses me, since Benaloh is referred to as an "extension" or "...
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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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What does it mean for a number to be “in the order of” a prime number?

In some papers I'm looking at there is some language that says things like "Choose a random number in the order of prime q" and I see some syntax that it is referring to that looks s = Zq (where q is ...
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Are the RFC3526 MODP groups Schnorr groups?

I was wondering if a group like the 1536-bit MODP Group from RFC 3526 was a Schnorr group? A Schnorr group must apparently have: $p$ and $q$ being primes $p = q\cdot r+1$ $1 < h < p$ $h^r\not\...
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Is it possible to construct a multiplicative group from $\mathbb{Z}_n$ if $n$ is not a prime number?

With $n$ being a prime number I know we can generate groups over multiplication. Is it possible the other way around ($n$ not being a prime)?
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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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Why do we use prime numbers apart from hard factorization?

Why are prime numbers important in cryptographic constructs? I am not interested in RSA examples where factorization is the hard problem itself, that makes sense. However wherever I go I encounter ...
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Efficient function/algorithm/method to do modular exponentiation

I was working on this project where I needed an RSA key, and I wondered if there was and more efficient way of doing $g^a \bmod n$ other than calculating $g^a$ and then finding the remainder when you ...
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Would calculating the nth prime ruin encryption?

I'm sure there is some proof that the nth prime can be found. But if we knew, would encryption that relies on primes be easy to decrypt?
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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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Deterministic procedure for mapping an arbitrary value into a 𝑝,𝑞 pair for public key cryptography

I want public key cryptosystem to used for re-encryption as describe in Can Paillier ,RSA or any other schemes be used for universal re-encryption like elGamal? Now i have little solution for ...
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Is there an algorithm for factoring N, which is just as simple as this one, but faster?

I found a simple algorithm for factoring semiprime numbers, you can read about it in Factoring Semiprimes and Possible Implications for RSA (paywall-free). It basically works like this: You reverse ...
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Paillier Complex Residuosity problem?

Paillier Cryptosystem depends on both the factorization where $n = p.q$ and the complex residuosity problem which is defined in the original paper as: The problem of deciding n-th residuosity, i.e. ...
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Question about using residue number system for repeated multiplications

I understand that when you are using RNS you need a co-prime moduli-set e.g. ${\{m_1, m_2, m_3\}}$, and the dynamic range is the product of each modulus in that set $M = m_1.m_2.m_3$. Also it's ...
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RSA Modulus doesn't match the primes [closed]

I generated a 2048-bit RSA key with ssh-keygen. When running: openssl rsa -in key -noout -text The result is: ...
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What is a method for hiding RSA prime values from .exe De-Compiling?

I've just been looking into RSA encryption with C++ and am wanting to make a program that can en/decrypt files of my own custom file extension. Obviously I need to choose two primes numbers (for my p ...
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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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Algorithm to find primes $q$ and $p$ with $q\, |\, p - 1$?

I understand that if $p$ is prime then $p-1$ must be composite (at least divisible by $2$ as it is even). But how does an algorithm find a prime $q$ such that $q \cdot r = p - 1$. I thought prime ...
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Are safe primes $p=2^k \pm s$ with $s$ small less recommandable than others as a discrete log modulus?

I take the definition of safe prime as: a prime $p$ is safe when $(p-1)/2$ is prime. Safe primes of appropriate size are the standard choice for the modulus of cryptosystems related to the discrete ...

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