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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 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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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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166 views

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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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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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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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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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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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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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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137 views

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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1answer
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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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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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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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177 views

Why does square root $\pmod n$ find $p$ and $q$ (as $n = p \cdot q$)?

Let $n = p*q$, with $p \neq q$ and $x^2=1 \pmod n$, $x+1 \neq 0 \pmod n, x-1 \neq 0 \pmod n$ (So x is a non-trivial square root mod n.) I don't see how $\gcd(x+1,n) \in \{p,q\}$ follows. I ...
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Why can every prime number be written as 6k±1? [closed]

I want to know how to be sure that each prime number can be written in the form $6k±1$. How I can find the prime number that exists after a composite number with this property of prime or other ...
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In practice, do large prime generators account for difference in gaps between primes? [duplicate]

It is my understanding that a method for creating large primes is to: pick a random large number, check the number against trial division of a set of smaller prime numbers, check with something like ...
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406 views

Distribution of safe primes generated using different techniques

Is there any difference in the distribution of safe primes generated by creating prime $q$ and testing $2q+1$ for primality, compared to generating a larger prime $p$ and testing $(p-1)/2$ instead? ...
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174 views

How should I address message size limits in RSA encryption?

I am making an end-to-end encryption software program in Java using RSA. I am using BigIntegers and its number theory methods. (I know this is a very slow approach, but I just want to learn to the ...
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RSA encryption, Number theory [duplicate]

In RSA algorithm we have the value of $e$ & $d$ exponent and also one of the prime numbers. My question is how to produce another prime number that is not equal to the first one and the digit of a ...
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55 views

Finding a prime field with n-th roots of unity

How can I find the smallest prime $p$, such that field $GF(p)$ has $n$-th roots of unity? For example, I know that for $p=2^{256} - 351 \times 2^{32} + 1$ there exit roots of unity for $n=2^{32}$. ...
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1answer
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Value of $\varphi(n)$ and $\lambda(n)$

Is it true that $\varphi(n)$ is generally larger than $\lambda(n)$ for the same $n$? If so, can anyone give me a proof?
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Problem with Chaum`s Untraceable Electronic Cash

For example according to protocol I need calculate this: $b=F^{(1/h)} \bmod pq.$ Where $p$ and $q$ are prime numbers. I have $F$ and $h$. But how can I calculate $b$? I tried to do this: $\text{...
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1answer
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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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How to verify if g is a generator for p?

For learning purpose, supposed I have a 16-digit prime which is 2685735182215187, how do I verify if g is a generator? (p is supposedly a special kind of prime)
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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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1answer
145 views

Does a Finite Field of 36 elements exist? [duplicate]

I'm having a little trouble understanding the finite fields theory, so I'm sorry if my question would seem a little stupid. I wanted to know if a finite field of 36 elements could exist. Basically, I ...
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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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81 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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150 views

Distinguishing generators of RSA primes and moduli

We are trying to distinguish two RSA prime generators: Prime generator 1 draw uniformly at random an integer $p$ with $2^{1023.5}<p<2^{1024}$, until it is prime output $p$. Prime generator 2 ...
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274 views

Are relatively prime numbers used in RSA

We know that the Totient function is multiplicative. Which means that when $p$ and $q$ are relatively prime, then $\varphi(p q)$ is equal to $\varphi(p) \varphi(q)$. My question is, are only prime ...
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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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105 views

Magic Number to calculate number of rounds for M-R in FIPS 186-4

In Fips 186-4, there is an algorithm in Appendix F (at page 117 in my copy of the 2013 version) to calculate the number of rounds of the Miller-Rabin primality test to random bases. Does anybody ...
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Random Primes Conjecture [closed]

I know it is believed that primes appear to be randomly distributed among the integers. Is there a formal conjecture or theorem that expressly states that the occurrence of the prime numbers is ...
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1answer
105 views

How can I generate large primes for Pedersen commitment?

I want to make a commitment on Shamir's Secret Sharing, based on the work of Pedersen, "Non-Interactive and Information-Theoretic Secure Verifiable Secret Sharing". To implement the commitment ...
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How to efficiently generate a random safe prime of given length?

A prime $p$ is said to be safe prime if $(p-1)/2$ is also a prime. How to efficiently generate a safe prime? I have written the following code in sagemath which generates a random safe prime of 1536 ...
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1answer
224 views

DSA: How to calculate 224-bit $q$ for 2048-bit $p$

$p$ is a 2048 bit prime number $q$ is a 224 bit prime number I know that $q$ is a prime divisor of $p-1$, thus $p=1 \bmod q$ but I couldn't write efficient code to calculate this. I can calculate ...
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How to formally say that the integers modulo $p$ for a prime $p$ gives results that are “more random” than for a composite $n$?

I'm doing a presentation on cryptography for non-experts. My main algorithm of the presentation is the Diffie Hellman key exchange. It uses modulo arithmetic for a prime $p$. During my presentation, I'...
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Swift and (very) large integers [closed]

I am trying to write a simple Mac program to factor primes. Nothing new, but I have a problem. The largest integer I can use seems to be 2^64, which is not that big. Did you hit any similar problem? ...
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560 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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What is the difference between these cyclic groups

$\mathbb{Z}^*_p$ vs $\mathbb{Z}^*_{p-1}$ vs $\mathbb{Z}^*_{p^2}$ vs $\mathbb{Z}^+_{p^2}$ I know $p$ is the value. The value create must be coprime to $p$. Does that mean that the value create must be ...
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How big are the primes used in modern cryptography?

I know that this depends on which techniques you're using, but roughly speaking, when modern cryptography makes use of so-called "large prime numbers", how large (in bits or digits) are these primes ...
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In RSA, is it possible to compute the private key if one knows one of the primes used to generate n?

Within RSA the private key can be derived if one knows both $p$ and $q$. Is it possible to derive the RSA private key or decrypt / learn something about encrypted messages if you only know one of the ...
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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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Public-key generation - primes reuse

Given the following Trap-door Commitment scheme Secret key receiver: $x_B \in_u Z_q$ Public key receiver: $y_B = g^{x_B} \mod p$ Here, $p=q*k+1$ for two primes $p,q$ and $k \in Z$. And $g$ is the ...