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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NIST implementation of the Lucas primality test

The NIST standard FIPS 186-4 describes an implementation of the Lucas primality test in section C.3.3. I can follow the algorithm but I am puzzled by step 6.2: $V_{temp}=\frac{V_{i+1}^2+DU_{i+1}^2}{...
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What are the possible cryptographic implications of Zhang's proof of the Twin Prime Conjecture?

Earlier this year, Yitang Zhang published a proof of a weakened form of the Twin Prime Conjecture. I'm wondering if any of the new mathematical machinery he developed has uses in cryptography or could ...
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58 views

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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167 views

Biased RSA moduli and ROCA

Say I can generate many 1024-bit RSA public keys $(N,e)$ with fixed public exponent $e = 65537$. They turn out to be heavily biased when computing $N \bmod x$ for small primes $x$. These congruences ...
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584 views

Modular Arithmetic in RSA

Consider the following the following RSA public key $pk = (N, e) = (1457, 1307)$. (a) Knowing that $187^2 \equiv 1 \pmod {1457}$ find the factorization of $N$. (b) Given the factorization of $N$ ...
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Finding public exponent e

I'm trying to create an algorithm to find the public exponent e given a plain (non-CRT) private key that doesn't include the public exponent, i.e. I've only got $n$ and $d$. A question has already ...
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1k views

Primality testing (deterministic vs. non-deterministic)

I noticed that non-deterministic primality testing algorithms are more commonly used in practice while there is a deterministic algorithm e.g., AKS which runs in polynomial time? I am right? if so, ...
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52 views

Proving that RSA encryption function with non-square free modulus is not a permutation

Here is a backgroung for the question on hand. While studying RSA I came up to the question about what happens if $p$ and $q$ involved in modulus computation are not actually primes? There is already ...
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228 views

Why does FIPS 186-4 seem to have an excessively complex q generation?

So in DSA you have two primes - p and q. q is N bits long (let's assume 160 bits) and p is L bits long (let's assume 1024 bits). Here's what FIPS 186-4 says about generating the q parameter for DSA: ...
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2k views

Generation of N bit prime numbers -> what is the actual range?

Short version: When generating a prime number of N bits, should I draw random numbers from the range $[0 , 2^n]$, or $[2^{(n-1)} , 2^n]$? Context: I'm trying to implement a toy-version of RSA as a ...
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105 views

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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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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Integer factorization based password authentication

After looking at this security issue at DjangoProject, I started to think in a password-based authentication that places the burden of PBKDF2 (or whatever is the hashing function) on the client. So I ...
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How can I find the prime numbers used in RSA?

I got this question in a local hacking event, but I couldn't solve it. Problem Statement ---- Continuing their snooping habit, NSA kept bugging Alice's communication. Resorting to the age old ...
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1k views

How to encrypt the number one using RSA?

for example: if we have public key : (5,221) and private key : (77,221) and we want to encrypt 1: c(m) = (m)^p mod n c(m)=(1)^77 mod 21 =1 so how to deal with ...
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Factors of RSA modulus

In the article A Method for Obtaining Digital Signatures and Public-Key Cryptosystems, the original RSA article, it is mentioned that Miller has shown that n (the modulus) can be factored using any ...
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Could Riemann hypothesis solve certainly RSA?

I don't have the background for dealing with Riemann hypothesis but is well known that covers the prime distribution below a specified number. In order to solve the RSA problem you have to factor the ...
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359 views

How did they factor RSA-704?

I don't understand the 'Wiedemann algorithm' works. Can someone explain the factoring of RSA-704 in an easy way?
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211 views

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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418 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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159 views

For large prime P, how often is (P-1) evenly divisible by 65537?

When calculating prime numbers $p$ and $q$ for an RSA private key, one of the requirements is that $\gcd(p-1,e)=1$ and $\gcd(q-1,e)=1$, where $e$ is the RSA exponent (typically 65537). I'm curious ...
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284 views

Is it hard to recover $p$ from $k \phi(p)$?

Given $k\phi(p)$, is it hard to recover $p$? Here, $p$ is a large prime, $\phi(\cdot)$ is Euler's totient function and $k$ is an unknown integer. Or what's the complexity to recover $p$ from $k \phi(...
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Random numbers for Rabin-Miller primality tests

I've implemented a Rabin-Miller primality test fuction following Wikipedia and the book Applied Cryptography. Now I'm using it for generating primes with a string seed. The book suggests the following ...
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Quadratic residue problem on composite integers

Its believed that the quadratic residue modulo $n=p·q$ for large primes $p$ and $q$ is intractable, which forms the basis of some cryptosystems. However, it is solvable if the factors of $n$ are know,...
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How can I use eulers totient and the chinese remainder theorem for modular exponentiation?

I'm trying to implement modular exponentiation in Java using Lagrange and the Chinese remainder theorem. The example we've been given is: Let $N = 55 = 5 · 11$ and suppose we want to compute $27^{...
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123 views

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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The problem of 'Common DH Primes'

NSA crack multiple encrypted channel by some tricks on 'Common DH Primes', which is a old news, but how actually they crack it? I still can't get the concept why using a 'Common DH Primes' will ...
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Modulo properties of two prime numbers

I am supposed to prove that $x = y \mod (pq) \iff x = y \mod p$ and $x = y \mod q$ with $p$ and $q$ are prime numbers. It somewhat sounds reasonable to me, but unfortunately I don't have any clue how ...
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61 views

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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70 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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466 views

Diffie Hellman unsafe prime vulnerability

I have done some research about how the DH key exchange is unsafe if an unsafe prime p is used (that is, $p-1$ has a lot of small factors). Many answers here on StackExchange claim that for any factor ...
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305 views

RSA: Given $n$ and $\varphi(n)$, find $e$ such that $e=d$

Say we have chosen $n=pq$, with $p=89,q=97 \implies n=8633$ and $\varphi(n)=88\times96=8448$. Also, if we want the encoding ($e$) and decoding ($d$) exponents to be equal, we need $$(e,8448)=1 \: \: \...
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159 views

Highest prime factor that is Safe for a particular scheme

My question is how many bits of prime number is secure so that it cannot be factored from very large number? Until today how large prime factor is found in large number? Quantum computing find ...
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How “hard” it is to take an e'th root mod p?

I know it's hard to find the $e$th root of a number mod $n=p_1*p_2$, and if it would be possible we could break RSA. But how hard it is to take an $e$th root mod $p$ where $p$ is a prime and $\gcd(e,p-...
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234 views

Pick faster private exponent

I recently tried to send 1536-bit modulus CSR to COMODO. They refused to sign the certificate. I later found out that it's because NIST mandated 2048-bit modulus on the SSL certificate. I think it's ...
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456 views

Given p and q of DSA how do you show they are prime?

I am given p = 4916335901 q = 88903 and am asked to show these are prime as well as q|(p-1) in DSA. I am unsure on how to ...
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272 views

Selecting a large NUMS Safe prime

Suppose I want to use the following simple hash function. For a mesage $m$, take some public $a$ and prime $p$ and raise $a^m \bmod p$ (never mind the computational expense of this operation). This ...
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2k views

What is the correct value for “certainty” in RSA key pair generation?

I'm creating an RSA key pair in Bouncy Castle and need to specify an int value for certainty. This Stack Overflow answer says it is a relative test for how prime the values are. There is another ...
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3k views

How are the primes used to generate RSA keys?

I am confused about how keys in RSA asymmetric encryption are generated and what the implications for open communications are. Textbooks say the one-way function is merely two primes (with some ...
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78 views

AugPake implementation Doubt K computation

try to implement the AugPake protocol in java using BigInteger. I am having some difficulty computing $K=Y^z \mod p$ because $z$ is always $0$, for $z={1\over x+(w*...
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Mathematics behind end-to-end encryption

I am interested in end to end encryption, recently I read some articles on its principles but I can't find anything on the maths behind the encrypting and decrypting functions that relates private and ...
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128 views

Largest number that could be factored in milli seconds

Considering a home pc/laptop as machine used (Say typical 2.4 GHz, 16GB RAM, 4 core processor) for running any factorization algorithm. What would be the largest number that could be factored into its ...
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1answer
138 views

Diffe-Helman Exchange result is always 1

I watched a video on Khan Academy explaining the Diffe-Hellman exchange. When I try to do an example problem, I get 1 all the time. Does the generator and prime modulus (or base on Wikipedia) have to ...
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89 views

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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98 views

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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318 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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76 views

Can we efficiently factor if we are given a Pocklington certificate of one of the prime factors?

I recently read Squeamish Ossifrage's answer on generating RSA keys from (short) randomness where they make the following comment: (You might want to keep the certificate secret too.) As the ...
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1k views

Why are cube and square roots of primes used as SHA constants?

Do square or cube roots have properties that make them a better choice over, for example, the primes themselves? Or is it arbitrary - would random numbers work as effectively?
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623 views

Sieving the sequence $x^2-n$ to recognize b-smooth numbers

I am currently programming the quadratic sieve and have several literature books / papers and will take an example out of [1] for my question: [1] An Introduction to Mathemtaical Cryptography by J. ...
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553 views

Good Encryption Exponent

I have placed a bet that I can create a public key such that my adversary will not be able to crack (decrypt) it for at least one week. For my primes $p$ and $q$, I chose very large numbers that are $...