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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factorization of an integer $N$ that is in special format

Suppose $p_0$ and $q_0$ are known prime numbers and define $p_i$ and $q_i$ as follows: $$p_{i+1} = next\_prime(p_i^2 + q_i^2), \qquad i \ge 0$$ and $$q_{i+1} = next\_prime(2p_iq_i), \qquad i \ge 0$$ ...
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How to generate 1000 prime number of 1024-bit with much less time?

I am generating thousand prime number of 1024 bit each. But it takes lots of time. My procedure is as follows. Generate prime number using ...
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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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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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Why does RSA need p and q to be prime numbers?

This is a clarifying thread after finding: What makes RSA secure by using prime numbers?. Despite the fact that this question was answered already, I am still struggling to really grasp the ...
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New Improved Probabilistic version of RSA

On the 2nd page of "New probabilistic public-key encryption based on the RSA cryptosystem" by Roman'kov (PDF), at last it says Alice can find "f" of order "l" with least probability of (1-1/l). I ...
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How to test implementation of primality tests like Miller–Rabin?

The Miller-Rabin primality test is an algorithm for checking if number is a prime. What would be best way to test implementation of such algorithm (or any primality test in general)?
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NIST Diffie-Hellman prime: how was it picked? Where did it come from?

According to this Matasano Crypto challenge, the NIST "likes" the following prime modulus, which appears to be expressed in hexadecimal: ...
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Are the prime numbers used for RSA encryption known? [duplicate]

I read that one reason why RSA is secure is because it uses a huge number that's called the modulus which is the product of two prime numbers. For maths reasons the prime numbers being prime numbers ...
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41 views

Security of Diffie Hellman in specific cyclic group

For some $k$, let's say $p = 1+ \prod_{j=1}^k q( j)$, where $q(1)=2$, $q(2)=3$, if $p$ is prime, the diffie-hellman key exchange is not secure in cyclic group $Z^*_p$. Why?
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Implications of pattern in finial digit of prime numbers [duplicate]

http://qz.com/639452/mathematicians-are-geeking-out-about-a-bizarre-discovery-in-prime-numbers/ What are the implications of this (very) new research on crytopgraphy? I would have thought this would ...
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“Prime conspiracy”'s effect on cryptography [duplicate]

Recent news reported about the discovery of a "Prime Conspiracy" which can be read about here. In summary, researchers have discovered that the last digit of prime numbers have a greater ...
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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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Do any cryptography algorithms work on numbers besides primes?

I know prime numbers are important for several algorithms and protocols. Are there any algorithms and protocols that don't require primes?
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Fast algorithm for reduction modulo a prime [closed]

If the prime is $p=2^a\cdot3^b+1$ , is there any fast reduction technique modulo this prime?
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How common are weak RSA keys?

There exist certain attacks that can be used against RSA keys whose prime factors are of specific forms, such as one by Coppersmith. How common are these RSA keys? If you generate primes randomly, ...
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Is RSA safe anymore? [duplicate]

Some people may have heard of Shor's algorithm. It allows for integer factorization on a quantum computer. This wasn't a problem a little while ago since we didn't have any quantum computers. Google ...
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Obtaining Diffie-Hellman generator

In the Wikipedia article on Diffie-Hellman, the algorithm calls for a large prime modulus, $p$, and a generator, $g$, which is a primitive root of $p$. As far as my knowledge of number theory goes, ...
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Difference between Pseudo Mersenne primes and Generalized Mersenne primes

The field prime numbers $p$ proposed by the NIST standards are referred to as Generalized Mersenne prime numbers [1] and as Pseudo Mersenne prime numbers [2]. Is there a difference between Pseudo ...
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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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RSA incorrectly translates for small keys [duplicate]

I've been learning about RSA and wrote my own implementation. I don't pretend to have intuitive understanding of RSA or that I understand why it works, but I believe to have some basic understanding ...
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Simple RSA Key Generation example [duplicate]

I have two prime numbers: p = 37 q = 41 And I need to find whether any of these prime numbers, 5, 7 or 11, can be used as a valid encryption key e? My working ...
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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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164 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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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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Find plaintext of RSA by solving extended euclidean algorith for two encrptions with two different exponents for same plaintext

This is my homework question (but I am not asking the answer to it): Suppose two users Alice and Bob have the same RSA modulus n and suppose that their encryption exponents eA and eB are ...
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Miller Rabin - Error probability of .5 a possibility?

I'm testing the property of Miller Rabin that the error probability is at most 1/4 when only a single base a is chosen and we iterate only one time. We are testing odd integers 90,000 to 100,000. I'...
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Why does the modulus of Diffie–Hellman need to be a prime?

I read a lot about Diffie-Hellman, but there is one thing I dont understand: why does the modulus p need to be a prime? What if it would not be a prime?
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Why does GnuPG save an array of remainders when generating prime numbers?

In looking at GPG's gen_prime() function, found within the cyphers/primegen.c file of ...
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Cryptographic random numbers for key generation

I am trying to understand how a cryptographic library works (for example, one that provides assymetric encryption such as RSA), but I'm running into a few problems about the key-generation. There are ...
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91 views

ssh-keygen DH Primality Testing

I'm pretty familiar with using ssh-keygen to create groups that go in the /etc/ssh/moduli file for the Diffie-Hellman Group Exchange in openssh. Reading over the man page, it says "By default, each ...
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Algorithm for factoring a number $n$ of a specific form given $n$ and $\varphi(n)$

Given the natural number $n$, which is in the form $p^2 \cdot q^2$ with $p$,$q$ prime numbers. Also $\varphi(n)$ is given. Describe a fast algorithm (polynomial time) that calculates $p$ and $q$. ...
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What if the p and q used in keys generation of Pailler cryptosystem are composite?

I've seen a few implementations of Paillier cryptosystem that uses probable primes to choose $p$ and $q$. Assuming that a keypair is generated with $p$ and $q$ that are coprime and that $pq$ is ...
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Requirements for the modulus in the Massey-Omura three pass protocol

In the Massey-Omura three pass protocol: How many bits long should the prime modulus $M$ be in order to be secure? Should the $M$ be secret? Should the $M$ be generated every time or it could be ...
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625 views

Why should the primes used in RSA be distinct?

The two primes $p$ and $q$ part of the public key need to be distinct. What's the reason for them to be distinct? Is it because factorization of $p^2$ where $p$ is a prime is relatively easier, or is ...
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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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What is the danger if a non-prime is chosen for RSA? [duplicate]

I was reading this question about generating primes for RSA keys. The answers point out that most implementations of of the algorithm use probabilistic prime-ness checking algorithms. The answer by @...
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Property of Multiplicative group of integers mod n

While practising on paper I've realized of a property of multiplicative group of integers mod $n$. First, let's define $G$ being $p$ a prime and $g$ a primitive root mod n or a generator of a ...
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Euler's Totient function for semiprime numbers

I have noticed, during the period I spent studying RSA, that Euler's Totient function can be calculated in another way than $ϕ(N) =(p-1).(q-1)$ Let me explain myself by pointing to a brief example: $...
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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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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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199 views

Why doesn't this defeat RSA?

Apologies for the obviously ridiculous question but I need to know where I'm going wrong here. For RSA, we compute $n=pq$ for primes $p$ and $q$. We then choose an $e$ such that $gcd(e, \varphi(n))=1$...
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If someone had a list of all primes, would it be possible for them to factor any integer in polynomial time? [duplicate]

For example, if they somehow got a function that would churn out any arbitrary amount of primes in a row. Could they break the RSA problem then?
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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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331 views

RSA modulus (N) from public key and calculating N from p, q not equal [closed]

I have a RSA public key in the form of public exponent and modulus as follows: ...
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389 views

Creating a random password based off of a prime number

So I am making an application that basically creates strings that must be encrypted before they are stored on a user's device. If the user blindly starts running the application without creating a ...
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Of what use is my code for finding prime numbers of a certain size?

I've developed a bit of Mathematica code that can find primes within a range of numbers. For example, if I wanted all the primes between one million and two million, it could do that. Of what use is ...
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Why would cryptography fall apart if there were a finite number of primes? [closed]

I vaguely know that prime numbers are very important in cryptography, but I assume for most encryption methods, they tend to stay rather 'small'. Are we really using massive prime numbers for ...
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126 views

Prime modulus for RSA and sharing a secret?

According to this paper entitled "Using Commutative Encryption to Share a Secret" they define their modulus to be a large prime p, which is public. Both exponents are private in this case. According ...
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Proof that $gcd(e, \lambda(N)) = 1 \hspace{1mm} \Longleftrightarrow \hspace{1mm} gcd(e, \varphi(N)) = 1$

What is the proof for the fact that $gcd(e, \lambda(N)) = 1 \hspace{1mm} \Longleftrightarrow \hspace{1mm} gcd(e, \varphi(N)) = 1$ Where: $N = P * Q$ where $P$ and $Q$ are both primes. $\varphi(N)$ ...