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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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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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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Is it possible to validate a Public Key in RSA?

If I have a 1024-bit number, and someone is telling me that it is in fact a valid RSA public key, is there any way I can quickly validate that it is indeed so (without cracking RSA)? (I suppose I am ...
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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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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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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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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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How are primes generated for RSA?

As I understand it, the RSA algorithm is based on finding two large primes (p and q) and multiplying them. The security aspect is based on the fact that it's difficult to factor it back into p and q. ...
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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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176 views

Is the strength of RSA over quadratic or other cyclotomic fields as strong as over the integers?

If we assume the strength of RSA is based on the difficulty of factoring (which I know we can't guarantee) and we compose the modulus of some other quadratic ring that is a unique factorization domain ...
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Factoring large numbers

I am trying to factor few integers that are each between 115 and 135 digits long. I was wondering if anyone knew of any efficient methods or any programs that I could use to find the two primes $p$ ...
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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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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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Is it safer to generate your own Diffie-Hellman primes or to use those defined in RFC 3526?

I was wondering if the prime numbers defined for use with Diffie-Hellman in RFC 3526 are more trustworthy than generating one's own, especially considering the recent Arjen Lenstra paper (Ron was ...
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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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64 views

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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101 views

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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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, ...
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271 views

Timelock puzzle improvment

I came across this question with this answer about a cryptographic timelock-puzzle that needs approximately 30 years to be solved. There is also an explanation with source code for that puzzle ...
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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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2answers
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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 ...
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Why is it a bad idea to use Diffie-Hellman with a prime such that $p - 1$ is smooth?

I recently stumbled upon another question in which somebody menitoned that using custom primes for DH is safe if you keep in mind, that $p-1$ should not be a smooth number. Why is that?
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phi(P*Q) = (P-1) * (Q-1)

I was trying to understand RSA when I encountered the Euler Function. I do understand this: $\phi(P)$, where $P$ is a prime is $P-1$. However it seems that for a number $N$ such at $N=P\cdot Q$ where ...
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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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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. $\varphi(N)$ ...
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State of the art RSA key generation

I would like to know if there is an algorithm to generate a RSA key at the state of the art of the present cryptanalysis. Beside the key lenght I know there are some weakness in the choice of prime ...
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133 views

Why is factoring $p-1$ easy when $p$ is a safe prime?

A paper states: [...] $(p,g,y)$ is a correct ElGamal public key if $g^x=y\pmod p$. To verify this the order of $g$, and thus the factorization of $p-1$, is needed. This is easy for safe primes ...
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Finding strong primes

Wikipedia lists the following conditions for a prime to be strong: $p-1$ has large prime factors. That is, $p = a_1 q_1 + 1$ for some integer $a_1$ and large prime $q_1$. $q_1-1$ has large prime ...
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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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How much (home PC) CPU time is required to generate a prime number of a given size?

How much CPU time is required on a typical home computer to generate a prime number of size 100 bit, 200 bit , 512 bit and 1024 bit using given random bits of the respective sizes? Please note that ...
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197 views

Sophie Germain primes and safe primes

I am trying to find a list or table of safe prime numbers i.e. the ones that are based on the Sophie Germain primes i.e. $N = 2p + 1$ where $p$ is also prime. All I found till now is this database. ...
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Prime Numbers in Discrete Log

I am implementing a security protocol based on discrete log. I came across the equation $p = kq + 1$. Understand that based on number theories that both $p$ and $q$ should be large enough to be ...
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Why are primes important for encryption

Why are primes so important? Why can't we just use a random number? My guess is that it's because finding a random prime require more computing power, than finding a random number. Can anybody confirm ...
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Significance of 3mod4 in squares and square roots mod n?

Why do most literature while discussing squares or square root modulo a prime P, consider P to be congruent to 3 mod 4?
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Fermats Little Theorem, primitive root [closed]

So I am studying for finals and I am not able to solve the problem: Let $ p = 3 * 2^{11484018}- 1 $ be a prime with 3457035 digits. Find a positive integer $x$ so that $2^x\equiv 3\pmod p$ Any ...
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Generation of strong primes

It seems that this is pretty difficult to find large (above 1024 bits) strong primes, or at least such primes p where (p-1) has a very large prime factor. Is there any information regarding the ...
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Is there a way to systematically calculate the public exponent $e$ in RSA?

I'm learning RSA in one of my classes and we were given a problem: $p = 5$, $q = 11$ I have done the following steps: $n = 5 \cdot 11 = 55$ $\phi = (5-1)\cdot(11-1) = 40$ I know that to ...
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special public keys and modulo n

I just picked up cryptography and have some questions on RSA cryptosystem: Say there are two public keys (n, e1), (n, e2), e1 is coprime to e2. They share the same n. Is it possible to find the ...
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Small Prime Difference in RSA

In RSA, the $p$ and $q$ should be randomly generated, and they are the same size. The difference between $p$ and $q$ should not be small. Suppose that $u=|p-q|<20$ and $p \times q ...
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How does one calculate a primitive root for Diffie-Hellman?

In the Diffie-Hellman key exchange, one of the steps involves calculating a primitive root of a prime number $p$. How would one go about doing so, considering that $p$ could be very large? Is there ...
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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: ...
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In RSA, why does Alice's $N$ need to be relatively prime to Bob's $N$?

I was asked this by my professor and I didn't understand the reasoning behind it. If Alice has a the key pair $(p_a, n_a)$ and Bob has the key pair $(p_b, n_b)$, why do $n_a$ and $n_b$ have to be ...
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Equal length of primes in paillier cryptosystem

In the key generation step of paillier cryptosystem , In order to satisfy $\gcd(pq,(p-1)(q-1))=1$ , we can take equal length primes. Instead of taking(length as parameter to generate $p,q$) equal ...
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How can convert affine to Jacobian coordinates?

I have a point in affine coordinates: $(x,y)$. What should I do when I want to convert to $(X,Y,Z)$ in Jacobian coordinates? I need it for calculating ECC in a prime field.
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What is the use of Mersenne Primes in cryptography

There is an international search for Mersenne Primes. The project is huge. But what is the use of Mersenne Primes in cryptography? Do they have any other properties other than the $2^n-1$ form?
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Compare two approaches for cracking RSA key

I came across these questions while studying for a crypto course, does anyone have any ideas on how to answer these? (a) Random prime numbers of size 1536 bits are chosen to generate an RSA modulus ...