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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87
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6answers
80k views

How can I generate large prime numbers for RSA?

What is the currently industry-standard algorithm used to generate large prime numbers to be used in RSA encryption? I'm aware that I can find any number of articles on the Internet that explain how ...
68
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5answers
31k views

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. ...
32
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6answers
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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 ...
25
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6answers
20k views

Why does RSA need p and q to be prime numbers?

Despite having read What makes RSA secure by using prime numbers?, I seek a clarification because I am still struggling to really grasp the underlying concepts of RSA. Specifically, why can't we ...
25
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5answers
21k views

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 ...
25
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3answers
1k views

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 ...
24
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1answer
14k views

In layman's terms, how does Shor's algorithm work?

I've just been reading up on Shor's algorithm, and I find it both fascinating and baffling. I don't really understand much about it, other than that it can factor semiprimes in polynomial time. Could ...
23
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1answer
4k views

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 ...
21
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3answers
18k views

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 ...
17
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2answers
3k views

Can I select a large random prime using this procedure?

Say I want a random 1024-bit prime $p$. The obviously-correct way to do this is select a random 1024-bit number and test its primality with the usual well-known tests. But suppose instead that I do ...
17
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2answers
1k views

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, \...
16
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4answers
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Is Wiener's attack on RSA extendable to larger keys with low hamming weight?

Using small private exponents with RSA improves performance. However, it has been shown (Wiener, 1990) that if $\log d \leq \frac14 \log N$, the private exponent $d$ can be reconstructed from the ...
16
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4answers
10k views

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 ...
16
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2answers
32k views

What makes RSA secure by using prime numbers?

I am just learning about the RSA algorithm. Looking at the first two steps: Choose two distinct prime numbers $p$ and $q$. Compute $n = pq$. I have some probably stupid questions: Why do $p$ and $...
15
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2answers
2k views

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: ...
15
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4answers
7k views

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 ...
15
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1answer
3k views

Necessity for finite field arithmetic and the prime number p in Shamir's Secret Sharing Scheme

Shamir's original paper (PDF, 197kb) describing a threshold secret sharing scheme states: To make this claim more precise, we use modular arithmetic instead of real arithmetic. The set of ...
13
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3answers
2k views

Is it reasonable to assure that p-1 and q-1 aren't smooth?

I came across the requirement that, in RSA, $p-1$ and $q-1$ shouldn't be smooth, shouldn't consist of lots of small factors. Therefore my question: How complicated is it to check whether $p-1$ is ...
11
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4answers
9k views

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 ...
11
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1answer
191 views

Is it easy to factorize a number of the form $n = t^{2} \cdotp p$?

Is it easy to factorize a number of the form $n = t^{2} \cdotp p$, where $t$ and $p$ are large primes?
10
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4answers
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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?
10
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2answers
11k views

How large should a Diffie-Hellman p be?

In a Diffie-Hellman exchange, the parties need to agree on a prime p and a base g in order to continue. Assuming some ...
10
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1answer
5k views

Why does the PKCS1 RSA private key structure contain more than just exponent and modulus?

The ASN.1 spec for the PKCS1 RSA private key format is as follows: ...
10
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3answers
5k views

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 ...
9
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1answer
6k views

Generating Random Primes

Although this has been extensively discussed around here, I'm curious whether my approach makes sense, or I should just stick to "the standard version". I'm implementing some homomorphic encryption ...
9
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2answers
2k views

Why are elliptic curves constructed using prime fields and not composite fields?

I come across this: Numbers mod composite number does not form a field rather it forms a ring and every number has a multiplicative inverse under integer mod prime Maybe these are the reasons ...
9
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2answers
1k views

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 ...
8
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2answers
1k views

Is there a pseudo message that will encrypt and decrypt correctly if one of the primes is a pseudo prime in RSA

For prime number generation, one can use a probabilistic prime number generation algorithm like the Miller–Rabin primality test that will yield a composite as a probable prime with probability $\frac{...
8
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2answers
1k views

Trial divisions before Miller-Rabin checks?

I'm trying to understand prime number generation (more correctly, the primality checking) as described in Handbook of Applied Cryptography. The context is circa pages 145 - 150, and specifically ...
8
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2answers
2k views

How to better generate large primes: sieving and then random picking or random picking and then checking?

I'm writing an RSA algorithm, and am wondering what is the best and/or usual way to choose the initial prime numbers (p and q). I know of two methods to achieve this, one based on a prime number ...
8
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1answer
439 views

In RSA, rationale for prime $p$ with $p-1$ having prime factor $u$ with $u-1$ having large prime factor?

In the 1978 RSA paper, it is recommended, among other things, to choose primes $p$ such that $(p-1)$ has a large prime factor $u$. This was motivated by Pollard's p-1 algorithm. Further, the authors ...
8
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1answer
239 views

Solving hard problems in $\mathbb Z_{p}^{*}$ when $\mathbb p$ is close to $\mathbb 2^{n}$

Suppose, for some security parameter $n$ you choose a prime $p$ such that $p = 2^n+c$ for some relatively small $|c| < 2^m << 2^n$. I have seen such primes being called Pseudo-Mersenne Primes ...
7
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2answers
2k views

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 ...
7
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1answer
410 views

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 ...
7
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1answer
213 views

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 ...
7
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1answer
137 views

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 ...
7
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3answers
5k views

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?
6
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2answers
2k views

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 =...
6
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2answers
3k views

Does the secret key used in AES have to prime?

I’ve looked around this site and the web quite a bit, but can’t find a definitive answer on whether or not the secret key $k$ used in the AES crypto-system has to be a prime number? Or can you just ...
6
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1answer
1k views

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. ...
6
votes
2answers
4k views

What exactly is inside a private key?

May sound stupid to many, but I would like to have some pointers on what exactly is contained inside a private key. I have decent understanding of public/private keys/certificates (have created them ...
6
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2answers
655 views

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)?
6
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1answer
700 views

Is it possible to fool Miller-Rabin test?

It is well known that it's possible to fool Fermat test with Carmichael numbers. But, is it possible to deliberately fool many-rounded Miller-Rabin test by constructing some special number without ...
6
votes
1answer
286 views

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 ...
6
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1answer
394 views

How are trapdoor functions developed/found and where can I find existing ones?

Trapdoor functions are a fundamental part of public key cryptography. An example of the most common trapdoor is Prime Factorization, used in cryptosystems such as RSA How are these trapdoor ...
6
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1answer
409 views

Why does Schnorr's Digital Signature scheme necessitate two prime numbers?

One of the necessary components to the Schnorr Digital Signature scheme is a pair of prime numbers $p$ and $q$ such that $q$ divides $p-1.$ However, there is never a modular inverse taken of q so why ...
6
votes
1answer
170 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 ...
5
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3answers
2k views

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)
5
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2answers
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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 ...
5
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2answers
1k views

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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