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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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 ...
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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 ...
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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 ...
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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 ...
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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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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 $p$ ...
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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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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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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 ...
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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 ...
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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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Choosing primes in the Paillier cryptosystem

In the first step of key generation phase in Paillier cryptosystem given here. It's given that ( length($p$) == length($q$) )$\implies$ gcd$(pq,(p-1(q-1)))$=1 where length($k$) = # bits in ...
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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 ...
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3answers
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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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1answer
375 views

Security of Pohlig-Hellman exponentation cipher?

I am looking into implementing Pohlig-Hellman exponentation 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. ...
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1answer
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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 ...
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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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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 ...
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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 ...
6
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1answer
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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 ...
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1answer
144 views

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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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 ...
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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 ...