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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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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348 views

RFC 3526 - What does pi mean?

In RFC 3526 there are a series of primes listed as standard parameters used for Diffie-Helman. The primes are list in two formats. One is the long format, where the number is given in hex. For example:...
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104 views

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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344 views

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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643 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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73 views

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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738 views

ABC Conjecture's Impact on RSA Encryption

A recent proof of the ABC Conjecture has been released by one Shinichi Mochizuki. Now, I'm not well versed in mathematics but it would appear that this proof implies that finding prime factors could ...
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270 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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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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95 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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421 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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460 views

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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627 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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2k views

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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156 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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212 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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182 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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65 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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99 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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167 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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392 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 ...
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223 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 $...
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501 views

Prime number theorem - RSA

I am having trouble understanding the prime number theorem. As part of some revision for an exam, I am trying to answer the following questions (but seeing as I don't understand the concept of the ...
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490 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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356 views

Choosing good parameter for Lenstra's elliptic curve factorization

In Wikipedia, there is an article explaining Lenstra's factorization algorithm. As far as I got it, we choose some $e \in \mathbb{N}$ and a point $P$ on the curve and then calculate $eP$. While ...
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74 views

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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91 views

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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47 views

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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520 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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352 views

RSA: What happens to restrictions of plaintext n, dependent on p and q?

Ok - i will try to ask my question as clear as possible. Im getting a little deeper into the RSA-cryptosystem. At one point i'm a little confused. We have a plaintext $x$ and ciphertext $y$, with $x ...
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1answer
53 views

How can I evaluate the congruency of an AKS primality test?

Despite the fact primality test is a mathematical issue, it plays a part on the security of many cryptosystems such as RSA. I was trying to understand how it works until I came to the following ...
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208 views

Proof of work for determining whether a number is prime?

I have an idea for a system that would be outsourcing some brute force calculations to many users in hopes of finding divisors of a number. However, there is a possibility that a given number would be ...
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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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106 views

Are there public $p$ and $q$ numbers for use in DSA?

There are many RFC documents giving large primes to use in Diffie-Hellman. However, I couldn't find standards on the $p$ and $q$ large primes used in the DSA signature scheme. This is proving to be a ...
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182 views

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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173 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 (i....
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645 views

Is there an algorithm for factoring N, which is just as simple as this one, but faster?

I found a simple algorithm for factoring semiprime numbers, you can read about it in Factoring Semiprimes and Possible Implications for RSA. It basically works like this: You reverse the digits in $...
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291 views

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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440 views

why AKS is so slow in practice? [closed]

This note (http://maths-people.anu.edu.au/~brent/pd/primality4.pdf) states that AKS is not practical. However, it is known that AKS runs in polynomial-time, and I cannot understand where the slowness ...
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130 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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151 views

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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449 views

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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204 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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904 views

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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1k 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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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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126 views

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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410 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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299 views

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 ...
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219 views

Are the RFC3526 MODP groups Schnorr groups?

I was wondering if a group like the 1536-bit MODP Group from RFC 3526 was a Schnorr group? A Schnorr group must apparently have: $p$ and $q$ being primes $p = q\cdot r+1$ $1 < h < p$ $h^r\...