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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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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287 views

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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294 views

How costly is to find millions of large prime numbers for RSA?

Consider I need to assign a large distinct prime number to each element in a large set. This must be deterministic so the function always gives me the same prime to the same value. What is the most ...
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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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Modulo properties of two prime numbers

I am supposed to prove that x = y mod (p*q) <=> x = y mod p and x = y mod q with p and q are prime numbers. It somewhat sounds reasonable to me, but unfortunately I don't have any clue how to prove ...
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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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88 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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181 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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188 views

Choosing primes in the Paillier cryptosystem

In the first step of key generation phase in Paillier cryptosystem given here. It's given that $$\operatorname{length}(p) = \operatorname{length}(q) ) \implies ...
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754 views

Algorithm for proving Carmichael numbers

I have an application for determining if a number is a prime or not, currently I'm getting a random number, then doing the Fermat primality testing to find out if the number is probably prime (so this ...
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222 views

ECDSA - point order criterion

i am creating some primitive demostration for ECDSA over small curve (p < 229). But my implementation have some weird issues. Verify process return false even if the signature is correct. Because I ...
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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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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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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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26 views

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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88 views

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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Hill cipher key space

Key space is the set of all possible keys that can be used to generate a key. We using the number of valid key to describe it. I've given a hill cipher of block size $k$ over alphabet of size $p$, ...
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228 views

Best group if one wants the discrete log problem to be hard?

Suppose one is implementing a cryptographic scheme over a group where one needs the discrete logarithm to be hard - what is the recommended group to use? I'm looking for a group where calculations are ...
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739 views

Decrypting a message without the private key using CRT

I am given 5 different encryption modulus, $N$, each ranging from 78 to 88 numbers long. Then for the encryption exponent, each has the same which is 5. Then I am given 5 different encrypted messages, ...
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74 views

Why is it impractical to generate a semiprime dictionary? [duplicate]

This might be a very simple question. However, I am just learning the concept, so just excuse me. I am wondering why there is not any attempt to generate all semiprime numbers? (as an dict. attack to ...
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168 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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135 views

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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277 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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What is the difference between, and security of $Z_p$ and $F_p$?

While reading some papers, I found that sometimes $Z_p$ was used but sometimes $F_p$ was used ($p$ is prime). Usually, the author will only use their subgroups like this: choose a big prime factor of ...
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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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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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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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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. ...
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120 views

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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105 views

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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125 views

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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How can i calculate prime of Elliptic Curve?

In many articles i have found directly the calculation of prime elliptic curve. How can i calculate this prime $p$ ? For example if I consider NIST P-256, $ p = 2^{256}-2^{224}+2^{192}+2^{96}-1$. Why ...
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Solving congruences using PARI

I'm having trouble finding info in the docs about how to solve a system of congruences. The closest I can find is 'matsolvemod' in here: ...
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124 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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151 views

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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306 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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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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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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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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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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120 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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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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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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170 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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From Factorisation of semiprimes to breaching confidentiality

If someone or some group found an efficient way to factor large composites with two distinct prime divisors, would this make it easier to decode any messages?
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How can I convert numbers into prime numbers?

I'm working with one-way accumulators, but I'm not knowledgable in cryptography. Is there an easy peasy way to hash numbers (or whatever) into prime numbers? Obviously I'd like it to be collision ...
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why RSA uses Semiprime numbers? [duplicate]

Why does RSA use semiprime numbers? Why not just use any big number ?? What is the advantage of the two original numbers being prime? Because factoring any big number will be difficult
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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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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 ...