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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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 ...
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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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178 views

RSA example-calculation: Public Key = Private Key (e = d)

I am a bit confused. I just calculated manually the single steps of RSA for an implementation with small numbers and suddenly $d$ was equal $e$. Please help me understand what I am doing wrong. ...
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53 views

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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190 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 ...
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145 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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86 views

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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115 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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114 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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233 views

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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2answers
459 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 ...
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46 views

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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60 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 $p = q\cdot r+1$ $1 < h < p$ ...
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193 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 ...
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107 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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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 ...
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288 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 ...
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82 views

Modular exponentiation with Chinese Remainder Theorem

I'm learning modular exponentiation with Chinese remainder theorem. I found a great answer from below How can I use eulers totient and the chinese remainder theorem for modular exponentiation? But I ...
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151 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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114 views

Generating Diffie-Hellman parameters efficiently

I am working on an Android project for school and I am supposed to do a DHKE (Diffie Hellman Key Exchange). Everything works well. The problem is that it takes a lot of time (really a lot) to ...
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326 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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2answers
157 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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705 views

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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130 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 ...
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171 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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209 views

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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143 views

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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137 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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306 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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66 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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161 views

Would this program be useful in cryptography?

I know nothing of encrypting. I'm not even sure how to tag this. I wrote a program that can calculate this pretty quickly on my macbook pro 2.3GHz IntelCore i7. The two exponents are Mersenne primes, ...
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175 views

Factoring large numbers

I am trying to factor few integers that are each between 115 and 135 digits long. I have just, little over a month ago, began my study of Cryptography. I was wondering if anyone knew of any efficient ...
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1answer
129 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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762 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 ...
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168 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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44 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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108 views

Why are huge prime numbers important in cryptography?

I read an article the other day about the search for prime numbers. According to the article and several online sources the biggest prime number is over 17 million digits! This made me wonder why ...
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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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How to calculate y value from ((y*y) mod prime) efficiently

i am working ECC-224 bit. can any one tell me, how to calculate y value from ((y*y) mod prime) efficiently for large bit numbers.
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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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138 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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130 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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176 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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311 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 ...
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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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162 views

Can you help me clear a confusion about small subgroup attack on HMQV?

First,i want to show you with a picture how the HMQV works. There are some notations you might not familiar, it doesn't matter. I just want to show you the procedure. Next it's an attack on HMQV ...
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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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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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228 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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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 ...