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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Is the matrix step of GNFS still the hardest part?

When the factorization of RSA-768 was announced in December 2009: the sieving took about 24 months and the matrix step took 119 days (4 months). So sieving took about 6 times as long. This is despite ...
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Generalized Benaloh cryptosystem with $r=2$

Benaloh cryptosystem requires $\gcd(r, (q-1))=1$ which is impossible if $q>2$ (since it needs to be a large prime) and $r=2$. This confuses me, since Benaloh is referred to as an "extension" or "...
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Problem with Chaum`s Untraceable Electronic Cash

For example according to protocol I need calculate this: $b=F^{(1/h)} \bmod pq.$ Where $p$ and $q$ are prime numbers. I have $F$ and $h$. But how can I calculate $b$? I tried to do this: $\text{...
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232 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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35 views

How large a product out of 3 close-by factors need to be to avoid factorization?

For encryption a prime $P = 2 \cdot Q \cdot R \cdot S +1$ was used. An adversary want to solve the discrete log problem $m \equiv g^i \bmod P$. For this he want to use the Pholig-Hellmann algorithm. ...
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46 views

Question about using residue number system for repeated multiplications

I understand that when you are using RNS you need a co-prime moduli-set e.g. ${\{m_1, m_2, m_3\}}$, and the dynamic range is the product of each modulus in that set $M = m_1.m_2.m_3$. Also it's ...
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484 views

See the RSA's $n = p q, e \text{ and } d$ from ssh-keygen

For learning purposes, how to get the RSA's parameters (I use standard notation): $n = p q $, where $p$ and $q$ are prime and $e$, $d$, such that $ed = 1 \ (mod\ \varphi (n)) $ when doing ...
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77 views

Algorithms to test vs. generate primes

What would be the value in having a true prime number generator algorithm instead of a prime number test algorithm? Also, if such a thing existed, what would be the impact on not only cryptography?
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497 views

How to generate 1000 prime number of 1024-bit with much less time?

I am generating thousand prime number of 1024 bit each. But it takes lots of time. My procedure is as follows. Generate prime number using ...
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3k 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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369 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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1k 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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19 views

Three Pass Protocol Question!

Alice and Bob have agreed to use the Three Pass Protocol. p=1009 Alice chooses the encryption exponent e_A = 101 Bob chooses the encryption exponent e_B = 209 Now Alice and Bob send three ...
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38 views

About the irreducible test for a polynomial over $GF(2^m)$ (its coefficients belong to $GF(2^m)$)

For the Miller-Rabin probabilistic primality test, we can apply it to test a big number whether a probably prime number or not. Does there exist a method to test a higher-degree polynomial over $GF(2^...
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81 views

Yukel's Sieve - Factorization of Numbers into a Square Sieve

https://www.youtube.com/watch?v=liTTGeitpGQ https://www.youtube.com/watch?v=2nOwgiweyqc https://www.youtube.com/watch?v=rGwFsOG27DQ I came across these videos explaining a pattern that is found in ...
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68 views

Finding two primes that satisfy the conditions

I want to implement partially blind signature, and I looked at Zheng Gong's scheme. (Title: Efficient Partially Blind Signature Scheme with Provable Security) In the scheme, there is a initialization ...
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237 views

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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47 views

Deterministic procedure for mapping an arbitrary value into a 𝑝,𝑞 pair for public key cryptography

I want public key cryptosystem to used for re-encryption as describe in Can Paillier ,RSA or any other schemes be used for universal re-encryption like elGamal? Now i have little solution for ...
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170 views

A quick way to justify whether $a^m \equiv 1 \pmod{n}$?

Say I would like to justify whether $10^{28} \equiv 1 \pmod{29}$. I know according to Fermat's little theorem that $a^{p-1} \equiv 1 \pmod{p}$ when $a$ is a primitive root modulo $p$. What about ...
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1answer
144 views

Integer Factorisation

If I have a set of numbers of the form $\{ {kp+r}:k\geq0\}$ with p a prime or product of primes k large in $\in Z^+$ and r fixed, is it computationally feasible to find a factorisation for any one ...