# 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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### 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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### 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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### 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 ...
### 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 ...
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 ...