# 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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### 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 ...
1k views

### 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: http://pari.math.u-bordeaux.fr/dochtml/html.stable/Vectors,...
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### Prime Factorization in RSA always leads to the product of two primes?

Lets prime factorize $30$: $$30 = 3 \cdot 10 = 3 \cdot 2 \cdot 5$$ We see that the number $30$ is a product of $3$ primes. But in RSA, when factorizing huge numbers, we always seem to only get two ...
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### Finding the prime factors of $n'$ given the previous $n$ and its prime factors $p$ and $q$

Say I have a very large number $n$ = $pq$ where $p$ and $q$ are prime. A new number $n'$ is generated by increasing these prime factors. How can I find this new number's prime factors?
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### Trial division + Miller-Rabin complexity

I was reading a paper that shows time complexity of Trial division + Miller-rabin test for creating prime number. T = N(T_rnd + T_td + T_mr) T_rnd denotes time for making rnd number, T_td denotes ...
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### If there is an algorithm A can calculate the modular square root of input n, How to use it to get prime factors?

Suppose you are given an algorithm $A$ which takes $y \in \{0, 1, \ldots , N ā 1\}$ as input, and outputs $x \in \{0,1,\ldots,N ā 1\}$ such that $x^2 \equiv y \pmod{N}$. Design an efficient, ...
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### In practice, do large prime generators account for difference in gaps between primes? [duplicate]

It is my understanding that a method for creating large primes is to: pick a random large number, check the number against trial division of a set of smaller prime numbers, check with something like ...
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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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### 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 @...
269 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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### RSA and Prime numbers distribution law [duplicate]

I have read a few articles and watched some Youtube videos explaining the algorithm of RSA. It seems that RSA is mainly based on a mathematical trick (Prime Factorization). I am wondering though what'...
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### are all elements of ZpxZp in ECC definite over Zp

are all elements of ZpxZp in ECC (elliptic curve) definite over Zp ? otherwise: assume G a base point of ECC and n the order of G. why n is equal or nother to p*p ? (p a prime number). (Think to a ...