A residue of order 2. A number $a$ for which the congruence $x^2 ≡ a \pmod m$ has a solution is called a quadratic residue modulo $m$.

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### What is the restriction on $k$, for the $k$th composite residuosity problem to be hard?

The paper “Residuosity Problem and Its Applications to Cryptography” considers the exponent to be an odd integer. When $k = 2$, it is called the quadratic residuosity problem (mod $n$, where $n$ is ...
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Why doesn't Cocks IBE use the hash function H from ID space to quadratic residue set $\mathbb{QR}_N$ in $\mathbb{Z}/N\mathbb{Z}$ to reduce the ciphertext expansion by half? I think it is also IND-ID-...
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### Help with next step in the Quadratic Sieve

So I am at the same step as someone from math.stackexchange but he never recieved an answer so I will copy-paste his question here: Say, for N = 90283, I compute bound 𝐵=𝑒(12+𝑜(1))(ln(𝑛)ln(ln𝑛√))...
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### Cryptographic Counter

Good morning, I state that I am not an expert in cryptography. I'm studying the feasibility of a project which looks like requires a kind of cryptographic counter that behave similarly to the one in ...
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Good evening, I'm about to write my own quadratic sieve implementation in C using GMP library for large numbers. I'm facing an issue while attempting to do the last factorization step for the number: ...
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### Using quadratic residue to learn the sign of a field element

Given $x' \in \{-x, x\} \bmod q$ (where $q$ could be any prime of my choice), $s$ is a random element in the field, $y = x'\cdot s$ and $y' = \pm\sqrt{(x\cdot s)^2}\bmod q$ (i.e., both solutions to ...
When $n = pq$ where $p$ and $q$ are primes, we can generate random numbers until we get $a$ and $b$ such that $a^2 \equiv b^2 \pmod n$. This implies $n$ has some common factor with $a^2-b^2$, and then ...