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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### 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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### Compact encoding of an elliptic curve point

I'm working on a project with elliptic curve cryptography (ECC), I'm using the secp256k1 library (the one that's used in bitcoin). My goal is to create the most compact platform-independent ...
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### Is a composition of computational hardness problems still hard?

It is well known that both $g^x$ and $x^2$ are computational hardness problems in certain rings. But I wonder if the composition of them is still hard? Namely, given $(g, g^x, x^2)$ in a ring $Z_n$ ...
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### why doesn't quadratic residue attack work with Elgamal encryption with decisional diffie Hellman assumption?

I was reading this notes http://www.cs.umd.edu/~jkatz/gradcrypto2/NOTES/lecture4.pdf It's given that discrete log assumption is not enough for semantic security, I'm assuming there maybe chance of ...
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### Checking both Quadratic residuosity and Jacobi symbol simultaneously and efficiently

I have to randomly generate a number $u$ such that $u \in J(N)-Q(N)$ where $J(N)$ denotes the set of elements less than $N$, whose Jacobi symbol value is equal to 1; and $Q(N)$ denotes the set of ...
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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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### Algorithm for computing square roots in $GF(2^n)$

Short question: is there an algorithm for efficiently computing square roots in $\mathbb{F}_{2^n}$?
Theorem: Let $y$ be a quadratic residue in $\mathbb{Z}_N$* where $N=pq$. There are exactly four integers $x_1, x_2, x_3, x_4$ where $0 < x_1 < x_2 < \frac{N}{2} < x_3 < x_4 < N$ ...