Tagged Questions

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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2
votes
2answers
49 views

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 ...
1
vote
0answers
35 views

What is the restriction on k, for the kth composite residuosity problem to be hard

This paper considers the exponent to be an odd integer. When k = 2, it is called the quadratic residuosity problem (mod n where n is composite) which is hard and can be solved if the factorization of ...
4
votes
3answers
144 views

Algorithm for computing square roots in $GF(2^n)$

Short question: is there an algorithm for efficiently computing square roots in $\mathbb{F}_{2^n}$?
4
votes
2answers
394 views

Exactly two of the four roots must be greater than N/2

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 $0 < x_1 < x_2 < \frac{N}{2} < x_3 < x_4 < N$ ...
1
vote
1answer
40 views

Computational Diffie-Hellman problem over the group of quadratic residues

Suppose that $N=pq$ where $p$ and $q$ are safe primes. $\mathbb{QR}_N$ is the group of quadratic residues which is a cyclic group with order $\frac{\phi(N)}{4}$. Let $g$ be the generator of ...
5
votes
2answers
73 views

What does it mean that $BW_N$ is a permutation over the squares mod N?

Let $BW_N$ be a function such that $BW_N:\mathbb{QR}_{N} \mapsto \mathbb{QR}_{N}$ and let if be defined as follow: $BW_N(x) = x^2 \pmod N$ where $N=pq$ and p and q are primes and $p=q=3 \pmod 4$. I am ...
3
votes
2answers
82 views

Possible to check if $a \in \mathrm{QR}_n$?

It is possible to check $a \in \mathrm{QR}_p \text{ iff } a^{(p-1)/2} \equiv 1\ (\bmod\ p)$ if $p$ is a prime. $n$ is a large RSA modulus. Is it also possible to check if $a \in \mathrm{QR}_n$ if the ...
1
vote
1answer
57 views

Are those two distributions indistinguishable?

The Decision composite residuosity problem problem states that is impossible two distinguish between those two ensembles: $\{x^N \mod {N^2} | x \in \mathbb{Z^*_{N{^2}}}\}$ and $\{r \in ...