Questions tagged [quadratic-residuosity]

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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Given a deterministic oracle that calculates square roots modulo n, factor n

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 ...
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Question on the Quadratic Residuosity Assumption

I am reading the Handbook of Applied Cryptography and on page 99 the authors write , after showing that $QRP \le_P FACTORING$: It is believed that the $QRP$ is as difficult as the problem of ...
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What makes the quadratic residuosity problem hard?

The quadratic residuosity problem is the problem of determining whether, for given $r$, $m$, $\exists a.a^2\equiv r\mod m$. This problem's believed to be hard to solve in general (e.g. an efficient ...
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Understanding $\varepsilon$-advantage in cryptographically secure deterministic RNG

In Efficient and Secure Pseudo-Random Number Generation by Vazirani & Vazirani, it is stated that every pseudorandom number generator which satisfies the XOR Condition can securely output $\log n$ ...
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Difficulty understanding 'random square root' in enhancement to fiat-shamir

This was published in 1986 and I'm trying to reproduce it in an assignment. It's a small variation on fiat-shamir, by the original author, which does away with a public key (and supposedly drastically ...
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About Cocks IBE

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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Elgamal problem on $\mathbb{QR}_p$ with $p$ a safe prime

I need some orientation to solve the following problem: Let $p = 2q+1$ be a safe prime and $s(x)$ the smallest of the two square roots of $x$ modulo $p$. Then: Determine the distribution of $s(g^{ab}...
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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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58 views

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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Why work in a subgroup for Naor and Pinkas oblivious transfer?

In section 4 (protocol 4.1) of the paper by Naor and Pinkas [1], why did the authors decide to operate in a subgroup? When they say "the messages are in the subgroup" does that mean $x, y, z_0, z_1$, $...
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Quadratic residue zero knowledge proof - simulator with identical distribution

I am looking at the zero knowledge proof for quadratic residues and am confused when it comes to showing a simulator that can give a transcript of the proof with the same distribution as the proof ...
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Elliptic Curve Cryptography calculation of $y^2 \equiv x^3 + x + 1 \pmod{23}$

Learning the basics of elliptic curve cryptography. The question is a mathematical one. While finding the points in the elliptic group $E_{23}(1,1)$,this is how one proceeds : How is $y^2= 7$ ...
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Issue Factorization Quadratic Sieve

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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In the Quadratic Sieve, why restrict the factor base?

In the Quadratic Sieve, when factoring a number $N$, many descriptions and most implementations select as the factor base the set of small primes $p_j$ less than some bound $B$ restricted to having ...
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Get $a$ such that quadratic residue has a solution (Rabin)

My task is to implement Rabin signature. I have trouble with choosing padding a such that $$x^2 \equiv a \pmod n$$ has a solution. In that context, $n=p\cdot q$ is composite, where $p$ and $q$ are ...
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Decrypting an RSA message given $a^2 \equiv 1 \pmod n$

I need help with a practice problem for an upcoming test. I've learned the answer to the problem is "well done", but don't know how to get there. Any help is greatly appreciated. Suppose that the ...
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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 ...
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Working on subgroup of $\mathbb{Z}^*_p$ in practice

It is said that, given a group $\mathbb{Z}^*_p$, we can always have a subgroup whose order is prime. To this end, for a safe prime $p=2q+1$, compute $x_i^2 \bmod p$ for all $x_i \in \mathbb{Z}^*_p$. ...
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How to prove the hardness of Rabin's function?

I am unable to prove the following theorem: If for a $1/(\log(n))$ fraction of the quadratic residues $q\pmod n$ one could find a square root of $q$, then one could factor $n$ in random polynomial ...
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For which RSA moduli, precisely, is $e=d$ for all $e$?

This question shows that there are at least two valid RSA moduli $n$, namely $35$ and $91$, such that for any $e$ coprime to $\lambda(n)$, $$e^2\equiv1\mod\lambda(n)\text.$$ Reading the linked ...
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Cocks IBE Scheme: why is -a a quadratic residue mod n?

In the Cocks IBE scheme it is required for the hash function, that the Jacobi symbol of its output and the universally available moduls $n = p*q$ is $+1$, so: $\Big(\frac{H(ID)}{n}\Big) = \Big(\frac{...
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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 $\mathbb{F}_{2^n}$?
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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 $0 < x_1 < x_2 < \frac{N}{2} < x_3 < x_4 < N$ ...
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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 $\mathbb{...
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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 ...
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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 ...
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Are those two distributions indistinguishable?

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

How can one show how to reduce the quadratic residuosity problem to an integer factorization?