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 $m$.

53 questions
Filter by
Sorted by
Tagged with
51 views

• 1
94 views

Properties of Sums of Legendre Symbols

Context An unknown modulus N with 8 unknown prime factors $p_1, p_2, p_3, p_4, p_5, p_6, p_7, p_8$ a plaintext $m$ is encrypted with the formula $c = 2^m \mod N$ the only things the attacker know are ...
1 vote
20 views

Does exist an Elliptic analogue of Benaloh encryption scheme?

The definition of Benaloh encryption scheme can be found here. Does exist an elliptic analogue of this scheme? I want to use this scheme but the length of the key ...
• 187
1 vote
50 views

Hiding and binding property of Goldwasser-Micali like bit commitment scheme

Let $N=pq$ be an RSA modulus, that is, $p$ and $q$ are large, distinct primes. Let $J_{N}=\{y\in\mathbb{Z}^{*}_{N}:(\frac{y}{N})_{J}=1\}$ denote the set of all integers in $\mathbb{Z}^{*}_{N}$ with ...
• 438
55 views

exponent bit-length for hard DL (128-bit security)

Following up on my previous post, I thought I might get a more concrete answer if I gave a more concrete question. I require 128-bit security so I choose a 3072-bit RSA modulus ($\ell_n=3072$). ...
1 vote
29 views

Honest verifier zero knowledge property for this protocol

This is zero-knowledge proof that show x is not a quadratic residue. I am trying to verify Honest verifier zero knowledge property. My steps were these: Let S be a simulator that does not know how to ...
• 173
95 views

Hardness of DL in group of Quadratic Residues (product of safe primes)

A protocol I am working with requires $\ell_n$-bit RSA modulus and $\ell_\Lambda$ such that computing $\ell_\Lambda$-bit discrete logs is hard in $QR_n$ (technically $n$ is $\ell_n+2$ bits in the ...
141 views

Take n = 4633 and B = {−1, 2, 3}. Note the b-smooth numbers as {67, 68, 69}. Find the factor of n

This question is from Quadratic Sieve Factorization Method. Didn't find the solution on the web also. And not aware of how to solve such questions.
54 views

QR-PKE not CCA secure

Due to a comment stating "... QR-PKE is secure (CPA)..." I've been thinking of how to prove that it's not CCA secure, and would like to understand whether my proof is correct. Here's the QR-...
• 403
131 views

Can functional encryption encrypt and decrypt negative integers?

I want to use DCR-based functional encryption for encryption and decryption purpose. However, I'm unsure whether DCR-based functional encryption can support negative integers. Is there any ...
• 21
196 views

Quadratic Sieve: Sieving with prime powers

I am trying to understand the Quadratic Sieve algorithm. Currently I am stuck at the sieving part. Let's say the number to be factored is 9788111. I decide to look for 50-smooth factors. My initial ...
• 2,201
1 vote
196 views

How to prove correct decryption in Goldwasser-Micali cryptosystem

In How to prove correct decryption in Paillier cryptosystem, it was asked whether Alice (in sole possession of the secret key) can convince Bob that a given plaintext is the decryption of a ciphertext ...
• 11
148 views

How the coin flipping protocol prevent Alice from generating $n$ from many primes?

This is a question from reading the paper 'Coin Flipping by Telephone - a protocol for solving impossible problems'. The fact that the coin is unbiased is based on the fact that if n is a product of ...
• 131
1 vote
726 views

Adversary for attack on one variant of ElGamal

I came by the following question: Consider the following variant of ElGamal encryption. Let $p= 2q+ 1$, let $G$ be the group of squares modulo $p$ so $G$ is a subgroup of $Z_p^*$ of order $q$, and ...
• 41
1 vote
535 views

ZKP for product of two primes

I'm struggling to understand the intuition of the zero knowledge-ness of this proof from the following paper. The proof is a 2 round where the verifier asks the prover to extract square roots of ...
• 209
1 vote
537 views

How do Quadratic Residues work and how do they relate to Diffie Hellman?

Can someone explain Quadratic residues to me in english? I keep reading forums with all the math symbols and it's hard to follow. And how are they incorporated in Diffie-Hellman? Is there a difference ...
1 vote
46 views

Is there a quick method of listing certain elements of a cyclic group?

I'm studying applied cryptography and stumbled upon the following question to practice the knowledge about Congruence, Groups etc. "List all Elements $x$, where $x^2 = 2$ in $\mathbb{Z}_{31}$ Okay, ...
• 13
1 vote
518 views

Is it possible to perfom quadratic residues attack on elgamal cryptosystem?

I understand how this attack work on mental poker, but i am unable to see how can i apply it in Elgamal.
• 13
1 vote
31 views

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√))...
• 111
1 vote
212 views

Identify the cryptosystem where $\ m = c^2 \bmod n$?

I came across with cryptosystem whose decryption method is: $\ m = c^2 \bmod n$. It's exact opposite of Rabin's, where's the same formula is used for encryption. What is the name of this ...
• 25
89 views

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 ...
92 views

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 ...
• 617
858 views

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 ...
• 191
1 vote
158 views

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 $\log n$ ...
58 views

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 ...
110 views

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-...
353 views

• 1,365
573 views

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 ...
• 225
552 views

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 $y^2= 7$ ...
• 3
1 vote
188 views

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: ...
• 11
2k views

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 ...
• 141k
1 vote
607 views

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 ...
• 121
302 views

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 ...
1 vote
83 views

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 ...
• 101
360 views

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 $x_i \in \mathbb{Z}^*_p$. ...
• 835
184 views

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 ...
143 views

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 ...
1 vote
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 ...