Questions tagged [paillier]

A public-key cryptosystem invented by Pascal Paillier in 1999.

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Paillier Decryption?

Let $c_1$ and $c_2$ two encryptions of $m_1$ and $m_2$ using the Paillier Cryptosystem. $c_1= E(m_1,r_1) = g^{m_1} r_1^n \bmod n^2$ and $c_2= E(m_2,r_2) = g^{m_2} r_2^n \bmod n^2$ Paillier ...
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Proof of lemma 1 Paillier encryption

In the original paper of Paillier, lemma 1 shows why $n$ must divide the order of $g$. What I don't understand in the proof of this lemma is why $g^{x_2-x_1}(y_2/y_1)^n$ implies $g^{\lambda(x_2-x_1)}$....
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Paillier scheme and noise growth

Does the problem of noise growth exist in the Paillier homomorphic scheme ?
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Do any probabilistic hashing algorithms have additive homomorphism?

What I am looking for is a function that meets the following criteria: For each possible input (assume integers from [0, 255]), there must be trillions of possible outputs so as to prevent preimage ...
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Zero knowledge proof for Paillier addition under multiple keys

Suppose $m_0, m_1, m_2 \in \mathbb{N}$ such that $m_0 = m_1 + m_2$, $m_i > 0$ (none of them can be 0 or lower) Under a Paillier cryptosystem, set $e_0 = E(m_0, r_0)$ for a public key $(g_0, n_0)$ ...
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Homomorphic encrypted streams (Paillier)

Situation: Alice (violin) and Bob (drums) play music together and want to (real-time) stream the concert to Carol. In order for Carol to save bandwidth, the stream is sent through a server which ...
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72 views

random mask reversible after homomorphic encryption

I would like to know if this process is feasible under homomorphic encryption, ideally under paillier or any other additive scheme Apply a mask X to obfuscate a message A ie. Am = A (op) X where (op)...
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A modified question of Hazay & Lindell's Efficient Secure Two-Party protocols Book

Based on the question proposed on page 27, we propose a modified question as follows: Suppose the protocol is based on Paillier cryptosystem and $P_2$ has generated related public and private keys ($...
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108 views

Can Paillier ,RSA or any other schemes be used for universal re-encryption like elGamal?

ElGamal, RSA, and Paillier cryptosystem have homomorphic property, and can be used fro re-encryption purposes. I want to use the encryption to re-encrypt ciphertext(as in proxy re-encryption but ...
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Paillier's Cryptosystem - Homomorphism

I'm lacking quite some mathematical knowledge here, but could anyone please explain to me why the Paillier cryptosystem is still (additive/multiplicative) homomorphic despite introducing a random ...
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Paillier Private $\mu$ and $\lambda$

The Paillier CryptoSystem has a public key that $(g,n)$ and the private key which can be exclusive to $\lambda$, where the decryption scheme is: $m = L(c^\lambda \bmod n^2)/L(g^\lambda \bmod n^2) \...
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Order of g in Paillier Scheme

I'm trying to understand the Paillier Scheme but there's something I can't understand in the keyGen algorithm: Ensure $n$ divides the order of $g$ by checking the existence of the following ...
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65 views

Is it possible to calculate the random factor $r$ from a encrypted message and the private key in a Paillier cryptosystem?

I have already done my research and found various sources that state that it is possible but there are also a lot of them that says it is not possible to recover $r$. This Q/A on this site for example ...
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Deterministic procedure for mapping an arbitrary value into a 𝑝,𝑞 pair for public key cryptography

I want public key cryptosystem to used for re-encryption as describe in Can Paillier ,RSA or any other schemes be used for universal re-encryption like elGamal? Now i have little solution for ...
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Paillier Complex Residuosity problem?

Paillier Cryptosystem depends on both the factorization where $n = p.q$ and the complex residuosity problem which is defined in the original paper as: The problem of deciding n-th residuosity, i.e. ...
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285 views

Comparison of values in Paillier homomorphic encryption

For a project, I am using homomorphic encryption with the Paillier cryptosystem, and I have to compare two values... Can this be done using homomorphic encryption? And I know subtraction can be done ...
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What are some disadvantages of homomorphic encryption schemes?

I'm doing some self-teaching / research for my own benefit in homomorphic cryptography. I've studied both additive and multiplicative schemes (Paillier and RSA respectively), but all I can seem to ...
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Weakening of Paillier cryptosystem due to ciphertext equivalence and order in CryptDB

The Paillier cryptosystem is probabilistic in nature and IND-CPA secure. By design given two ciphertexts one cannot distinguish whether decrypting those two ciphertexts will result in same or ...
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Why is the Paillier cryptosystem not considered fully homomorphic encryption? [duplicate]

Paillier is an additive homomorphic encryption system that can achieve the encrypted version of its sums. However, we can calculate the encrypted version of their multiplications by raising an ...
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Compute ln function of a Paillier encrypted value [closed]

If I have an encrypted value $Enc(x)$ with Paillier cryptosystem, is it feasible to compute an encryption form of $\ln(x)$ or its approximation using homomorphic properties? The input $x$ is always ...
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Can Paillier Encryption has independent decryption key?

As Pailliear cryptosystem secret key $\lambda$, depends on primes $p$ and $q$. As $\lambda = \operatorname{lcm}(p-1,q-1)$. I want decryption key to independent from $p$ and $q$. It can be possible ...
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How Paillier cryptosystem can be used practically to encrypt and decrypt big messages “m”?

I want to use the Paillier cryptosystem for encryption and decryption purposes in my research work. But i haven't found a way to encrypt big input messages; As i want to encrypt the message i,e m : <...
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Is there any relationship between Paillier Cryptosystem's random r and other factors

I want to try run an example of Paillier cryptosystem(Algorithm), So i just started with some basic examples, but cannot obtain correct result/decryption. I just change random factor ...
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In which public key encryption algorithms are the private and public key not reversible?

The RSA public key encryption system has the characteristic that the public key and the private key can be reversed. That is, information encrypted with the public key can be decrypted with the ...
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Homomorphic/Paillier crypto system for use case?: overflow for multiple counter exponent possible? Different cipher factor needed all the time?

Recently I read about homomorphic cryptosystem. They might solve a problem. To do this there need to be some modifications from standard version. Using Paillier here but a solution for other also ...
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155 views

Zero-knowledge proof for Paillier parameters

For RSA one can give a non-interactive zero-knowledge proof that RSA with parameters $(e,N)$ form a permutation and a proof of knowledge of the associated RSA secret key. For example, such a proof can ...
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homomorphic division with scaling/Paillier

Let's say that I have to use fractions instead of integers and I am using Paillier cryptosystem. So, I use scaling to obtain integers. Assume that I have a secure division protocol. What happens if ...
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Pseudorandomness based on homomorphic multiplication property of Paillier cryptosystem

Given the instance $(n, g, \lambda)$ of Paillier cryptosystem with $\text{ord}(g) = n \lambda$ (symbols have their usual meaning), and $c = g^{na}$, is it possible to distinguish (computationally) $c' ...
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Prove that some Cyphertext C encrypts some plaintext D

I have a Paillier Cyphertext C and a counterparty that controls the keypair that was used to encrypt the data D to arrive at C. How can they prove to me that the Cyphertext C is actually the ...
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How to calculate random factor in Paillier cryptosystem?

I am currently learning paillier cryptosystem,and have two questions about random r.I use the characteristics of homomorphic addition to obtain the product of two ciphertexts C and the corresponding ...
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Homomorphic properties of Paillier

I'm curious about the homomorphic properties of Paillier. So, basically if I have $\textsf{Dec}(\textsf{sk}, \textsf{Enc}(\textsf{pk}, \alpha) \cdot \textsf{Enc}(\textsf{pk}, \alpha^{-1}))$, I will ...
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Can the CRT speed-up Paillier decryption by more than a factor of two?

In the Pailler cryptosystem, decryption goes $m\gets\displaystyle\left\lfloor\frac {\left(c^\lambda\bmod n^2\right)-1}n\right\rfloor\mu\bmod n$ with $\mu<n$ being a part of the private key just ...
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228 views

Applying Chinese RemainderTheorem and Paillier Homomorphic encryption

I'm trying to optimize the decryption process for Paillier Homomorphic Encryption (PHE) using the Chinese Remainder Theorem (CRT). However, I want to check if there's a different way of applying CRT ...
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Combining share decryption on Paillier threshold scheme

I am trying to implement the Paillier threshold scheme described by Fouque, et al, but I am having an issue when combining share decryptions. The scheme calculates the plaintext $M$ with the formula: ...
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Bijective encryption function in Paillier cryptoststem

In Paillier cryptoststem many ciphertexts can correspond to the same plaintext. How can I modify the scheme so to make the correspondence between ciphertexts and plaintexts a one to one correspondece? ...
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How can I do minus on plaintexts in the Paillier cryptosystem?

We have $E(a)$, $E(b)$ encrypted under the same Paillier key. As we all know, we can get $E(a+b)$ by calculating $E(a)*E(b)$. But can we get $E(a-b)$, by calculating $E(a)/E(b)$? I tried to ...
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Paillier encryption problem when q or p divides r

I am having a problem with Paillier encryption as described on Wikipedia. It says to pick $0 < r < n$, where $n=pq$ for large, equally sized primes $p$ and $q$. However, I've been testing ...
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In a specific Paillier implementation, why is r prime?

I have a question about an implementation of the Paillier cryptosystem. In the description of Paillier above, encryption of a plaintext message $m$ on $\mathbb {Z}_{n^{2}}$, $0\leq m<n$, proceeds ...
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367 views

Paillier: guessing the message when knowing the cipher and the random number

I cannot get my head around this. In Paillier, the ciphertext is calculated using $c = g^m.r^n\ mod\ n^2$ where $(n,g)$ forms the public key and $r$ is a random number $0<r<n$. Assuming an ...
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135 views

Paillier addition with plain text

$A$ sends $B$ the encryption $E_{pkA}(m)$. $B$ computes $R=xE_{pkA}(m) + y$ and sends $R$ back to $A$, but tells him nothing about the parameters $x$ and $y$. $A$ performs $D_{pkA}(R)$ and recovers ...
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Question about Damgård–Jurik crypto system

I am reading the paper about Damgård–Jurik cryptosystem. In the proof, I found this equation $c^d = (g^mr^{n^s})^d = (\boxed{(1+n)^{j m}x^m}r^{n^s})^d = \boxed{(1+n)^{j md\pmod{n^s}}}(\boxed{...
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Verification in Threshold RSA or Threshold Paillier

In the key generation of the threshold version of RSA or the threshold version of the Paillier cryptosystem (e.g. "Shoup - 2000 - Practical threshold signatures" or "Fouque et al. - 2000 - Sharing ...
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The way to calculate $r$ from $c=r^t\mod{n}$ where $(c,t,n)$ is known

I want to know if there is a easy way to calculate $r$ from $c=r^t\mod{n}$ where $(c,t,n)$ is known and $t=pq$ is an RSA? If $n=t^2$, is it more easier?
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Is there any way to prove two numbers that are equal after Paillier encryptions?

I have two numbers $x_1$ and $x_2$, and there are two Paillier homomorphic encryption (public, private) key pairs $(p_1,r_1)$ and $(p_2, r_2)$. I only know $p_1, r_1$ and $p_2$. Suppose $C_1=E_{p_1}(...
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Paillier's scheme generalisation

Paillier's scheme assume has message and ciphertext space equal to $\mathbb{Z}_N$ with $N=pq$, that is $N$ is the product of two different primes. Is there a way to generalise this for $N$ that is ...
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Is it possible to manage an encrypted dataset for face recognition?

Im in my final year of PhD in computer vision and my supervisor has given me a task that I am not very familiar with. So I am teaching myself homomorphic encryption everyday. This question is mostly ...
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388 views

Calculating distance between two vectors using Paillier homomorphic encryption [closed]

I am basically trying to perform euclidean distance calculation in the encrypted domain (Paillier encryption). Using the homomorphic properties of Paillier, the squared euclidean distance formula can ...
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190 views

How to compute Lambda in a Threshold Paillier scheme

I am evaluating a threshold Paillier scheme as described in the paper: Ivan Damgard, Mads Jurik, Jesper Buus Nielsen, "A Generalization of Paillier’s Public-Key System with Applications to Electronic ...
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242 views

Paillier subtraction for negative result

I am trying to figure out subtraction on Paillier. From what I read so far, given $m_1$ smaller than $m_2$ ($m_1<m_2$) I can compute $E(m_2-m_1)$ as $E(m_2)\cdot E(m_1)^{-1}$ where $E(m_1)^{-1}$ ...
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Is Paillier secure from known plaintext attack for single character message?

Assuming I have three messages m1,m2,m3 where m1=m2=m3=1 and I compute c1,c2 and c3. Does that mean that c1=c2=c3 in cipher-text from ? If not, how many times can I encrypt a message m=1 and still ...