Questions tagged [paillier]

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

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42 views

can we know the sign of a ciphertext (homomorphic)

Is it possible to know the sign (positive or negative) of an homomorphic ciphertext particularly under paillier scheme ?
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Possibility of computing a and b values from the ciphertext?

Using paillier encryption, $N$ is the product of two large prime numbers, $s$ is sampled randomly from $Z_{N^2}$ we get $ C \leftarrow g^ms^N \bmod N^2 $ where $g=1+N$, By multiplying the cipher $c$ ...
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Elliptic curve subgroup with $p$ elements in field of characteristic $p$

Are there any elliptic curves defined over a finite field $\mathrm{GF}(p^k)$ with a subgroup of order $p$ where the discrete log (and preferably DDH) problem is hard? Elliptic curve with prime ...
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Identify the value of the plaintext from a noised expression

Having the following instructions: Using Paillier encryption to encrypt $m$. So, we get $Enc(m)$ Multiply $Enc(m)$ and $A$ to Get $C$. So, $C = Enc(m).A $ Decrypt $C$ using Paillier Decryption ...
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Can a partially homomorphic encryption scheme be made CCA1 Secure? [duplicate]

Can a partially homomorphic encryption scheme be made CCA1 Secure? like paillier ? what is the highest security for any PHE?
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How to store the ciphertext generated by the paillier encryption in a pixel?

Goal: I want to implement a paillier algorithm to encrypt images and hide some data reversibly. This is the paper that proposed this algorithm: https://www.sciencedirect.com/science/article/abs/pii/...
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74 views

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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130 views

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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51 views

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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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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82 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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44 views

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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90 views

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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115 views

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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61 views

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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70 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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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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49 views

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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116 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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32 views

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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121 views

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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445 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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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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162 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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111 views

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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125 views

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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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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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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161 views

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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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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242 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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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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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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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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368 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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145 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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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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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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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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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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432 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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128 views

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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195 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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298 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 $E(m_1)^{-1}$ ...