# Questions tagged [paillier]

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

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### Efficient multiplication modulo a square

Can anyone point me to techniques for efficient computation of modular multiplication/exponentiation modulo a square, as comes up, e.g., in the context of Paillier encryption? The standard references ...
65 views

### How to calcualte the average of several plaintexts with Paillier homomorphic encryption?

I am trying to do some calculations with the Paillier Homomorphic encryption scheme. Specifically, given three plain text integers, x1, x2,and ...
1 vote
55 views

### Threshold Paillier encryption key generation time

I have used the threshold version of Paillier encryption without a trusted dealer in an application. I have tested the key generation phase with different security key sizes such as 80, 512, 1024, and ...
264 views

### Generating suitable prime numbers for Paillier key pair in GG18

I am working on MPCs (multi party computation) in crypto, and now I am developing a implementation of GG 18. In sign phase, algorithm needs MtA (Multiplicative to Additive) and uses a Paillier key ...
63 views

### To prove equality/inequality of plaintexts of 2 ciphertexts encypted under different encryption schemes

We have 2 ciphertexts, one encrypted using Paillier and another encrypted under Elgamal encryption schemes. Is there a way to design ZK-proof to prove equality of the underlying plaintexts of these 2 ...
43 views

### Variant of CCA security for Paillier with blinded decryption oracle

Consider a variant of the Paillier encryption scheme where the message space is restricted to $\mathbb{Z}_q$ such that the RSA modulus $N$ of the Paillier cryptosystem satisfies $N > q + q^2$. I am ...
1 vote
103 views

### Reencrypting in Homomorphic Encryption

I am getting started with Homomorphic Encryption libraries. I am trying to port a codebase written in Python to Haskell. I see that there is a re-encryption operation happening in one part of the code....
156 views

### Paillier cryptosystem break with random number

In the Paillier cryptosystem we choose $n=p\,q$ where $p$ and $q$ are primes, $g=n+1$, $\lambda=\phi(n)$, $\mu=\lambda^{-1}\bmod n$. The public key is: $(n,g)$. The private key is: $(\lambda,\mu)$. ...
101 views

### Zero Knowledge Proofs for Microsoft SEAL (Homomorphic Encryption)

I am working on a system involving multiple parties performing homomorphic cryptographic operations using Microsoft SEAL (BFV). Because of the nature of the system, it would be preferable for the ...
160 views

### How to proof Two CipherTexts holding same value with Zero Knowledge Proof?

I am a beginner in cryptography. I wonder is it possible to create by Zero Knowledge Proof, to prove Two cipherTexts that are encrypted by same Public Key with Paillier Encryption have the same value ...
31 views

### Question about homomorphic property of Paillier cryptosystem breaks after modulus n are taken

Hi I am watching this slide about threshold signature: http://cyber.biu.ac.il/wp-content/uploads/2021/11/Threshold_Sinature_Schemes_Rosario_Gennaro.pdf And in page 39 it said that the homomorphic ...
79 views

### How to factorize $N$ in OU cryptosystem under chosen ciphertext attack?

OU cryptosystem: $N = p^2q$, the secret key is the factorization of $N$, the public key is $g\leftarrow \mathbb{Z}_N$, $g^{p-1} \neq 1\mod p^2$, to encrypt an element $m$ $\in$ $\mathbb{Z}_p$, ...
50 views

### Sigma protocol for proof of encrypted content

Suppose I have a device which encrypt my fix plaintext (On request) using Paillier algorithm. So this device every time generate a new random number (used in Paillier algorithm) and a new ciphertext (...
1 vote
32 views

### Paillier cryptosystem for verifiable shuffles

I am looking for some algorithm or implementation on Pailler cryptosystem-based verifiable shuffles for mixNet. So far, all the verifiable shuffles and mixNet are available for ElGamal cryptosystems ...
40 views

### How to prove an RSA ciphertext matches a Paillier ciphertext?

Suppose I know an RSA public key $(n,e)$ and I create two ciphertexts: An RSA ciphertext $C_1 = m^e \mod n$ and a Paillier ciphertext $C_2 = g^m \cdot r^n \mod n^2$. Is there a known efficient method ...
1 vote
79 views

### Is it possible to use Batching with any of the Partial Homomorphic Cryptosystems?

I'm familiar with the concept of batching in the context of (Fully) Homomorphic Encryption - whereby many values can be encrypted as a single ciphertext and operated on simultaneously in an SIMD/...
1 vote
85 views

### Additive Homomorphic Encryption scheme with very small ciphertext size

I'm looking for a Additive Homomorphic Encryption scheme which can allow a ciphertext of size smaller than 128 bit. I started studying this topic very recently so I don't have much knowledge, but I ...
1 vote
53 views

### Can I get individual data by paillier decryption if we send aggregated encrypted data?

Suppose there are different users data for $n$ Smart meters. We encrypt all the individual data and then add them together. After decrypting the summation, how can we retrieve the original data? We ...
1 vote
111 views

### Working with Paillier and ECDSA - Order issue

I'm trying to implement two party computation for ECDSA signing using Paillier cryptosystem. But my problem is that the order of Paillier is different from the order of the curve (secp256k1 in my case)...
155 views

https://en.wikipedia.org/wiki/Paillier_cryptosystem Paillier cryptosystem exploits the fact that certain discrete logarithms can be computed easily. If I were to select $g \in \mathbb{Z}_{n^2}^*$ ...
1 vote
35 views

### Internal direct product of group of invertible elements in a Paillier modulus

Let $p$ and $q$ are Sophie-Germain primes such that $p=2p'+1$ and $q=2q'+1$. Also let $n=pq$ and $n'=p'q'$. In Section 8.2.1 of this paper, the internal direct product of $\mathbb{Z}_{n^2}^*$ is shown ...
200 views

### How to prove that paillier encryption is positive (zero-knowledge)? [closed]

Is it possible that the plaintext encrypted in a ciphertext using paillier encryption is positive without using a zero knowledge range proof?
101 views

### Is this a safe zero knowledge proof that two paillier encryptions are equal?

We have encryptions $c_1$ and $c_2$, the person who knows the plaintext and randomness in both wants to prove that they know it. Let $r_1$ and $r_2$ be the randomness values in $c_1$ and $c_2$ ...
48 views

### In paillier homomorphism, how is the randomness r changed during addition?

Two add the plaintexts encrypted in a ciphertext, you would just multiply the ciphertext and modulo it. However, how does the randomness value of the new ciphertext change? Assuming you the encryptor ...
78 views

### Why is asked that gcd(pq,(p-1)(q-1))=1 in the Paillier encryption scheme?

I don't see this property $\gcd(p\,q,(p-1)(q-1))=1$ used in the scheme. And in Paillier's original paper, I don't find this requirement. Is it required just for the difficulty of factoring $n$? Or is ...
50 views

### Searching in Paillier Cryptosystem

I have implemented Paillier Cryptosystem. Lets say, I have an encrypted array E(x) = [2,4,5,10,0,20] and I want to find that if 0 exist in that array. Due to the limitations of Paillier cryptosystem I ...
1 vote
86 views

### Short randomness in ElGamal and Paillier

In the Paillier cryptosystem the encryption of $m \in \mathbb{Z}_N$ with randomness $r \in \mathbb{Z}_n^*$ is $c = g^m r^n \bmod{n^2}$. My question is, what if short (E.g. 512bits) $r$ is used? ...
1 vote
48 views

### Generation of the order $\lambda$ (which is lcm((p-1),(q-1))) element g in modified paillier, why $-a^{2n}$?

As the question states, in variants of paillier cryptosystem, such as CS01 and DT-PKC, when they want an element $g$ of order $\lambda$, they choose a random number $a$ from group $Z^*_{n^2}$ and ...
123 views

### Paillier cryptosystem safety

I am working on system which can calculate average salary for different positions in large companies I want to use Paillier schema to do such calculation. I have 3 fields which I want to encrypt: ...
1 vote
141 views

### Do we need to consider overflow in paillier encryption?

Homomorphic multiplication of plaintexts in Paillier cryptosystem can be constructed as follow: Dsk(E(x1)^x2 mod N^2) = x1x2 mod N. So after the decryption, we get the result of multiplication x1x2. ...
129 views

### $n=pq$ and $n=p^2q$. How to take the value of two $n$ is the same in security

For example, Paillier's RSA modulus is $n=pq$, but OU's RSA modulus is $p^2q$. I think when two $n$ are the same, the security of the two cryptographic schemes must be different. So for example, if I ...
492 views

### How to calculate the n in n-bit security of a crypto algorithm?

I think I'm likely missing the term because searching for this is coming up with not so precise results. I'm looking to calculate the n-bit security of say Paillier vs ElGamal vs EC ElGamal, when I ...
418 views

### Paillier versus Lifted ElGamal for homomorphic addition for e-voting

I'm looking to create an anonymous e-Voting system which will assign a certain number of bits to each candidate during a vote, e.g. 010000 for Alice, 000100 for Bob, and 000001 for Charlie. It works ...
1 vote
84 views

### Encrypting full word using Paillier Scheme

I am using Paillier scheme to encrypt a message however, I have divided the words into alphabets and then convert each alphabet to ASCII code encrypting the final result. It works fine, but I want to ...
53 views

### How to know the exact result in Paillier cheaper-constant multiplication

The encryption function $E_{k^+}: Z_n \rightarrow Z_{n^2}$. The decryption function $D_{k^-}: Z_{n^2} \rightarrow Z_n$. $m_1 = 42, k = 15, n=77$. After encryption, exponentiation and decryption, I get:...
143 views

### ElGamal and Paillier combined

currently I am working on some secure accounting application. I have few requirements: Application should store items which can be searched by name. Every item has price and tax. Every invoice ...
429 views

### What is the difference between Paillier additive homomorphic property and addition of two paillier ciphers

Paillier has additive homomorphic property that states: if two ciphers c1 and c2, are multiplied, and the result is decrypted, it is equal to addition of the two plaintexts. D(c1*c2)= m1+m2 ...
1 vote
157 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 ...
126 views

### Paillier scheme : Encoding floats into integers impact on computations

In Privacy Preserving Processing Over Encrypted Images, I could understand that appropriate encoding of floats into integers (required in Paillier) only incur negligible error in computations. Any ...
1 vote
119 views

### Is Paillier a stream or block encryption

Does Paillier follow a stream encryption or block encryption technique. If it’s a block encryption then what is the size of the block in bits or bytes.
210 views

### Efficiently prove the correctness of Paillier encryption in or "outside" a zk-SNARK

I'm working with a zk-SNARK library  that allows me to prove the correctness of arbitrary arithmetic circuits, and I now want to use these zk-SNARKs to prove that some Paillier  ciphertext $c$ ...
101 views

### Evaluate the time of Paillier decryption

If I have 4 kilobytes of Paillier encrypted data, how can I know the time needed to decrypt it?
1 vote