# Tagged Questions

ElGamal is a public key encryption scheme with security based on the discrete logarithm problem.

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### One-wayness proof of Elgamal

Here in the proof of Elgemal one-wayness: Consider an adversary V that can invert random Elgamal encryptions with $ε$ probability. $V *$ : Adversary to compute the Diffie-Hellman function: ...
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### Elgamal and Cramer-Shoup encryption: Given private key and ciphertext, can public key be determined?

I have a potential use case for Elgamal or Cramer-Shoup encryption which would effectively reverse the visibility of the public and private keys. The ciphertext and private key would be published and ...
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### Getting a key size of DSA/Elgamal certificate

I've tried to get a public key size of DSA/Elgamal PGP certificate. Using the Cleopatra I find out that there is actually two public keys - one with DSA with key ...
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### El Gamal special construction

The algorithm: I have to determine if the following algorithm is secure: $p,q \in \mathbb{P}$ and $N = p \cdot q$ $e \in [2, N - 1]$ such that $\gcd(e, N) = 1$ $r_1, r_2$ are random numbers ...
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### Combining Elgamal encryption with the Elgamal signature scheme - Security issues?

For didactic (not real-world) purposes, I would like to teach other students the Elgamal encryption system and the Elgamal signature scheme. Are there known security issues when combining them? (i.e. ...
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### Is ElGamal encryption still secure if the randomness is known to be even?

In ElGamal encryption $(g^r, g^mg^{kr})$, if the randomness $r$ is always chosen from even numbers, and the attacker knows about this, is it still provable secure?
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### Elgamal with secret key equal to zero possible?

From various sources (e.g. this paper, page 3), the key generation algorithm of Elgamal samples the secret key $x$ from $\mathbb{Z}_q$, which is identifiable to $\{0, 1, 2, \dots, q-1\}$. My ...
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### Attacks on schemes based on elliptic curves when the transmitted points are not on the curve

Some elliptic curve schemes require to send a curve point during the normal execution of the protocol. For example, ElGamal encryption and ElGamal signature require this. On the other hand, ECDSA does ...
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### ElGamal - What if the Decisional Diffie-Hellmann problem could be solved?

I have read that if the DDH problem could be efficiently solved, the IND-CPA would not hold for ElGamal. I don't see why it makes ElGamal less secure if you have $g^a$, $g^b$ and $g^c$
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### How to find element g order q, given 2 large primes p and q where q|p-1

i have calculated 2 large primes, $p$ (minimum 2048 bits) and $q$ (minimum 224 bits), where $p-1 \mod q = 0$ by using SageMath. $p =$...
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### Why do the subexponential algoriths for the DLP not work for the ECDLP?

Elliptic curve cryptography is much more secure for the same parameters because attacks that work on the DLP do not work on the ECDLP. Why do the attacks fail in the latter case?
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### Is there an additive homomorphic encryption that supports exponentation

For example say we have two numbers a and b. Now is there any partial homomorphic encryption scheme that allows to compute (a-b)^2 over the ciphertexts of a and b without round trips.
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### Simulation of a custom build network security algorithm with ElGamal Cryptosystem using Elliptic Curve

I am trying to build an algorithm to encrypt and decrypt text using ElGamal Cryptosystem using Elliptic Curve. My algorithm generation is done. But at simulation part I stuck. My algorithm steps are ...
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### Elliptic curve ElGamal with homomorphic mapping

I am interested in ElGamal due to the fact that you can achieve some degree of homomorphic properties. I became interested in applying ElGamal to elliptic curves, and found this other question with an ...
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### An MPC protocol from Elgamal is a good solution a homomorphic multiplication?

I want to compute a multiplication between many secret values and then distribute the result to everyone involved. For this, I thought about an MPC protocol built from Threshold Homomorphic Elgamal. ...
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### Elliptic ElGamal Public Key Cryptosystem doubt

I need an example of Elliptic ElGamal Public Key Cryptosystem. I have been trying with some values but I don't get the right solution. I have $p=13$, the elliptic curve $E:y^2=x^3+11x+7$ and a point ...
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### Which values are used for an elgamal cryptosystem public key?

I know that – within the elgamal cryptosystem – the values of $a$ and $b$ are public. But which values are used to create public keys?
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### How is El Gamal different from Diffie Hellman Key Exchange [duplicate]

I am Reading RSA and Public-Key Cryptography by Mollin and I can't make out how El Gamal is different from Diffie Hellman Key Exchange. Any thoughts?
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### ElGamal in a different group

Can Elgamal be secure in $\bmod {n^2}$? Meaning instead of using a prime order group to use a group where DCR assumption holds?
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### Random Coin Flip

Quoting an old question: Consider the following protocol for two parties A and B to flip a fair coin (more complicated versions of this might be used for Internet gambling): A trusted ...
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### How to calculate mapping in bilinear

I am trying to read a paper in cryptography. In key generation phase, paper give a definition for bilinear like G and Gt be two cyclic groups of prime order p $e: G * G \to G_t$. be a map with the ...
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### How to compare the efficiency of public key cryptosystems, i.e., RSA vs El Gamal?

As part of my Mathematics degree I'm taking a Discrete Mathematics module which partially covers Public Key Cryptography but does not at all enter it in depth. I'm currently working on a project that ...
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### How to prove that a commitment hides the decryption of an ElGamal ciphertext?

I've decided to remove a previous unanswered question of mine and break it down into smaller pieces so it's not such a loaded question. For this question I need to prove that I've committed to a ...
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### ElGamal Signatures

I know various applications of RSA signatures. I wonder, is there any real-world applications of ElGamal signatures and encryption?
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### Batch ElGamal Scheme

I know about ElGamal scheme and its key generation, encryption, and decryption methods. I'm not sure about what is meant by batch ElGamal. Can someone please clarify it. How it differs from ElGamal ...
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### Do $v_1=\alpha\cdot r_1$ and $v_2=\alpha\cdot r_2$ leak information about $\alpha$

Please consider we have finite field $\mathbb{F}_p$ for large prime number $p$. We have a fixed field element $\alpha$. By $r_i\leftarrow \mathbb{F}_p$ we mean we pick $r_i$ uniformly random from the ...
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### Is Cramer-Shoup backdoorable?

We recently had the question whether it's possible to have multiple private keys with one public key for the cramer-shoup cryptosystem. There it was stated that finding such "secondary" private keys ...
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### ElGamal in $Z^*_{p^n}$

If $p$ is an odd prime and $n$ natural,it is known that the group $Z^*_{p^n}$ is cyclic.Explain why the selection-choice of the group $Z^*_{{3^{1000}}}$ for the construction of a cryptosystem ElGamal ...
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### ElGamal signatures systems

Let $p$ prime number $q/p-1$ prime and $g\in (Z/pZ)^*$ element of order $q$.Also $a\in \{1,...,q-1\}$ the private key and $y\equiv g^a\pmod p$ the correspoding public key.For each of the following ...
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### Ephemeral key is held constant in ElGamal

Is there a known algorithm to recover the sender's messages in ElGamal $\mathbb{F}_p^*$ if the ephemeral key is held constant in two or more transmissions, assuming the messages are always distinct? ...
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### ElGamal and Paillier key sizes for short messages

I am using ElGamal and Paillier schemes to encrypt a large number of short messages: typical 4-byte integers. I do this for the homomorphic properties of these schemes. However, the way the ...
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### Brute forcing the secret key in Elgamal encryption

Crypto noob here, I am attempting to do this programming challenge. I do not have the secret key that is used to decrypt the message. However, the key is small enough for a brute force approach. I am ...
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### Is ElGamal IND-CCA1?

We all know that textbook ElGamal falls due to chosen ciphertext attacks, because of its multiplicative homomorphic property ($E(A)*E(B)=E(AB)$). However these attacks require the ciphertext ($E(A)$ ...
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### ElGamal against chosen ciphertext attacks

In Hoffstein, Pipher, and Silverman's book An Introduction to Mathematical Cryptography, the authors make the following remark: An attack in which Eve has access to an oracle that decrypts ...
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### ElGamal scheme attack when one message is known and ciphertexts are intercepted

Let $G$ be a finite cyclic group of order $p$ and let $pk = (g, h=g^a)$ and $sk=(g,a)$ be Bob's ElGamal public/secret key pair in $G$. To encrypt a message $m$, a random number $r$ is selected and a ...
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### How to perform homomorphic multiplication in ElGamal?

How can I compute homomorphic multiplication in ElGamal? That is: Given two ciphertexts $(R_1,c_1)$ and $(R_2,c_2)$ corresponding to plaintexts $m_1$ and $m_2$ under some public key; how can I compute ...
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### What does L_n is the bit length of the group order n states need for my calculation in ECDSA algorithm?

I am doing the program of implementing ECDSA for which I am trying to solve the equation scenario. In ECDSA that signature generation algorithm which states as hash value from the SHA-1 and where $l$ ...
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### Secure ElGamal with OAEP

Is it possible to make ElGamal IND-CCA2 using OAEP or OAEP+? (OAEP+ from: "OAEP Reconsiderd" by Shoup) The reason I ask is that I recently answered this question and it came to my mind that OAEP or ...
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### Zero knowledge proof relation + number is binary

I want to develop a ZKPK for the following problem: $$Y=g_0^{r_Y} \prod_{i=1}^n g_i^{s_i}$$ and $$Z=h_0^{r_Z} \prod_{i=1}^n h_i^{s_i}$$ I want to proof knowledge of $r_Y,r_Z$ and $s_i$ which I have ...
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### Can the same random number be used in encryption and signing?

In several public key algorithms, the person running the algorithm must generate a random number (that's separate from the key). Can this random number be the same for an encryption and a signature? ...
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### Reusing the random exponent for ElGamal encryption with different plaintexts

In the basic ElGamal encryption scheme, we encrypt a message $m$ as $(g^r, h^r m)$, where $g$ is the group generator and $h$ is the public key of the recipient. If the sender has another message $m'$ ...
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### Drawback of ElGamal encryption

I am trying to find out some disadvantages of ElGamal cryptography but I'm not able to figure out what's wrong with the algorithm. The only one I found is that a known-plain text attack is possible in ...
Let $G$ be any group where group operations, as well as finding the inverse of an element, can be performed quickly, but the discrete log problem is difficult. Consider the following cryptosystem: ...
$GF(2^m)$ Galois fields are widely used in different cryptographic algorithms, for example, in Rijndael. However, $GF(p^m)$ fields are possible with any prime $p$, not only 2, but $GF(2^m)$ fields ...