# Tagged Questions

Elgamal is a public key encryption scheme and signature scheme with security based on the discrete logarithm problem.

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### How to protect from Silver–Pohlig–Hellman algorithm

I read that Silver–Pohlig–Hellman algorithm solves the discrete logarithm with prime module $p$ in $O(\log^2(p))$ if $p-1$ is a smooth number. This seems pretty fatal for cryptography, since it is a ...
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### Difference between Pedersen commitment and commitment based on ElGamal

Does any of you know what is the difference between the Pedersen commitment and the commitment that uses the ElGamal encryption scheme? For the sake of completeness, I recall what both of them look ...
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### Recovering the random number r

For a padded message, M, using the El Gamal encryption schema, how can we determine the random number $r$, when we are given $p$, the prime number, $g$ which is the primitive root of $p$, $b$ and $x$ ...
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### Generating a valid signature on El-Gamal without knowing the private key

Suppose we are given $p$, the large prime, $g$ which is the primitive root for $p$, $b$ which is calculated as $b=g^x$ mod $p$ where $x$ is the private key and $0<x<p-1$. Also suppose we know ...
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### Solving a discrete logarithm using GDlog

I am trying to calculate an x, such that t = g^x mod p (I need to crack a weak elgamal encryption for university). I found gdlog, but I cant figure out how I can use the input to calculate my x. Here ...
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### About Elliptic Curve ElGamal, 3 simple problems I have trouble with

In Elliptic Curve ElGamal, why are a=b=1 always legal for primes whose lengths are no shorter than 11(2) bits long? Is there any reason why the Point at Infinity can always be encoded as (0,0)? ...
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### How does an oblivious test of plaintext equality work?

Assume an ElGamal Cryptosystem. Assume a set of three players, $P_1$, $P_2$ and $P_3$. The private key $x$ is shared among the players. The player $P_1$ has a piece of the private key $x_1$, $P_2$ has ...
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### In Elgamal, the generator g is always quadratic non-residue modulo p where p is a safe prime and inverse of g can be also generator?

In Elgamal, the generator $g$ is always quadratic non-residue modulo $p$, where $p$ is a safe prime and the inverse of $g$ can also be generator? Can I prove it? I can't come up with it at all.
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### Are any of the major asymmetric ciphers distinguishable (EG, RSA, ECC)?

Related to "Is it possible to derive the encryption method from encrypted text?". Given ciphertexts generated by any of the major asymmetric ciphers (RSA, ElGamal, ECC, etc..) can these ciphertexts ...
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### Is there an effective way for public key generation in homomorphic encryption schemes?

Is there any way to effectively generate valid distributed public key in schemes like ElGamal or Benaloh for a number of participants? There's no need in private keys since there's no intent to ...
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### ElGamal signature without calculating the inverse

I stumbled upon this question in some textbook. Propose a variant of ElGamal signature scheme such that there is no need to calculate the inverse $k^{-1}$ as it is usually done using the EEA. ...
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### Does ELGAMAL Encryption create different key for each sending block?

I ask about ELGAMAL algorithm. Is ELGAMAL algorithm used new key for each encryption process. in other word it shold be use new key for each chunk. sample examble If we have message has four block ...
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### What does SSL use? RSA? El-Gamal? Elliptic curves?

I'm not sure what SSL uses to share the symmetric key to both end users, i.e. at the beginning of the communication. Is it RSA? Or El-Gamal? Or something else? Thanks!
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### chaining rsa with ecies

In an answer to a previous question it was suggested that one way to protect your asymmetrically encrypted AES-256 keys, from say a solution to prime factorization, would be to chain asymmetric ...
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### DSA vs. ElGamal vs. Schnorr

It has been well known for a long time that NSA played a primary role in the development of the Digital Signature Algorithm (DSA). DSA is a sort of hybrid of the ElGamal and Schnorr signature ...
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### Proof of correct construction of a private key in distributed cryptography

In an exponential ElGamal encryption scheme where the key generation is done in a distributed way among $n$ trustees we have that each trustee $i$ (where $1 \leq i \leq n$): Selects a private key ...
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### Proof of correctness of a homomorphic ElGamal sum

Let's suppose we are using the exponential ElGamal as a public-key encryption scheme, so that we encrypt $g^m$ instead of $m$, for some generator $g$. Let $x$ be the private key, and $h=g^x$ be the ...
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### ElGamal with elliptic curves

I've searched some information on ECC, but so far I have only found Diffie-Hellman key-exchange implementations using ECC, but I don't want to exchange keys, I want to encrypt & decrypt data like ...
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### Can Elgamal be made additively homomorphic and how could it be used for E-voting?

Elgamal is a cryptosystem that is homomorphic over multiplication. How can I convert it to an additive homomorphic cryptosystem? How can I use this additive homomorphic Elgamal cryptosystem for ...
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### Why Elliptic curve cryptography are not popular in practice

RSA and ElGamal can be implemented using the technique of Elliptic curves. I am confused on why the it seems that Elliptic curves are not so popular in cryptographic applications since they provide ...
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### Why elgamal is still discussed

As I understand, if two users have a shared key, it is possible to encrypt a message using symmetric key encryption. So when a secret key is shared through Diffie-Hellman asymmetric key exchange, it ...
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I'm performing ElGamal encryption algorithm and using the additive homomorphic property so the product of two ciphertexts is the encryption of the sum of the plaintexts. The problem is that I need to ...
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### What is meant by this notation in the ElGamal key generation process?

Alice chooses i) A large prime $p_A$ (say 200 to 300 digits) ii) A primitive element $\alpha_A$ modulo $p_A$, iii) A (possibly random) integer $d_A$ with $2 \le d_A \le p_A-2$. Alice ...
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### Can ElGamal encryption and ElGamal signatures be used together sharing the same key-pairs?

I'm working on a encryption system where each party can store exactly a single ElGamal private key in a device. This is a hardware limit. The system must be expanded to support signatures and ...
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### calculating beta for elgamal elliptic curves [duplicate]

Suppose we use elgamal elliptic curves for secure communication. Bob selects a prime $p$, an elliptic curve $E$, a point $\alpha$ on $E \pmod p$, and a secret integer $f$. Suppose that Bob has ...
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### Why is the discrete log problem easy when the exponent comes from a binomial distribution?

I read in http://epubs.surrey.ac.uk/7219/2/esorics06.pdf that in exponential El Gamal the discrete log problem for recovering $m$ from $g^m$ can be made tractable when $m$ is drawn from a binomial ...
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### Chosen ciphertext insecurity in an ElGamal variant

I'm trying to prove something and if I can show that there is a simple way to calculate $(g^a \bmod p)^k$ if I know both $g^k \bmod p$ and $g^a \bmod p$, then (I think) it will help me prove it, but ...
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### ElGamal: Generation of “g” value?

I haven't been able to find a clear explanation on this (and I'm probably just confused in general). I have the following parameters of ElGamal signatures: $p$ = safe prime $q = (p-1)/2$ $g$ = an ...
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### How is ElGamal not secure under chosen ciphertext attack, but semantically secure in some cases?

I know that you can create a ciphertext c' using c and then find the corresponding m' for c' which you can use to find m for c. So, doesn't this mean that it is not semantically secure? But I also ...
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### Efficient algorithm for remainder calculation over prime field for ECC implementation?

I am working on 224-bit elliptic curve cryptography. In this 224-bit * 224-bit multiplication results 448-bit output. I am reducing 448-bit into prime field range( prime number $2^{224}-2^{96}+1$) ...
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### Use curve25519 for ElGamal crypto

DJB described curve25519 in his paper which can be found here: http://cr.yp.to/ecdh/curve25519-20060209.pdf. It seems that the main purpose was for Diffie-Hellman key exchange. I think this means that ...
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### Is Guillou-Quisquater existentially unforgeable against adaptive message attack under a random oracle model?

First of all, the Guillou-Quisquater digital signature scheme is: Note everything is $\bmod n$. Message is denoted by $m$. Private key: $s$ Public key: Hash function $H$, $e$, ...
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### Elgamal test vectors

I know that Elgamal algorithms have fallen out of fashion, and they are not included in any widespread cryptographic standard nowadays. However, they are still present in several popular crypto ...
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### elgamal based decryption mixnet

Can anybody help me to understand elgamal based decryption mixnet? Pure elgamal can be used in mixnet (re-encryption mixnet) and its very easy, and I have implemented decryption mixnet using RSA also. ...
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### Why can't I break ElGamal encryption by brute-forcing the secret exponent?

I am doing a course on cryptography on coursera and one of the topics covered was the ElGamal Encryption system. I am using the terms as defined in Wikipedia. Alice publishes $g$ and $g^x$. ...
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### ElGamal Signature Scheme: Recovering the key when reusing randomness

Show how if Alice uses the same value of $k$ to sign two different messages $m_1$ and $m_2$, using the ElGamal signature scheme, Eve can recover the value of $a$ from the corresponding signatures ...
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### Need for Twin Elgamal encryption

What is the need for two private/public keys in twin elgamal? I'm relatively new and would need some help.
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### What's a good P to use in ElGamal key generation?

I need to generate a 1024 bit ElGamal key on Android, and key generation takes forever (~10 minutes on a Galaxy Nexus), I suspect because it take so long to generate a safe prime. Is it dangerous for ...
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### How can I prove in zero knowldege that an ElGamal shuffle is correct for a special setting? [closed]

In a special ElGamal encryption scheme, every user has an ElGamal encryption key-pair using the same cyclic group $G$ and generator $g$. The system has a special function :  ...
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### Besides key and ciphertext sizes what are other advantages of elliptic curve versions of various protocols?

There are elliptic curve variants of Diffie-Hellman, ElGamal, DSA and possibly other protocols/algorithms. I know that these elliptic curve variants have smaller key and ciphertext sizes which will ...
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### Existing works on pre-computing ElGamal ephermal keys

I was playing around with a problem in e-voting schemes that use additive homomorphic encryption to tally votes, namely that at the end of the day somebody (or somebodies, if the secret material has ...
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### How to securely use Elgamal encryption in libgcrypt?

How secure is libgcrypt's Elgamal implementation of encryption (how different it is from textbook Elgamal), and how can I tweak padding and other preprocessing actions? For example, I know that usign ...
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### Additive ElGamal cryptosystem using a finite field

I'm trying to implement a modified version of the ElGamal cryptosystem as specified by Cramer et al. in "A secure and optimally efficient multi-authority election scheme", which possesses additive ...
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### Using bad generator in ElGamal Encryption

Suppose Alice chooses a random Prime $p$ and a random private Key $a \in \mathbb{Z}^*_p$. By accident, she also chooses a random number $g \in \mathbb{Z}^*_p$, which is not a generator of ...
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### Safe generator for ElGamal signature

What are the properties a generator $g$ should have to be secure for ElGamal signatures (original scheme)? I am aware that it is poorly chosen and not secure when $g|p-1$ or $g^{-1}|p-1$, where $p$ ...
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### When to use RSA and when ElGamal asymmetric encryption

If i am not wrong in cryptography there are 2 basic cryptographic schemes for public key cryptography. RSA encryption whose security is based on the infeasibility of solving the factoring of big ...
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### ElGamal: Multiplicative cyclic group and key generation

Here on the ElGamal wikipedia page http://en.wikipedia.org/wiki/ElGamal_encryption Alice generates an efficient description of a multiplicative cyclic group G, of order q, with generator g. How ...
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### Malleability of ElGamal and Hashed ElGamal

Question: Suppose A encrypts a number $x$ which indicates her bid on a contract, using ElGamal encryption. Say that the encryption of $x$ produces a ciphertext $c$. Explain how E can modify $c$ to ...