# Tag Info

Accepted

### Zero knowledge proof for sign of message value

What you are looking for is called a range proof. There has been a vast body of research on the topic recently - so vast, in fact, that it can be quite hard to know what is the state of the art, and ...
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### What is largest prime factor in Diffie-Hellman?

How to determine if the largest prime factor of $p-1$ is in fact large? Most often, it is not determined from $p$ that $p-1$ has a large prime factor. Rather, a large prime factor $q$ is chosen, ...
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### Exponentiation in ECC

In a group, where there is by definition only one operation, exponentiation means repeated application of the group operation, whatever that is. That is, if the group operation is noted $\circ$, $g$ ...
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### ECIES/ ECDHE/ EC-ElGamal encryption comparison

Fix a group $G$ of order $q$ in which discrete logs are hard, and fix a standard base point $g \in G$. Fix an authenticated cipher $E_k$ of bit strings. In (EC)IES, roughly: A public key is a point ...
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### Do $v_1=\alpha\cdot r_1$ and $v_2=\alpha\cdot r_2$ leak information about $\alpha$

Question: Given $n$ values $v_1=\alpha \cdot r_1 \bmod p,..., v_n=\alpha \cdot r_n \bmod p$ for a large $n$ can the adversary learn the value $\alpha$? Answer: assuming that the $r_i$ values are ...

### Time gap between Diffie-Hellman Key Exchange and ElGamal encryption?

Actually, if you read Diffie and Hellman's paper closely, you'll see that they explicitly talk about taking another's party value from a public file. Thus, it really already does public-key encryption....
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### ElGamal with elliptic curves II

I have a concern regarding the security of the scheme -- let's suppose that there are only two possible messages, $m_0$ and $m_1$. Then, the encryption is done by multiplying, in the field, the chosen ...
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### XOR instead of Multiplication in ElGamal encryption

One small question, from the DDH assumption we know $g^{xy}$ is random. Actually, that's not true. What the DDH assumption says is that we cannot distinguish $g^{xy}$ from $g^z$; however it does not ...

In general, proving that $g$ is a primitive root (often called a generator) of a cyclic group is fairly simple. Note this holds true for non prime modulo as well Step 1: Verify that $0\leqslant g \... 7 votes Accepted ### Why is it claimed that ElGamal is worse than RSA? What the notes remark is that the ciphertext in ElGamal encryption in$\mathbb Z_p^*$is about twice as large as the ciphertext in RSA, when working with$p$and$N$of about equal size; that's ... 7 votes Accepted ### ElGamal CPA secure So let's go through the IND-CPA game, shall we? Pick two messages$m_0$and$m_1$arbitrarily. Send them to the challenger who chooses$b\in\{0,1\}$uniformly at random and returns you$c=E(m_b)$. ... 7 votes Accepted ### ElGamal in$\mathbb Z^*_p$with$g$not a generator of$\mathbb Z^*_p$Because in the exercise$g$is is not a generator of$\mathbb Z^*_p$, the exercise would indeed be incorrect in a context explicitly giving a definition of ElGamal encryption that both requires$g$... 7 votes ### What is largest prime factor in Diffie-Hellman? My question is the following: how to determine if the largest prime factor of$p-1$is in fact large? Yes, showing that$p-1$is not smooth (terminology for "has a large prime factor") for random ... 6 votes Accepted ### On getting beyond LSB in discrete log If$p = 2q+1$where$q$is prime, and then we can efficiently solve for$x_0 \in \{0, 1\}$for$g^{2x+x_0} = h \pmod p$, if$g$is a generator. This is true; this is a special case of the observation ... 6 votes Accepted ### How many pairs of${b_i}^k \equiv g_i\pmod p$are enough to solve discrete logarithm problem? Having multiple sets$b_i^k = g_i$with a common solution$k$do not help you recover$k$; it cannot make the problem much easier than if you had the simple equation$b^k = g$. Here's the proof; ... 6 votes ### How to create an EC point from a plaintext message for encryption There is also a variant of Koblitz's approach * Let the message units$m$be integers$0<m<M$, let$\kappa$be large enough integer so that we are satisfied with error probability$2^{-\kappa}$, ... 5 votes Accepted ### One-wayness proof of Elgamal You are confusing two different security notions. The proof you describe is regarding the Onewayness under chosen-plaintext attacks (OW-CPA). Basically, this notion covers the idea that it shouldn't ... 5 votes ### ElGamal in$Z^*_{p^n}$The order of$(\mathbf{Z}/3^{1000}\mathbf{Z})^*$is$\varphi(3^{1000}) = 2\times 3^{999}$, which is a highly composite number, and hence the discrete logarithm in this group is highly vulnerable to ... 5 votes ### Which values are used for an elgamal cryptosystem public key? Let$p$be a prime such that the Discrete Logarithm problem in$({\mathbb Z_p}^*,.)$is infeasible, and let$\alpha \in {\mathbb Z_p}^*$be a primitive element. $$\beta =\alpha^a \bmod p$$ The ... 5 votes Accepted ### How is El Gamal different from Diffie Hellman Key Exchange The difference is purely conceptual. That is, when Diffie-Hellman published their paper, they equated between public-key encryption and trapdoor functions. Thus, they did not think that they had ... 5 votes ### How is a re-encryption done with elGamal? First, I can't find a copy of the RSA mental poker report, so I cannot say for sure what kind of "commutative encryption" they wanted to use, but one type is the Pohlig-Hellman cipher, where you ... 5 votes ### Is it insecure using addition instead of multiplication in Elgamal encryption? In the generic sense of an abstract group, this is a problem since addition may not be defined. However, when working modulo a prime$p$, addition is certainly defined. However, it is not secure. In ... 5 votes Accepted ### EL gamal cryptosystem Ok, so now the question makes more sense to me. Yes you are correct. Sadly, you cannot solve the problem without solving the DLP. This is simply because in the problem he didn't share his secret x ... 5 votes Accepted ### How to securely map messages to points on an elliptic curve If I do have to implement a mapping for security, how would you reccomend I map message values to points, and how should I figure out the maximum message size based on my curve? The default ... 5 votes Accepted ### Verification of encryption for ElGamal cryptosystem Actually, an ElGamal ciphertext$(a,b)$is "valid" iff$(a,b) \in G^2$, because for any message$m\in G$and any$\mathsf{pk}\in G$, the encryption of$m$will be in$G^2$, because for any$r$,$(g^r,...
The standard approach for this goes as follows, which I think is usually attributed to this paper by Koblitz: Suppose you have a curve over an $k$-bit prime field. Also suppose you want to encode a ...