From 79e20e2e68b060167c067dcbd53f501b377c3554 Mon Sep 17 00:00:00 2001 From: abell1689neo Date: Sat, 7 Dec 2024 00:20:39 +0900 Subject: [PATCH] ELGamalre (#75) * ELGamalre * revised --------- Co-authored-by: user --- content/Basic Cryptography/ElGamal.md | 37 +++++++++++++++++- .../Basic Cryptography/images/ElGamal(1).png | Bin 0 -> 218610 bytes 2 files changed, 36 insertions(+), 1 deletion(-) create mode 100644 content/Basic Cryptography/images/ElGamal(1).png diff --git a/content/Basic Cryptography/ElGamal.md b/content/Basic Cryptography/ElGamal.md index d97a962..d641b58 100644 --- a/content/Basic Cryptography/ElGamal.md +++ b/content/Basic Cryptography/ElGamal.md @@ -1,4 +1,39 @@ -The ElGamal system utilizes the [Discrete Logarithm Problem]. +The ElGamal system utilizes the [Discrete Logarithm Problem](../Basic%20Algebra/Discrete%20logarithm%20problem.md) as its primal measure obtain security. +[Diffie-Hellman](./Diffie-Hellman.md) is used as fundamental, and is converted into public key encryption to embody ElGamal system. + + +## Basic Setting +![[ElGamal.png]] +$\mathcal{E}' = (\text{Gen}', \text{Enc}', \text{Dec}')$ is a private-key encryption with $K = G$ + +- Bob computes $k = u^y$ and encrypts $m$ using the private-key encryption $\mathcal{E}'$. + Resulting ciphertext is $c= (v,c')$ +- Alice computes $k= v^x$ and derives $m$ by decrypting $c'$ + + +### Key Generation in ElGamal Encryption: +Computed by Alice (reciever) +- A large prime number $x$ is selected from group $Z_q$ as a private(secret) key, and public key $u = g^x$ is made public +- can be represented as equation below + $\text{Gen}: sk=x \leftarrow \mathbb{Z}_q$ and $pk = u \leftarrow g^x$ + +### Encryption Process of Message m: +Computed by Bob (encryptor) +- random number $y \leftarrow \mathbb{Z}_q$ is chosen +- $v = g^y$, $k=u^y$ is computed, using public key generated by Alice(reciever) +- encryption is done by computing $c' \leftarrow \text{Enc}'_k(m)$, which is a private key encrytion of message m, via key $k$ +- output $c=(v,c')$ as a final ciphertext + +### Decryption Process: +- utilizes private key $x$ to compute $k=v^x$ (shared key generated by Bob) +- using shared key $k$ obtained above, ciphertext $c$ can be transfromed into the original plaintext $m$ through $m \leftarrow \text{Dec}'_k(c')$ + +In RSA, encrypting the same message always produces the same ciphertext, whereas the ElGamal system uses random numbers, resulting in different ciphertexts for the same message each time it is encrypted. Therefore, RSA-OAEP is used in commercial systems. Additionally, the ElGamal system offers security with smaller private key sizes compared to RSA. + +## Security of ElGamal +let $\mathcal{E}' = (\text{Gen}', \text{Enc}', \text{Dec}')$ be a private key encryption with $K = G$. +If $\mathcal{E}'$ is CPA-secure and the DDH assumption holds, the ElGamal is CPA-secure. + Key Generation in ElGamal Encryption: diff --git a/content/Basic Cryptography/images/ElGamal(1).png b/content/Basic Cryptography/images/ElGamal(1).png new file mode 100644 index 0000000000000000000000000000000000000000..c8ac5a0807dc87664e5046232268e13b071a4cd3 GIT binary patch literal 218610 zcmeEtWmg=*7VY33+(~fP!3TE;5PXoJ!9#F@y9NtxgOlJc0fIwt7zPM#gS)%K%l+`y zy7v#fPjB~!>Q%jZb*-w?b5y2L=B{!g6;BVsm zd!?g`q!0p{M`R^`S84o6f3%^d;W5?^3cs_0nFYA{$&?fUb=a~%S-eouP(t#6ZY;cY zNLOTV&jvdZ(j{`3{$Ewph~p}3Y=UMyb2cf-vt}M1dr{K_{SZ9aj zUi9Q-Vca}C&hi$E!I4;00bs!6Ob#tI`V_e;k$Bu z9YV%d21Mm%nBg+Sm~hG@9%Pmz#2h(IBGy|=!(;1NA3mq5dmd`*D(_YJ)z#tc;`i1W ztp--j6s72Q$T;m4YS+0F(jm6B6*c!0dXnV_Av#%p19G 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