Diffie-Hellman as public key cryptosystem

Most People know ElGamal as a Diffie-Hellman-based public key cryptosystem. But internally ElGamal uses a symetric cypher using multiprecision arithmetic. So why not using an efficient symetric cipher instead.

1. Diffie-Hellman as Public-Key Introducing a public-key cryptosystem based on Diffie-Hellman key-exchange (maybe a relaxed generalized ElGamal algorithm)
2. Diffie-Hellman key exchange given(G , p) a=random() A=Ga mod p (G , p , A) b=random() B=Gb mod p K= Ab mod p (B) K =Ba mod p Alice Bob
3. Diffie-Hellman key exchange given(G , p) a=random() A=Ga mod p (G , p , A) b=random() B=Gb mod p K= Ab mod p (B) K =Ba mod p Alice Bob ( p ,a)
4. A more advanced Usecase of Diffie-Hellman given(G , p) a=random() A=Ga mod p (G , p , A) b=random() B=Gb mod p K 1 =Ab mod p (B) K 1 =Ba mod p c=random() C=Gc mod p K 2 =Ac mod p (C) K 2 =Ca mod p
5. A more advanced Usecase of Diffie-Hellman given(G , p) a=random() A=Ga mod p (G , p , A) b=random() B=Gb mod p K 1 =Ab mod p (B) K 1 =Ba mod p c=random() C=Gc mod p K 2 =Ac mod p (C) K 2 =Ca mod p DH can create multiple shared secrets using the same secret
6. Creating a public key cryptosystem Aside the Diffie-Hellman primitives, we need a symetric encryption function
7. Creating a public key cryptosystem Aside the Diffie-Hellman primitives, we need a symetric encryption function crypt(key ,message)=ciphertext
8. Creating a public key cryptosystem Aside the Diffie-Hellman primitives, we need a symetric encryption function given(G , p) a=random() A=Ga mod p Pub=(G , p , A) Priv=( p ,a) And we need to define, what the public/private key is crypt(key ,message)=ciphertext
9. Sumary of the cryptosystem crypt(key ,message)=ciphertext decrypt(key ,ciphertext)=message The given symetric cipher: The key-pair generation given(G , p) a=random() A=Ga mod p Pub=(G , p , A) Priv=( p ,a) (G , p , A)=Pub b=random() B=Gb mod p K=Ab mod p c=crypt(K ,m) C=(B ,c) The encryption algorithm encrypt(Pub,m)=C (B ,c)=C ( p ,a)=Priv K=Ba mod p m=decrypt(K ,c) The encryption algorithm decrypt(Priv,C)=m
10. How the encryption would look like given(G , p) a=random() A=Ga mod p Pub=(G , p , A) Priv=( p ,a) Pub=(G , p , A) b=random() B=Gb mod p K=Ab mod p c=crypt(K ,m) C=(B ,c) K=Ba mod p m=decrypt(K ,c) Alice Bob Priv=( p ,a)
11. Whats about ElGamal? ElGamal is a subset of the cryptosysten i showed, wich uses a specific symetric encryption algorithm crypt(K ,m)=(K∗m)mod p decrypt( K , c)=(K−1∗m)mod p
12. So why not ElGamal? Like RSA and some other common-use Public-Key encryptions ElGamal uses numerics for encryption. This is Inefficient and not that secure like full-featured symetric ciphres.