In the ElGamal public key encryption scheme [1] 〈g〉 is a finite cyclic group of large enough order. A value q (a multiple of) the order of g, denoted as ord(g) (not necessarily a prime), is public. In the original ElGamal scheme, \(\langle g\rangle=Z_p^{\,\ast}\), p a prime and \(q=p-1\).
If Alice wants to make a public key, she chooses a random element a in Z q and she computes \(y_A:=g^a\) in the group 〈g〉. Her public key will be \((g,q,y_A)\). If a group of users uses the same g and q, the public key could be shorter. Her secret key is a.
If Bob, knowing Alice's public key \((g,q,y_A)\), wants to encrypt a message \(m\in\langle g\rangle\) to be sent to Alice, he chooses a random k in Z q and computes (c 1, c 2) := (g k, m·y k A ) in the group and sends \(c=(c_1,c_2)\). To decrypt Alice (using her secret key a) computes \(m':=c_2\cdot (c_1^a)^{-1}\) in this group.
The security of this scheme is related to the Diffie-Hellman problem. A non-malleable variant of this scheme was...
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References
ElGamal, T. (1985). “A public key cryptosystem and a signature scheme based on discrete logarithms.” IEEE Trans. Inform. Theory, 31, 469–472.
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Desmedt, Y. (2005). ElGamal Public Key Encryption. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_129
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