Implement the homomorphic encryption in circuit

LayerXcom / zero-chain

A privacy-preserving blockchain on Substrate

https://layerxcom.github.io/zerochain-book/

GNU General Public License v3.0

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Implement the homomorphic encryption in circuit #46

Closed osuketh closed 5 years ago

osuketh commented 5 years ago

For simplicity, we should add the homomorphic encryption like ElGamal or lifted-ElGamal, paillier instead of pedersen commitment.

osuketh commented 5 years ago

Add the lifted-ElGamal. It will encrypt the 32 bits integer and might need the suited statistical algorithm for the decryption of the transfer value. Anyway, the lifted-ElGamal is not expensive rather than paillier.

osuketh commented 5 years ago

The algorithm of Lifted-Elgamal

Encryption

Enc(m) = (mP + r(sP), rP)
where m: message, P: the point on the curve, s: secret key, sP: public key, r: random value

Decryption
```
(mP + rsP) - s(rP) = mP
```
m should be 32-bits num.

Homomorphic property

Enc(m) + Enc(m')
= (mP + rsP, rP) + (m'P + r'sP, r'P)
= ((m+m')P + (r + r')sP, (r + r')P)
= Enc(m + m')

osuketh commented 5 years ago

TODO: https://github.com/LayerXcom/zero-chain/blob/47aaa49df4e58f8be20e89c68c260da8b5daaad8/proofs/src/circuit_transfer.rs#L212-L219