The challenge of modern cryptography is to find efficient but secure encryption schemes. The Elliptic Curve Integrated Encryption Scheme (ECIES) is a widely used encryption scheme which is specified in many standards.
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It is used for example in Ethereum’s Whisper. While Bitcoin Core does not implement ECIES, it can be employed to send secure messages using Bitcoin keys. Together with the Decentral Jaxx project we contributed with the OpenPGP.js project that follows this approach.
Most encryption schemes rely on the intractability of some variants of the Diffie-Hellman problem.
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ECIES security assumes, amongst other things, that the hashed decisional Diffie-Hellman problem over the specified elliptic curve is of exponential complexity. Although this problem is easy for a specific class of elliptic curves, the rest can currently only be solved via discrete logarithms. This is in general a harder task which is believed to be of exponential complexity for carefully chosen elliptic curves.
In our paper here we summarize the ECIES scheme and its security when paired with the secp256k1 curve of the SEC Group which is currently used in Ethereum. (For more information about the complexity of the discrete logarithm problem on elliptic curves please see our previous article: ECDSA Security in Bitcoin and Ethereum: a Research Survey)