Abstract
Many privacy preserving blockchain and e-voting systems are based on the modified ElGamal scheme that supports homomorphic addition of encrypted values. For practicality reasons though, decryption requires the use of precomputed discrete-log (dlog) lookup tables along with algorithms like Shanks’s baby-step giant-step and Pollard’s kangaroo. We extend the Shanks approach as it is the most commonly used method in practice due to its determinism and simplicity, by proposing a truncated lookup table strategy to speed up decryption and reduce memory requirements. While there is significant overhead at the precomputation phase, these costs can be parallelized and only paid once and for all. As a starting point, we evaluated our solution against the widely-used secp family of elliptic curves and show that we can achieve storage reduction by 7x–14x, depending on the group size. Our algorithm can be immediately imported to existing works, especially when the range of encrypted values is known, such as in Zether, PGC and Solidus protocols.
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Notes
- 1.
Similar compression techniques are also discussed in [3], but our work focuses on collision-free tables per group to completely avoid false positives.
- 2.
While we assume the binary representation of \(g^x\) is random, we can employ a hash function if g has some special property.
- 3.
Note that the cost of a hashmap lookup is insignificant compared to elliptic curve (EC) point addition (about 40 times in our implementation), while a scalar to EC point multiplication is around 32 times more expensive than EC point addition using the double-and-add method for small 32-bit scalars.
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Chatzigiannis, P., Chalkias, K., Nikolaenko, V. (2022). Homomorphic Decryption in Blockchains via Compressed Discrete-Log Lookup Tables. In: Garcia-Alfaro, J., Muñoz-Tapia, J.L., Navarro-Arribas, G., Soriano, M. (eds) Data Privacy Management, Cryptocurrencies and Blockchain Technology. DPM CBT 2021 2021. Lecture Notes in Computer Science(), vol 13140. Springer, Cham. https://doi.org/10.1007/978-3-030-93944-1_23
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