Paper 2021/1273

OpenSquare: Decentralized Repeated Modular Squaring Service

Sri AravindaKrishnan Thyagarajan, Tiantian Gong, Adithya Bhat, Aniket Kate, and Dominique Schröder


Repeated Modular Squaring is a versatile computational operation that has led to practical constructions of timed-cryptographic primitives like time-lock puzzles (TLP) and verifiable delay functions (VDF) that have a fast growing list of applications. While there is a huge interest for timed-cryptographic primitives in the blockchains area, we find two real-world concerns that need immediate attention towards their large-scale practical adoption: Firstly, the requirement to constantly perform computations seems unrealistic for most of the users. Secondly, choosing the parameters for the bound $T$ seems complicated due to the lack of heuristics and experience. We present Opensquare, a decentralized repeated modular squaring service, that overcomes the above concerns. Opensquare lets clients outsource their repeated modular squaring computation via smart contracts to any computationally powerful servers that offer computational services for rewards in an unlinkable manner. Opensquare naturally gives us publicly computable heuristics about a pre-specified number ($T$) and the corresponding reward amounts of repeated squarings necessary for a time period. Moreover, Opensquare rewards multiple servers for a single request, in a sybil resistant manner to incentivise maximum server participation and is therefore resistant to censorship and single-points-of failures. We give game-theoretic analysis to support the mechanism design of Opensquare: (1) incentivises servers to stay available with their services, (2) minimizes the cost of outsourcing for the client, and (3) ensures the client receives the valid computational result with high probability. To demonstrate practicality, we also implement Opensquare's smart contract in Solidity and report the gas costs for all of its functions. Our results show that the on-chain computational costs for both the clients and the servers are quite low, and therefore feasible for practical deployments and usage.

Note: Published at ACM CCS 2021

Available format(s)
Cryptographic protocols
Publication info
Preprint. MINOR revision.
Time-Lock PuzzlesRepeated Modular SquaringSmart Contracts
Contact author(s)
t srikrishnan @ gmail com
2022-03-16: revised
2021-09-24: received
See all versions
Short URL
Creative Commons Attribution


      author = {Sri AravindaKrishnan Thyagarajan and Tiantian Gong and Adithya Bhat and Aniket Kate and Dominique Schröder},
      title = {OpenSquare: Decentralized Repeated Modular Squaring Service},
      howpublished = {Cryptology ePrint Archive, Paper 2021/1273},
      year = {2021},
      note = {\url{}},
      url = {}
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