Explore Circle STARKs

In recent years, the trend in STARKs protocol design has been to shift towards using smaller fields. The earliest STARKs implementations used 256-bit fields, but this design was less efficient. To improve efficiency, STARKs began using smaller fields such as Goldilocks, Mersenne31, and BabyBear.

Using small fields can significantly improve proof speed. For example, Starkware can prove 620,000 Poseidon2 hashes per second on the M3 notebook. However, small fields also bring some challenges, such as how to achieve sufficient randomness within a limited value space.

To solve this problem, one can use methods such as multiple random checks or expanding fields. Expanding fields are similar to plurals, introducing new value α such that α^2 equals a specific value, thereby creating more complex mathematical structures.

Circle STARKs is a clever scheme that efficiently implements the FRI protocol over small fields such as Mersenne31. It utilizes a special circular group structure that has properties similar to a two-to-one mapping. This structure allows us to perform efficient polynomial reductions over small fields.

Circle STARKs also support similar FFT operations, but they deal with Riemann-Roch spaces instead of strict polynomials. This introduces some differences in details, such as the construction methods of quotient operations and vanishing polynomials.

In general, Circle STARKs provide developers with a way to build efficient STARKs over small fields without having to focus excessively on the underlying mathematical details. It combines the computational efficiency of small fields with sufficient security, making it a promising optimization direction for STARKs.

Future STARK optimizations may focus on: optimizing the arithmetic of basic cryptographic primitives; improving parallelism through recursive construction; enhancing the arithmetic of the virtual machine to improve the developer experience. We are approaching the limits of efficiency for the STARK foundation layer, and future optimizations will concentrate more on these directions.

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