Showing papers by "Swastik Kopparty published in 2022"

Scalable and Transparent Proofs over All Large Fields, via Elliptic Curves

Abstract: Concretely efficient interactive oracle proofs (IOPs) are of interest due to their applications to scaling blockchains, their minimal security assumptions, and their potential future-proof resistance to quantum attacks. Scalable IOPs, in which prover time scales quasilinearly with the computation size and verifier time scales poly-logarithmically with it, have been known to exist thus far only over a set of finite fields of negligible density, namely, over “FFT-friendly” fields that contain a sub-group of size $$2^\mathsf{{k}} $$ . Our main result is to show that scalable IOPs can be constructed over any sufficiently large finite field, of size that is at least quadratic in the length of computation whose integrity is proved by the IOP. This result has practical applications as well, because it reduces the proving and verification complexity of cryptographic statements that are naturally stated over pre-defined finite fields which are not “FFT-friendly”. Prior state-of-the-art scalable IOPs relied heavily on arithmetization via univariate polynomials and Reed–Solomon codes over FFT-friendly fields. To prove our main result and extend scalability to all large finite fields, we generalize the prior techniques and use new algebraic geometry codes evaluated on sub-groups of elliptic curves (elliptic curve codes). We also show a new arithmetization scheme that uses the rich and well-understood group structure of elliptic curves to reduce statements of computational integrity to other statements about the proximity of functions evaluated on the elliptic curve to the new family of elliptic curve codes.

Elliptic Curve Fast Fourier Transform (ECFFT) Part II: Scalable and Transparent Proofs over All Large Fields

Abstract: Concretely efficient interactive oracle proofs (IOPs) are of interest due to their applications to scaling blockchains, their minimal security assumptions, and their potential future-proof resistance to quantum attacks. Scalable IOPs, in which prover time scales quasilinearly with the computation size and verifier time scales polylogarithmically with it, have been known to exist thus far only over a set of finite fields of negligible density, namely, over “FFT-friendly” fields that contain a sub-group of size 2. Our main result is to show that scalable IOPs can be constructed over any sufficiently large finite field, of size that is at least quadratic in the length of computation whose integrity is proved by the IOP. This result has practical applications as well, because it reduces the proving and verification complexity of cryptographic statements that are naturally stated over pre-defined finite fields which are not “FFT-friendly”. Prior state-of-the-art scalable IOPs relied heavily on arithmetization via univariate polynomials and Reed–Solomon codes over FFT-friendly fields. To prove our main result and extend scalability to all large finite fields, we generalize the prior techniques and use new algebraic geometry codes evaluated on sub-groups of elliptic curves (elliptic curve codes). We also show a new arithmetization scheme that uses the rich and well-understood group structure of elliptic curves to reduce statements of computational integrity to other statements about the proximity of functions evaluated on the elliptic curve to the new family of elliptic curve codes. This paper continues our recent work [BCKL21] that used elliptic curves and their subgroups to create FFT-based algorithms for polynomial manipulation over generic finite fields. However, our new IOP constructions force us to use new codes (ones that are not based on polynomials), and this poses a new set of challenges involving the more restricted automorphism group of these codes, and the constraints of Riemann–Roch spaces of strictly positive genus. *StarkWare Industries Ltd. {eli,dancar,david}@starkware.co †Department of Mathematics and Department of Computer Science, University of Toronto. swastik.kopparty@gmail.com

Lecture 3: Hadamard codes, BCH codes, Reed-Solomon codes

Abstract: Proof. Suppose f1, f2 ∈ F2 with f1 ̸= f2, but f1M = f2M . Then (f2 − f1)M = − → 0 . But if f2 ̸= f1 then f2− f1 has at least one nonzero coordinate. Suppose the i’th coordinate of f2− f1 is nonzero. By construction, M has a column with all zero entries except the i′th coordinate, call that the j′th column. Then the j′th coordinate of (f1 − f2)M is nonzero, contradicting our above conclusion. Thus for any f1, f2 ∈ F2 with f1 ̸= f2, we have f1M ̸= f2M . It follows that |C| = |F2| = 2t.

Lecture 6: Fourier Analysis and The Rate/Distance Trade-off for Distance Near 1/2

Abstract: We will now focus on the case where δ = 1−ε 2 . Note: In our discussion, we are first letting n→ ∞ and then only δ → 1−ε 2 . We know there exist codes with rate Ω(ε2). For all codes, rate is at most O(ε). This is quite a big gap. In this lecture we will see that the first bound is nearer the truth. This was first proved by McEliece, Rodemich, Rumsey and Welch in the 70s through their “Linear Programming bound”. We will see a simple proof (that only works for linear codes) of this result for the δ close to 1/2 asymptotic. This proof is a translation to Fourier analysis of a proof by Alon and an improvement by Schectman and Shraibman.