??:????”???????? Zk-stark”?????,?????????,???????zk-stark ???????????????:Arithmetization????????:Low Degree Testing???????????,???? zk-stark ????????????;???????,?????????????????????????(?????????????????,???????),??????????
??????,??????,????????????????????????????????????,?????????? LDT ??;????????????????,????????????,???,?????????????????????,????????,????????????????,????? FRI ???????,??,???????? FRI ?????,
FRI??
??,???????? FRI ??????FRI,? Fast RS IOPP,?? Fast Reed-Solomen Interactive Oracle Proofs of Proximity,??????? proximiary ????,??????????????????????????,??????????????????????,???????FRI????,???????????,
- ???? F ?,??????? L0,???? 2^n;
- ??,??????? f0:L0–>F ??? RS[F,L0,?] ?????????,? f0 ????????? d
??,? f0=P ?,?? IFFT ??,?? P1?P2 ? deg(P1?P2) < 1/2?2^n,??:
f0(x) = P(x) = P1(x^2) + x * P2(x^2) (1)
? Q(x, y) = P1(y) + x * P2(y),???? Q(x, y)?? x ?? d < 2;?? y ?? d < 1/2?2^n;??,?????????? x0 ??????,???
f1(y) = Q(x0, y) = P1(y) + x0 * P2(y) (2)
?? f1(y),y=x^2,?? x ??????1????,?? x^(2^n) = 1 ==> (x^2)(2^(n-1)) = 1 ==> y(2^(n-1)) = 1,?y?????? L1,? L1 ?????:
- ???? 2^(n-1);
- ? L1 ???????? L0 ?????,?? L1 ??? y,? L0 ???? x ?(-x)mod F,?? x^2 mod F = y && (-x)^2mod F = y;
??,????????? f1(y) ?? d < 1/2?2^n,???????? f1 ? f0 ????,???????????:
- ??????? L1 ?? L0 ????? y,s0,s1?? s0!=s1 && s0^2 = s1^2 = y
- ????? f0(s0),f0(s1),f1(y)???
- ????? f0(s0),f0(s1) ??????? x ? d<2 ???? g(x)
- ????? g(s0) = f1(y),???,???
?????:???? f1 ????? f0 ????,???? (1) ???? P1(x^2) ?P2(x^2)??? (2) ???? P1(y) ? P2(y) ????,???????? d < 1/2?2^n,???????? 2^(n-1),???????????????,????????? 1/2?2^n / 2^(n-1) = ??? ? coderates,?? ? = 2^(-8),?????????????? 1/256,????????,??????????????? 0?
????,??? f1 ???????,??fr????????????,????????????,
??,?????FRI???????,?? 1 ??:
FRI ????????:Commit ??? Query ??,????????????,???????,??:
- ???????? x0
- ????????? f1
- ???????
FRI ???????? 2 ???? Commit ??,??3????? Query ??,?? Commit ??,??????? f0~fr,r ??????,??? Query ??,????,
??,????? Commit ? Query ??????????????,???????????,
Commit:
- Common input
- R RS ????
- i ??????,?? {0~r}
- r ???? ?? k0-R/?
- ? ?????? x–>x^(2^?)
- L0 ??? 2^k0
- RS[F,Li,?] ????[ ???,???,???? ]
- q0(x) = x^(2^?)(???????,??????),L(i+1) = q0(Li),??? Li ?? L(i+1) ? 2^? –> 1 ??
- Prover input
- fi ? i ????????
- Li ? i ?????,?? 2^(n-i)
- RSi fi ???????
- LOOP i <= r
- 1
- xi ?????????
- 2
- Sy ? L(i+1)?????????? Li ??????
- f(i+1)(y) ?? f(i+1)?? L(i+1)??????
- 3 i==r
- ?? fr ? 2 ????
- ??? P(x)
- d ???? P(x) ??
- ?? d+1 ???? P(x) ??? a0~ad
- Commit ????
- 4 i < r
- ?? f(i+1) ??? 2 ??????
- ?? f(i+1)??,?? L(i+1)
- ???????
Query
- verifier input
- R /? /Li /RSi /xi /fi /P(x) ?Commit
- l query??
- ?? fr
- ?? a0~ad,??P`(x)
- ?? P`(x)?? Lr ??????,???? fr,? fr ?? RSr
- repeat l times
- i = {0~r-1}
- Si ?? s(i+1) = q0(x)? x ???
- i = {0~r-1}
- ? Si ?,??? Pi(x)
- round consistency check i = {0~r-1}
- f(i+1)(s(i+1)) = Pi(xi)
- ???,?????
??,? ? = 1(?,q0(x) = x^2)??,FRI ???????????? 2 ??:
?????????:
- ????????????,???????? f0 ???? d < ?*2^n
- ?????? l ?,??????????????
??
???? FRI ???????,????,??????????? r = Log2(?2^n),?????,????????? f0 ??? ?2^n ?,??? round consistency ??????????????????,??????? DEEP-FRI ??,??? FRI,DEEP-FRI ?????????????????,?????????? ????????????,?? ZK-STARK ?????:
- ???????:???? LDT
- ???????????????????? LDT ??
- LDT ???? FRI ??,?????????????????????
- ??????????????????????,????????????
- ???????????,???????????????,????????????
- ????,??????? CRS
- ????,?????????
??
??FRI????? https://medium.com/starkware/low-degree-testing-f7614f5172db
FRI paper https://eccc.weizmann.ac.il/report/2017/134/
DEEP-FRI paper https://arxiv.org/abs/1903.12243https://en.wikipedia.org/wiki/Reed–Solomon_error_correction
