Circle STARK Verifier in Bitcoin Script

This repository includes Bitcoin script implementations of various cryptographic primitives for STARK.

• M31, CM31, QM31, Circle Point
  • implementation of add, sub, mul of Mersenne-31 (M31) and its complex extension (CM31) and its degree-4 extension (QM31), which is imported from BitVM/rust-bitcoin-m31-or-babybear.
  • implementation of add, sub, mul of circle points, which are over the circle curve x^2 + y^2 = 1.
• CirclePoint over QM31
  • implementation of double of a circle point over QM31.
  • implementation of drawing a random point on the circle over QM31, which is useful for OODS.
  • implementation of addition of a circle point over QM31.
• Constraints on the circle curve over QM31
  • implementation of coset_vanishing() and pair_vanishing().
• Fiat-Shamir Transcript
  • aka "channel", which is the name used in Starkware's stwo library.
  • absorbing commitments and QM31 elements through OP_CAT + OP_SHA256.
  • squeezing random elements for Fiat-Shamir transform using hints and OP_CAT + OP_SHA256.
• Proof-of-Work Check
  • calculating a proof-of-work nonce for the "channel", based on specified security bits.
  • verifying the proof-of-work nonce and computing the new "channel" state.
• Merkle Tree
  • implementation of Merkle path verification using hints and OP_CAT + OP_SHA256.

The next step is to implement the FRI protocol, which reasons about the degree of a quotient polynomial.

---

Performance

These performance numbers are obtained from cargo test -- --nocapture over commit 6e5c211.

• M31, QM31
  • M31.add = 18 bytes, QM31.add = 84 bytes
  • M31.sub = 12 bytes, QM31.sub = 63 bytes
  • M31.mul = 1415 bytes, QM31.mul = 13321 bytes
  • M31.mul_by_constant = ~744 bytes, QM31.mul_by_m31_constant = ~2981 bytes
  • QM31.mul_by_m31 = 4702 bytes
  • QM31.commit = 7 bytes, QM31.from_hash = 250 bytes
  • 5M31.from_hash = 312 bytes
• CirclePoint over QM31
  • CirclePoint.double_x = 13505 bytes
  • CirclePoint.get_random_point = 40546 bytes
  • CirclePoint.add = 40542 bytes
  • CirclePoint.add_constant_point = ~21780 bytes
  • CirclePoint.add_constant_m31_point = ~9235 bytes
  • CirclePoint.add_x_only = 26791 bytes
• Constraints on the circle curve over QM31
  • Constraints.pair_vanishing_with_constant_m31_points = ~6195 bytes
  • Constraints.coset_vanishing(log_size=5) = 80827 bytes
  • Constraints.coset_vanishing(log_size=6) = 94332 bytes
  • Constraints.coset_vanishing(log_size=7) = 107837 bytes
  • Constraints.coset_vanishing(log_size=8) = 121342 bytes
  • Constraints.coset_vanishing(log_size=9) = 134847 bytes
• Fiat-Shamir Transcript
  • Channel.mix_digest = 2 bytes
  • Channel.mix_felts = 9 bytes per felt
  • Channel.squeeze_element_using_hint = 257 bytes (require 5 hint elements)
  • Channel.squeeze_5queries_using_hint = 1222 bytes (require 6 hint elements)
• Proof-of-Work Check
  • POW.verify_pow(1 bit) = 39 bytes
  • POW.verify_pow(2 bits) = 38 bytes
  • POW.verify_pow(3 bits) = 38 bytes
  • POW.verify_pow(4 bits) = 37 bytes
  • POW.verify_pow(5 bits) = 37 bytes
  • POW.verify_pow(6 bits) = 37 bytes
  • POW.verify_pow(7 bits) = 37 bytes
  • POW.verify_pow(8 bits) = 21 bytes
  • POW.verify_pow(9 bits) = 42 bytes
  • POW.verify_pow(10 bits) = 41 bytes
  • POW.verify_pow(11 bits) = 41 bytes
  • POW.verify_pow(12 bits) = 40 bytes
  • POW.verify_pow(13 bits) = 40 bytes
  • POW.verify_pow(14 bits) = 40 bytes
  • POW.verify_pow(15 bits) = 40 bytes
  • POW.verify_pow(16 bits) = 22 bytes
  • POW.verify_pow(17 bits) = 43 bytes
  • POW.verify_pow(18 bits) = 42 bytes
  • POW.verify_pow(19 bits) = 42 bytes
  • POW.verify_pow(20 bits) = 41 bytes
• Merkle tree
  • MT.verify(2^12) = 263 bytes (require 11 hint elements)
  • MT.verify(2^14) = 309 bytes (require 13 hint elements)
  • MT.verify(2^16) = 356 bytes (require 15 hint elements)
  • MT.verify(2^18) = 404 bytes (require 18 hint elements)
  • MT.verify(2^20) = 452 bytes (require 20 hint elements)

---

Channel

The channel is used for Fiat-Shamir transform. It absorbs elements that are either prior knowledge of the verifier or provers' messages, and it can be squeezed to produce pseudorandom elements. There are five operations.

• new(digest) -> channel: initialize a new channel using an initial digest
  • channel := digest
• mix_digest(channel, digest) -> channel': update the channel with a digest
  • channel' := SHA256(channel || commitment)
• mix_felts(channel, [qm31]) -> channel': update the channel with QM31 elements
  • channel' := SHA256(channel || commit(qm31)) for each qm31
• squeeze(channel) -> (qm31, channel'): squeeze a QM31 element out of the channel
  • hash := SHA256(channel || 0x00)
  • channel' := SHA256(channel)
  • qm31 := extract(hash)
• squeeze(channel, logn) -> (q1, q2, q3, q4, q5, channel'): squeeze five positions on a list of 2^logn elements
  • hash := SHA256(channel || 0x00)
  • channel' := SHA256(channel)
  • q1, q2, q3, q4, q5 := extract(hash, logn)

The constructions of commit and extract are as follows.

Commit. With OP_CAT + OP_SHA256, we can commit QM31 elements with a few bytes.

QM31 requires 7 bytes: commit(qm31) := SHA256(qm31.0.0 || SHA256(qm31.0.1 || SHA256(qm31.1.0 || SHA256(qm31.1.1))))

OP_SHA256 OP_CAT OP_SHA256 OP_CAT OP_SHA256 OP_CAT OP_SHA256

Extract. Since we do not have OP_SUBSTR, to extract a QM31 element or five positions from the hash, we need to use hints. The idea is to peel off the first few bytes of the hash and recreate a normalized QM31 element out of it. If we want to extract positions, the numbers are further adjusted to have only logn bits.

• For QM31, the hint includes five elements:
  • hash[0..4] in the minimal number encoding form
  • hash[4..8] in the minimal number encoding form
  • hash[8..12] in the minimal number encoding form
  • hash[12..16] in the minimal number encoding form
  • hash[16..32] as bytes, which is the tail
• For five positions, the hint includes six elements:
  • hash[0..4] in the minimal number encoding form
  • hash[4..8] in the minimal number encoding form
  • hash[8..12] in the minimal number encoding form
  • hash[12..16] in the minimal number encoding form
  • hash[16..20] in the minimal number encoding form
  • hash[20..32] as bytes, which is the tail

Due to the minimal number encoding form, the hint element, which represents a signed 32-bit integer, does not necessarily have four bytes. Our solution is to use OP_SIZE to detect its length and then use OP_CAT to pad it to four bytes. A subtlety here is that negative numbers, which occur when the original hash's last byte's most significant bit is 1, need to be handled differently, as it would first be padded with OP_LEFT (0x80) and then OP_PUSHBYTES_0 (0x00).

OP_IF
OP_PUSHBYTES_1 OP_PUSHBYTES_0
OP_ELSE
OP_PUSHBYTES_1 OP_LEFT
OP_ENDIF

Note that OP_PUSHBYTES_0 and OP_LEFT here are not opcodes but rather data that OP_PUSHBYTES_1 will push to the stack. One cannot directly write 0x00 and 0x80 in the interpreter, as they would become an empty string and 0x80 0x00, respectively.

It first uses OP_CAT to combine the hint elements together and compare it with the hash that is to be extracted from.

Each of the hint number, after peeling off the sign bit, becomes a non-negative 31-bit number, ranging from 0 to 2^31 - 1. However, in the rest of the computation, we want the number to be below 2^31 - 1. This is done by subtracting one from the resulting number unless the resulting number is zero.

OP_DUP OP_NOT 
OP_NOTIF OP_1SUB OP_ENDIF

After such adjustment, one obtains an element.

---

Proof of Work

The Proof-of-Work check accepts two inputs: channel and nonce, and checks whether the new channel state channel'=sha256(channel||nonce) has sufficiently many security bits, namely, n_bits.

Since we don't have OP_SUBSTR, the check also requires hints that change depending on whether n_bis is divisible by 8 or not.

If n_bits % 8==0, there is a single hint suffix, and the script checks that

0^(n_bits//8)||suffix==channel'

If n_bits % 8!=0, together with suffix, there is an additional byte-sized hint msb. Consequently, the script checks that

0^(n_bits//8)||msb||suffix==channel'

and also that msb starts with n_bits % 8 (which would be at least 1) zero bits.

---

License and contributors

This repository is intended to be public good. It is under the MIT license.

A portion of the code is contributed by L2 Iterative (L2IV), a crypto VC based in San Francisco and Hong Kong. The work receives support from Starkware, who is a limited partner in L2IV. For disclosure, L2IV has also invested into numerous companies active in the Bitcoin ecosystem, but this work is open-source and nonprofit, and is not intended for competition. The code is not investment advice.

Starkware contributes a portion of the code, including the original Rust FRI implementation (from stwo) and some Bitcoin scripts.

There are also community members contributing to the code and contributing to the ideas. Bitcoin Wildlife Sanctuary is a public-good project supported by many people.

Below we reiterate the contributors to this repository.

victorkstarkware 💻

Abdel @ StarkWare 🚧

Dependabot 🛡️

PayneJoe 💻

janusz 👀