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- [ ] # Keccak Halo2 implementation spec``
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## Acknowledgement
We port Konstantin Panarin's idea of [sparse representation](https://hackmd.io/xfgP5_uMTZyaEJJG4EJoRQ) and the [code examples](https://github.com/matter-labs/franklin-crypto/blob/dev/src/plonk/circuit/hashes_with_tables/keccak/gadgets.rs) to Halo2.
## Terminology
### Keccak
### Sparse representation
chunk
: a chunk is a bit
slice
: a slice is some chunks
The lane size is 64 bits. We split it into slices. In the binary to 13 base mapping, we work with 16 bits or 16 chunks per slice. A lane has 4 slices.
In the 13 base to 9 base mapping, we work with 4 bits or 4 chunks per slice.
first base
: the number 13
second base
: the number 9
## Introduction
### Overview
The original keccak
```python
# 1word = 64 bits
# 200 bytes, 25 words
state = [0 for i in range(200)]
for _input in split_every_17_word(raw_input):
# take rate=17 words of input
state ^= _input
# keccak_f(state)
for i in range(24):
# round_b
state = theta(state)
state = rho(state, rotation[i])
state = pi(state)
state = xi(state)
# add round constant to state[0][0]
state = iota(state, round_constant[i])
return state
```
How we do keccak in the circuit
```python
first_input, rest_inputs = split_every_17_word(raw_input)
# take
state = convert_base(first_input, _from=2, _to=13)
for _input in rest_inputs:
next_mixing = convert_base(_input, _from=2, _to=9)
state = keccak_f(state, next_mixing)
state = keccak_f(state, None)
# TODO: padding
def keccak_f(state, next_mixing):
for i in range(23):
# enter state of 25 lanes, each lane is 13 base 64 chunks
state = theta(state)
# enter with 13 base 65 chunks
state = rho(state, rotation[i])
# leave with 9 base 64 chunks
state = pi(state)
# first 23 rounds we add round_constant but no absorb
state = xi(state)
state = iota_b9(state, round_constant[i])
state = convert_base(state, _from=9, _to=13)
state = theta(state)
state = rho(state, rotation[23])
state = pi(state)
state = xi(state)
# last round:
if next_mixing is None:
# if we don't have new input to mix, just add the round_constant
state = iota_b9(
state,
convert_base(round_constant[23], _from=2, _to=9),
)
else:
# if we have new input to mix, add the round_constant after the base conversion.
# This is the case the coefficient of `state[0][0]` in the next theta could be 2.
state = absorb(state, next_mixing)
state = convert_base(state, _from=9, _to=13)
state = iota_b13(
state,
convert_base(round_constant[23], _from=2, _to=13),
)
```
### The first base and the second base
Why the first sparse base is 13? because in theta step we have
- at most 12 0~1 variables adding together.
- a rotate_left(, 1)
Note that we don't do full rotate_left(, 1), we do shift_left(, 1) instead and save the wrap over in the later step. After shift_left the sparse representation is at most 13^65 and is about 241 bits.
Why the second sparse base is 9? because in the xi and iota steps combined we have at most 8 0~1 variables adding together.
### Break down of the theta step
```
new_state[x][y] = state[x][y] + c[(x+4)% 5] + 13 * c[(x+1)%5]
```
```
state[x][y] s63 s62 s61 s2 s1 s0
c[(x+4)% 5] c4_63 c4_62 c4_61 c4_2 c4_1 c4_0
+) 13 * c[(x+1)% 5] c1_63 c1_62 c1_61 c1_60 c1_1 c1_0
---------------------------------------------------------------------
new_state[x][y] ns64 ns63 ns62 ns61 ... ns2 ns1 ns0
```
`new_state[x][y]` is a base 13 number with 65 chunks.
The full roation is supposed to give us a base 13 value `t` with 64 chunks. Let's denote it `t0`~`t63`.
The `t0` is supposed to be `s0 + c4_0 + c1_63`. But in `new_state[x][y]`, `s0 + c4_0` is at the `ns0` and `c1_63` is at the `ns64`.
We call the `ns0` "last chunk high value" and `ns64` "last chunk low value."
### Break down the rotate_and_convert
In the rotate_and_convert, we do the rotation and the conversion from base 13 to base 9 simultaniously.
For example, we want to rotate 5. Let's first observe the ground truth `t`
we assume `first base converter` has been applied on `t`, so that each chunk `ti` is either 0 or 1.
```
t t63 t62 t61 t60 t59 t58 ... t5 t4 t3 t2 t1 t0
rho(t, 5) t58 t57 t56 t55 t54 t53 ... t0 t63 t62 t61 t60 t59
```
```
t t63 t62 t61 t60 t59 t58 ... t5 t4 t3 t2 t1 t0
base 9 9^4 9^3 9^2 9 1 9^63 9^10 9^9 9^8 9^7 9^6 9^5
```
The sum-product of t and the base gives us a 64 chunks base 9 number.
But the `new_state[x][y]` from the theta step has the special chunks "chunk 0" and "chunk 64"
We can handle the rest of the chunks as usual, but the `ns0` and `ns64` needs special care.
```
nsxy ns64 ns63 ns62 ns61 ns60 ns59 ns58 ... ns5 ns4 ns3 ns2 ns1 ns0
base 9 9^4 9^3 9^2 9 1 9^63 9^10 9^9 9^8 9^7 9^6
```
add the sum-product with `keccak_u64_first_converter(ns0 + ns64)*9**5` and we get the correctly rotated base 9 number.
Example
```
nsxy ns64 ns63 ns62 ns61 ns60 ns59 ns58 ... ns5 ns4 ns3 ns2 ns1 ns0
example 1 7 6 0 5 8 10 9 2 1 12 3 11
# apply first base converter for the middle chunks
converted 1 0 0 1 0 0 1 0 1 0 1
base 9 9^4 9^3 9^2 9 1 9^63 9^10 9^9 9^8 9^7 9^6
```
Note that `ns64 + ns0` must be less than or equal to 12
`keccak_u64_first_converter(ns0 + ns64)` evalutes to 0
### Break down the rho step validation
## How do we implement Keccak steps in the circuit?
### Theta
Original
```python
# state in base 2
def theta(state: List[List[int]]):
c = [state[x][0] ^ state[x][1] ^ state[x][2] ^ state[x][3] ^ state[x][4] for x in range(5)]
d = [c[x - 1] ^ rotate_left(c[(x + 1) % 5], 1) for x in range(5)]
for x in range(5):
for y in range(5):
state[x][y] ^= d[x]
return state
```
In circuit
```python=
# state in base 13
# input state: coefficients of state[0][0] in 0~2, other state[x][y] in 0~1
# output state: coefficients in 0~12
def theta(state: List[List[int]]):
c = [state[x][0] + state[x][1] + state[x][2] + state[x][3] + state[x][4] for x in range(5)]
new_state = [[0 for x in range(5)] for y in range(5)]
for x in range(5):
for y in range(5):
new_state[x][y] = state[x][y] + c[(x+4)% 5] + 13 * c[(x+1)%5]
return new_state
```
#### The argument that each output lane's coefficients is at most 12
Note that in the middle of the rounds we might enter the theta step from previous iota step, which is a step that xors a round constant to `state[0][0]`. So that `state[0][0]` might start with 2.
Since `state[0][0]` might start with 2 and `c[0]` contains a `state[0][0]`, `c[0]` could be at most 6.
Taking a closer look to the number `new_state[x][y]` in base 13 system at line 9, we observe the largest possible value for coefficient at any given position.
For `x not in (0, 1, 4)`, we have `state[x][y]` at most 1, `c[(x+4)% 5]` at most 5, and `13 * c[(x+1)%5]` from the previous position that contributs at most 5. The total is at most 1+5+5 = 11.
For `x == 0`, we have `state[x][y]` at most 2, which happens at `state[0][0]`, `c[(x+4)% 5]` at most 5 since `state[0][0]` is not in it, and `13 * c[(x+1)%5]` at most 5 for the same reason. The total is at most 2 + 5 + 5 = 12.
For `x in (1, 4)`, `state[x][y]` is at most 1. One of `c[(x+4)% 5]` or `c[(x+1)%5]`, but not both, could be at most 6 since `state[0][0]` is in exact one of them. The total is at most 1 + 5 + 6 = 12.
#### Cost analysis
5 linear combination for c
25 linear combination for new_state
### Pi
Nothing too interesting here
```python
# state in base 9
def pi(state: List[List[int]]):
new_state = [[0 for x in range(5)] for y in range(5)]
for x in range(5):
for y in range(5):
new_state[x][y] = state[(x + 3* y) % 5][x]
```
#### Cost analysis
No cost in circuit???
### Rho
Original
```python
def rho(state: List[List[int]]):
for x in range(5):
for y in range(5):
state[x][y] = rotate_left(state[x][y], ROT[x][y])
return state[x][y]
```
In circuit
```python=
# input state[x][y] is in base 13, has shifted left 1, and thus has 65 chunks
# output state[x][y] is in base 9 and rotated properly. It has 64 chunks, each coefficient is 0~1.
def rho(state: List[List[int]]):
offset_map = {
1: [],
2: [],
3: []
}
new_state = [[0 for x in range(5)] for y in range(5)]
for x in range(5):
for y in range(5):
# in base 9, coefficients 0~1. Rotation has applied
new_state[x][y] = rotate_and_convert(state[x][y], ROT[x][y])
output_slices = []
output_coefs = []
# `acc` looks like the `raw` later. The difference is
# acc is witnessed and checked
acc = state[x][y]
last_chunk_low_value = state[x][y] % 13
# raw is computed by the prover but not witnessed
raw = state[x][y] // 13
last_coef = 9**ROT[x][y]
# coef of base 13 slice
cur_input_coef = 13
# coef of base 9 slice
cur_output_coef = 1 if ROT[x][y] == 63 else last_coef * 9
# which chunk are we at
cur_offset = 1
# Looping through slices
while cur_offset < 64:
# step could be 1, 2, 3, 4
step = get_step_size(cur_offset, ROT[x][y])
input_slice = raw % 13**step
raw //= 13**step
block_count, output_slice = first_to_second_base_converter_table.lookup(input_slice)
acc -= cur_input_coef * input_slice
output_slices.append(output_slice)
cur_offset += step
cur_input_coef *= 13 ** step
cur_output_coef = (
1 if cur_offset == 64 - ROT[x][y]
else cur_output_coef * 9 ** step
)
# keep track of non 4 step
if step < 4:
offset_map[step].append(block_count)
last_chunk_high_value = raw % 13
assert raw // 13 == 0, "No chunk should be left in the raw"
last_chunk_value = last_chunk_high_value * (13**64) + last_chunk_low_value
# we are looking up table to get this evaluation in the circuit
# keccak_u64_first_converter(last_chunk_low_value + last_chunk_high_value)
output_slice = of_first_to_second_base_converter_table.lookup(
last_chunk_value
)
assert acc == last_chunk_value
output_coefs.append(last_coef)
output_slices.append(output_slice)
# linear combination check
assert new_state[x][y] == sum([s*c for s, c in zip(output_slices, output_coefs)])
# offset_transformed = [0, 0, 1, 13, 170]
# for i in (1, 2, 3):
# # Perform a range check here
# assert sum(offset_map[i]) <= offset_transformed[i] * len(offset_map[i])
assert sum(offset_map[1]) == 0
assert sum(offset_map[2]) <= len(offset_map[2])
assert sum(offset_map[3]) <= 13 * len(offset_map[3])
return new_state
```
#### Cost analysis
- 25 lanes each needs
- for middle slices: approxmate 16 iterations (64 chunks /4 chunks) in the while loop, each needs
- a lookup to first_to_second_base_converter_table
- 1 linear combination check for current acc value
- for special chunk
- a lookup to of_first_to_second_base_converter_table
- check the remaining acc == last_chunk_value
- 1 linear combination check for new_state[x][y] and the sum product of output_slices and output_coefs
lastly we need 3 linear combination check for offset_map for each non-4 step
total:
- 25 * 16 lookups to first_to_second_base_converter_table lookup
- 25 lookups to of_first_to_second_base_converter_table
- 25*(16 + 1 + 1) + 3 = 453 linear combinations
### Xi + Iota step
Original
```python
# state in base 2
def xi_and_iota(state: List[List[int]], round_constant: int):
new_state = [[0 for x in range(5)] for y in range(5)]
# Xi step
for x in range(5):
for y in range(5):
new_state[x][y] = state[x][y] ^ ((~state[(x + 1) % 5][y]) & state[(x + 2) % 5][y])
# Iota step
new_state[0][0] ^= round_constant
return new_state
```
Original: Merging Iota step
```python
# state in base 2
def xi_and_iota(state: List[List[int]], round_constant: int):
new_state = [[0 for x in range(5)] for y in range(5)]
for x in range(5):
for y in range(5):
a = state[x][y]
b = state[(x + 1) % 5][y]
c = state[(x + 2) % 5][y]
d = 0 if x!=0 and y!=0 else round_constant
new_state[x][y] = a ^ (~b & c) ^ d
return new_state
```
We have the `keccak_u64_second_converter` function to map `2*a + b + 3*c + 2*d` to `a ^ (~b & c) ^ d`.
Depending if we are at the last round of round_b or last round of keccak_f, we'll use `d` to do
- iota step to add a round constant in base 9, or
- absorb next input to step, then add round constant in base 13 later.
In the circuit
```python
# state in base 9, coefficient in 0~1
def xi(state: List[List[int]]):
new_state = [[0 for x in range(5)] for y in range(5)]
# Xi step
for x in range(5):
for y in range(5):
# a, b, c, d are base9, coefficient in 0~1
a = state[x][y]
b = state[(x + 1) % 5][y]
c = state[(x + 2) % 5][y]
# coefficient in 0~6
new_state[x][y] = 2*a + b + 3*c
return new_state
```
```python
# state in base 9, coefficient in 0~6
# next_input is state in base 9, coefficient in 0~1
def absorb(state: List[List[int], next_input: List[List[int]):
for idx in range(17):
# state[idx] has 2*a + b + 3*c already, now add 2*d to make it 2*a + b + 3*c + 2*d
# coefficient in 0~8
state[idx] += 2 * next_input[idx]
return state
```
```python
# state in base 9, coefficient in 0~6
def iota_b9(state: List[List[int], round_constant_base9: int):
d = round_constant_base9
# state[0][0] has 2*a + b + 3*c already, now add 2*d to make it 2*a + b + 3*c + 2*d
# coefficient in 0~8
state[0][0] += 2*d
return state
```
```python
def iota_b13(state: List[List[int], round_constant_base13: int):
state[0][0] += round_constant_base13
return state
```
#### Cost analysis
25 linear combination check for `new_state[x][y]`
## Helpers
### get_step_size (check_offset_helper)
```python
# cur_offset begins with 1
def get_step_size(cur_offset: int, max_offset: int) -> int:
"""
Is the `check_offset_helper` in rust with
base_num_of_chunks = 4
"""
# near the start of the lane
if cur_offset < max_offset < cur_offset + 4:
return max_offset - cur_offset
# near the end of the lane
if cur_offset < 64 < cur_offset + 4:
return 64 - cur_offset
return 4
```
### get_steps
```python
def get_steps(max_offset: int):
cur_offset = 1
offsets = [cur_offset]
steps = []
while cur_offset < 64:
step = get_step_size(cur_offset, max_offset)
cur_offset += step
steps.append(step)
offsets.append(cur_offset)
print("=======")
print("max_offset", max_offset)
print("offsets", offsets, "length", len(offsets))
print("steps", steps)
```
```
get_steps(1)
get_steps(4)
get_steps(5)
get_steps(63)
=======
max_offset 1
offsets [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 64] length 17
steps [4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3]
=======
max_offset 4
offsets [1, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64] length 17
steps [3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4]
=======
max_offset 5
offsets [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 64] length 17
steps [4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3]
=======
max_offset 63
offsets [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 63, 64] length 18
steps [4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1]
```
### keccak_u64_first_converter
```python
def keccak_u64_first_converter(n: int) -> int:
"""
n is the sum of 12 different bits.
The theta step has 12 xor operations
If the sum is odd, then the 12 xor operations result 1
If the sum is even, then the 12 xor operations result 0
"""
assert n < 13
return n & 1
```
### keccak_u64_second_converter
```python
def keccak_u64_second_converter(n: int) -> int:
"""
n is the output of 2a + b + 3c + 2d, where a, b, c, d are bits
every possible n can be uniquely mapped to the output of a ^ (~b & c) ^ d
bit_table is the output of a ^ (~b & c) ^ d
"""
assert n < 9
bit_table = [0, 0, 1, 1, 0, 0, 1, 1, 0]
return bit_table[n]
```
```
f_logic = a ^ (~b & c) ^ d
f_algebratic = 2a + b + 3c + 2d
```
| idx | a | b | c | d | a ^ (~b & c) ^ d | 2a + b + 3c + 2d |
|-----|---|---|---|---|------------------|------------------|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 1 | 1 | 2 |
| 2 | 0 | 0 | 1 | 0 | 1 | 3 |
| 3 | 0 | 0 | 1 | 1 | 0 | 5 |
| 4 | 0 | 1 | 0 | 0 | 0 | 1 |
| 5 | 0 | 1 | 0 | 1 | 1 | 3 |
| 6 | 0 | 1 | 1 | 0 | 0 | 4 |
| 7 | 0 | 1 | 1 | 1 | 1 | 6 |
| 8 | 1 | 0 | 0 | 0 | 1 | 2 |
| 9 | 1 | 0 | 0 | 1 | 0 | 4 |
| 10 | 1 | 0 | 1 | 0 | 0 | 5 |
| 11 | 1 | 0 | 1 | 1 | 1 | 7 |
| 12 | 1 | 1 | 0 | 0 | 1 | 3 |
| 13 | 1 | 1 | 0 | 1 | 0 | 5 |
| 14 | 1 | 1 | 1 | 0 | 1 | 6 |
| 15 | 1 | 1 | 1 | 1 | 0 | 8 |
The idx is from 0~15
The a, b, c, d are binary decomposition
If we sort by the f_alg column and select f_logic column together, we get the following table. Don't worry about the duplicated rows, they are consistent since f_algebratic -> f_logic is well-defined
| 2a + b + 3c + 2d | a ^ (~b & c) ^ d |
| ----------------:| ----------------:|
| 0 | 0 |
| 1 | 0 |
| 2 | 1 |
| 3 | 1 |
| 4 | 0 |
| 5 | 0 |
| 6 | 1 |
| 7 | 1 |
| 8 | 0 |
If we implement that table it would look just like `keccak_u64_second_converter`
### rotate_and_convert
```python
def rotate_and_convert(base13_input: int, rot: int) -> int:
"""
Convert the base13 input to base9 output
base13_input is assumed having shifted left 1
the coefficient of base9 is the result of binary operations of base13 coefficient, which is 0~1
rot is applied on base9
"""
base = 9**rot
special_chunk = 0
raw = base13_input
acc = 0
for i in range(65):
remainder = raw % 13
if i in (0, 64):
# t0 and t64
special_chunk += remainder
else:
acc += keccak_u64_first_converter(remainder) * base
raw /= 13
base *= 9
if i == 64 - rot:
base = 1
acc += keccak_u64_first_converter(special_chunk) * 9**rot
return acc
```
### normalizer
```python
def normalizer(
_input: int,
input_base: int,
output_base: int,
transform_f:Callable[[int], int]
) -> int:
"""
Takes a _input in input_base, converts to output in output_base
"""
output = 0
base = 1
while _input > 0:
remainder = _input % input_base
output += transform_f(remainder) * base
_input /= input_base
base *= output_base
return output
```
## Tables
### from_binary_converter_table
Purpose: Convert 64 bits value from binary to base 13 in 16 bits chunks
```python
>>> import itertools
>>> list(itertools.product([1, 2, 3], [4, 5, 6]))
[(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)]
```
This table has 2**16 rows
```python
class FromBinaryConverterTable:
def __init__(self):
axies = [[0, 1] for i in range(16)]
for coefs in itertools.product(*axies):
assert len(coefs) == 16
# key is the binary evaluation of coefs
key = sum([coef*(2**i) for i, coef in enumerate(coefs)])
value0 = sum([coef*(13**i) for i, coef in enumerate(coefs)])
value1 = sum([coef*(9**i) for i, coef in enumerate(coefs)])
self.add_row(key, value0, value1)
```
### first_to_second_base_converter_table
Purpose: convert the middle slices, where the chunk size is 4
This table has 13**4 rows
```python
def block_counting_function(n: int) -> int:
"""
aka the function g.
The table is reverse engineered from the trace of `of_transformed`
See: Decyphering block counting function section
"""
table = [0, 0, 1, 13, 170]
return table[n]
class FirstToSecondBaseConverterTable:
def __init__(self):
axies = [list(range(13)) for i in range(4)]
for coefs in itertools.product(*axies):
assert len(coefs) == 4
# x0, x1, x2, x3 are 0~12
x0, x1, x2, x3 = coefs
key = x0 + x1 * 13 + x2 *13**2 + x3*13**3
fx0 = keccak_u64_first_converter(x0)
fx1 = keccak_u64_first_converter(x1)
fx2 = keccak_u64_first_converter(x2)
fx3 = keccak_u64_first_converter(x3)
# fx0, fx1, fx2, fx3 are 0~1
value = fx0 + fx1 * 9 + fx2 *9**2 + fx3*9**3
# could be 0, 1, 2, 3, 4
non_zero_chunk_count = 4 - len([i for i in coefs if i==0])
# could be 0, 0, 1, 13, 170
block_count = block_counting_function(non_zero_chunk_count)
self.add_row(key, block_count, value)
```
### of_first_to_second_base_converter_table
Purpose: convert the t0 and the t64 chunks from the arithmetic result of the sparse form to the bitwise operation result of the binary form.
The table size should be (13 + 1) * 13 / 2 = 91 rows
```python
class OfFirstToSecondBaseConverterTable:
def __init__(self):
for i in range(13):
for j in range(13 - i):
low = i
high = j
# 13 ** 64 is the base_b ^ offset
key = low + high * (13 ** 64)
value = keccak_u64_first_converter(low + high)
self.add_row(key, value)
```
### from_second_base_converter_table
Purpose: Convert from base 9 to base 13 or binary
This table has 9**5 rows
```python
class FromSecondBaseConverterTable:
def __init__(self):
axies = [list(range(9)) for i in range(5)]
for coefs in itertools.product(*axies):
assert len(coefs) == 5
# key is the binary evaluation of coefs
key = sum([coef*(9**i) for i, coef in enumerate(coefs)])
value0 = sum([coef*(13**i) for i, coef in enumerate(coefs)])
value1 = sum([coef*(2**i) for i, coef in enumerate(coefs)])
self.add_row(key, value0, value1)
```
## Sponges
Keccak use simple sponge
For Keccak256, the squeeze phase is a no-op
First we absorb the input bytes. Convert them to base 13 sparse representation using from_binary_converter_table.
We absorb at a rate of 16 bits. (could enlarge the table size to optimize this)
state is initialized 1600 bits.
We absorb 1088 bits of input at a time.
We run keccak_f1600 for 1600 bits state to get the next state.
When we finish the absorb, we xored 0x01 and 0x80 at specific place. Run keccak_f1600 again.
Return first 256 bits of the state.
## Cost analysis
- keccak1600 for 25 lanes (total 500ish lcs + 845 lookups)
- 100 lookups (=1600/16) for 16 chunks b2tob13 conversion (table is $2^{16}$ rows)
- 30 lcs in theta
- Rho
- 2800 lcs (running down (2) + running up (2) + block count acc (3)) *16 *25
- 400 lookups in Base13To9Table(28.5k rows)
- 25 lookups in specialChunkTable(170 rows) in rho
- final check
- range check 0~12
- range check 0~169
- 25 lcs in xi+iota
- 320 lookups (25 lanes * 64 chunks/5 chunks) for 5 chunks b9tob2 conversion (59k rows)
- sponge 1 absorb loop
- takes 1088 bits input (1088/16= 68 b2tob13 conversion)
- 1 keccak1600
- sponge padding when absorb complete
- 1 keccak1600
- for n bytes input
- x = 8*n / 1088 absorb phases
- x + 1 keccak1600
- for a 32 bytes word in EVM (x = 0)
- 0 absorb loop
- 1 padding keccak1600
## (WIP) Working with multiple inputs
> Han: yeah it’s determined at proving time. Naively we can handle keccak opcode by copying the memory value from bus mapping to another table, which could contains (hash, offset, u64_0, …, u64_16), then keccak circuit consume the table row by row to update the hash state, and it only resets the state (re-initialize) when hash != hash_prev hash == output
> To keep things as simple as possible, I take opcode SHA3 as an example and limit the keccak circuit’s access to bus mapping.
>1. When evm circuit sees a SHA3, it takes the offset and size from stack, then it starts a multi-step process to copy memory[offset:offset+size] from BusMapping to another table, let’s call it KeccakIO. The action *copy* here is actually do lookup twice to both tables and ensure thet have same row.
>2. Naively, we can have KeccakIO with 19 columns which are (hash, offset, data0, …, data16), where data* are decided by the absorbing rate
>3. Then in keccak circuit, say we have a chip keccak-f given (old_state, data0, …, data16) and will ouput (new_state, hash)
>4. What we need to do next is to repeat the keccak-f chip at much as possible, lookup the KeccakIO table to get some data to absorb, and update the state and see if output hash is matching to the one in table.
> 5. If it’s not matching, we keep absorb until matched.
> 6. If it’s matching, we reset the state for next chip, and start to verify next hash.
> 7. Somehow we need a mechanism like global_counter in BusMapping to avoid malicious insertion of invalid hash data pair.
## Appendix
### (WIP) Decyphering block counting function
What's known
- `g = |x| { of_transformed[x as usize] };`
- The value of `counts` is `[12, 12, 13]`. The first_base_num_of_chunks is 4, useually we lookup the 4 bit chunks. But there are exceptional cases, the `counts` means
- 12 cases in the OFFSETS, we have to deal with 1 chunks
- 12 cases in the OFFSETS, we have to deal with 2 chunks
- 13 cases in the OFFSETS, we have to deal with 3 chunks
- "of_transformed" is from "counts"
- printed log shows the value of "of_transformed" is [0, 0, 1, 13, 170]
- They have a unused constant MAX_OF_COUNT_PER_ITER= 50.
What's unknown
- The value of "of_transformed" does not match the code comments. The value of g(3) is 13 and is not at least 51
- what does "25 shifted rows so there are at most 50 overflows" mean?
https://github.com/matter-labs/franklin-crypto/blob/a78e17674dd1c99788259986cae11239953b50af/src/plonk/circuit/hashes_with_tables/keccak/gadgets.rs#L176
> we have 25 shifted rows so there are at most 50 overflows
let g(0) = 0, g(1) = 0, g(2) = 1
g(3) should be than at least 51
g(4) than should be g(3) * 50 + 1 = 2551
and so forth: g(i+1) = 25 * g(i) + 1
```python
# OFFSETS are the 64 - ROTs
OFFSETS = [
[64, 28, 61, 23, 46],
[63, 20, 54, 19, 62],
[2, 58, 21, 49, 3],
[36, 9, 39, 43, 8],
[37, 44, 25, 56, 50]
]
def get_step_size(cur_offset: int, max_offset: int) -> int:
"""
Is the `check_offset_helper` in rust with
base_num_of_chunks = 4
outputs either 1, 2, 3, 4
"""
# near the start of the lane
if cur_offset < max_offset < cur_offset + 4:
return max_offset - cur_offset
# near the end of the lane
if cur_offset < 64 < cur_offset + 4:
return 64 - cur_offset
return 4
def get_counts():
counts = [0, 0, 0]
for row in OFFSETS:
for max_offset in row:
cur_offset = 1
while cur_offset < 64:
step = get_step_size(cur_offset, max_offset)
if step != 4:
counts[step - 1] += 1;
cur_offset += step
print("counts", counts) # [12, 12, 13]
of_transformed = [0, 0]
for c in counts:
elem = of_transformed[-1]
of_transformed.append(c* elem + 1)
print("of_transformed", of_transformed) # [0, 0, 1, 13, 170]
get_counts()
```
## Chips
expose keccak256 as a gadget
impl keccak256 for keccak-f
keccak-f a trait,
gadget
rho, theta,
gadget is a group of traits
chips implements the traits
keccak-f as the unit interface
try learn from the poseidon example https://github.com/zcash/orchard/blob/main/src/circuit/gadget/poseidon.rs
## (WIP) Circuit as a Lookup Table
Because most `keccak256` in evm doesn't have fixed length input. If we want to handle all cases including the worst case by defining some max input length with a complete `keccak256`, we might have to fine-tune the circuit to not waste too much.
Another approach is to not have a complete `keccak256`, we instead have repeating `keccak_f` with some beginning and end condition, which seems more friendly for other circuit to use.
Naively each `keccak_f` could look like:
- size: 1~17, the current block size (for later padding use)
- offset: `if not is_final` then we have to bump the number of offset (either 1 or 17)
where `i_*` is input and the region inside waves we perform `keccak_f`.
Other circuits call it like `lookup(i_1, i_2, i_3, ..., i_17, hash, offset, size, is_final)` to ensure `keccak256` is correct. (would call multiple times if the input is more than 17 words)
Then keccak256 circuit will set **next input** to all 0 and check if `hash` is matching **next output** `(o_1, o_2, o_3, o_4)` when `is_final` is set to `1`.
The reason for **next input** and **next output** is because current algorithm kind of defer the `keccak_f` to next round with no input. If we are changing the order, the base 13 would not work anymore.
When the `is_final` =1, we do the padding.
> The approach to handle when `is_final` is set to `1` seems to have much room for optimization, but not have too much thought for now.
> [name=han]
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