EIP-198: Big integer modular exponentiation
|Authors||Vitalik Buterin (@vbuterin)|
Table of Contents
At address 0x00……05, add a precompile that expects input in the following format:
<length_of_BASE> <length_of_EXPONENT> <length_of_MODULUS> <BASE> <EXPONENT> <MODULUS>
Where every length is a 32-byte left-padded integer representing the number of bytes to be taken up by the next value. Call data is assumed to be infinitely right-padded with zero bytes, and excess data is ignored. Consumes
floor(mult_complexity(max(length_of_MODULUS, length_of_BASE)) * max(ADJUSTED_EXPONENT_LENGTH, 1) / GQUADDIVISOR) gas, and if there is enough gas, returns an output
(BASE**EXPONENT) % MODULUS as a byte array with the same length as the modulus.
ADJUSTED_EXPONENT_LENGTH is defined as follows.
length_of_EXPONENT <= 32, and all bits in
EXPONENTare 0, return 0
length_of_EXPONENT <= 32, then return the index of the highest bit in
EXPONENT(eg. 1 -> 0, 2 -> 1, 3 -> 1, 255 -> 7, 256 -> 8).
length_of_EXPONENT > 32, then return
8 * (length_of_EXPONENT - 32)plus the index of the highest bit in the first 32 bytes of
EXPONENT = \x00\x00\x01\x00.....\x00, with one hundred bytes, then the result is 8 * (100 - 32) + 253 = 797). If all of the first 32 bytes of
EXPONENTare zero, return exactly
8 * (length_of_EXPONENT - 32).
mult_complexity is a function intended to approximate the difficulty of Karatsuba multiplication (used in all major bigint libraries) and is defined as follows.
def mult_complexity(x): if x <= 64: return x ** 2 elif x <= 1024: return x ** 2 // 4 + 96 * x - 3072 else: return x ** 2 // 16 + 480 * x - 199680
For example, the input data:
0000000000000000000000000000000000000000000000000000000000000001 0000000000000000000000000000000000000000000000000000000000000020 0000000000000000000000000000000000000000000000000000000000000020 03 fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2e fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f
Represents the exponent
3**(2**256 - 2**32 - 978) % (2**256 - 2**32 - 977). By Fermat’s little theorem, this equals 1, so the result is:
Returned as 32 bytes because the modulus length was 32 bytes. The
ADJUSTED_EXPONENT_LENGTH would be 255, and the gas cost would be
mult_complexity(32) * 255 / 20 = 13056 gas (note that this is ~8 times the cost of using the EXP opcode to compute a 32-byte exponent). A 4096-bit RSA exponentiation would cost
mult_complexity(512) * 4095 / 100 = 22853376 gas in the worst case, though RSA verification in practice usually uses an exponent of 3 or 65537, which would reduce the gas consumption to 5580 or 89292, respectively.
This input data:
0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000020 0000000000000000000000000000000000000000000000000000000000000020 fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2e fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f
Would be parsed as a base of 0, exponent of
2**256 - 2**32 - 978 and modulus of
2**256 - 2**32 - 977, and so would return 0. Notice how if the length_of_BASE is 0, then it does not interpret any data as the base, instead immediately interpreting the next 32 bytes as EXPONENT.
This input data:
0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000020 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd
Would parse a base length of 0, an exponent length of 32, and a modulus length of
2**256 - 1, where the base is empty, the exponent is
2**256 - 2 and the modulus is
(2**256 - 3) * 256**(2**256 - 33) (yes, that’s a really big number). It would then immediately fail, as it’s not possible to provide enough gas to make that computation.
This input data:
0000000000000000000000000000000000000000000000000000000000000001 0000000000000000000000000000000000000000000000000000000000000002 0000000000000000000000000000000000000000000000000000000000000020 03 ffff 8000000000000000000000000000000000000000000000000000000000000000 07
Would parse as a base of 3, an exponent of 65535, and a modulus of
2**255, and it would ignore the remaining 0x07 byte.
This input data:
0000000000000000000000000000000000000000000000000000000000000001 0000000000000000000000000000000000000000000000000000000000000002 0000000000000000000000000000000000000000000000000000000000000020 03 ffff 80
Would also parse as a base of 3, an exponent of 65535 and a modulus of
2**255, as it attempts to grab 32 bytes for the modulus starting from 0x80 - but there is no further data, so it right-pads it with 31 zero bytes.
This allows for efficient RSA verification inside of the EVM, as well as other forms of number theory-based cryptography. Note that adding precompiles for addition and subtraction is not required, as the in-EVM algorithm is efficient enough, and multiplication can be done through this precompile via
a * b = ((a + b)**2 - (a - b)**2) / 4.
The bit-based exponent calculation is done specifically to fairly charge for the often-used exponents of 2 (for multiplication) and 3 and 65537 (for RSA verification).
Copyright and related rights waived via CC0.
Please cite this document as:
Vitalik Buterin (@vbuterin), "EIP-198: Big integer modular exponentiation," Ethereum Improvement Proposals, no. 198, January 2017. [Online serial]. Available: https://eips.ethereum.org/EIPS/eip-198.