# ECIP 1026: Modexp Precompiled Contract Source

Author Wei Tang Draft Standards Track Core 2017-06-28

## Rationale

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).

## Specification

### Parameters

• `GQUADDIVISOR: 100`

### Algorithm

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(max(length_of_MODULUS, length_of_BASE) ** 2 * 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.

• If `length_of_EXPONENT <= 32`, and all bits in `EXPONENT` are 0, return 0
• If `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).
• If `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` (eg. if `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 `EXPONENT` are zero, return exactly `8 * (length_of_EXPONENT - 32)`.

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:

``````0000000000000000000000000000000000000000000000000000000000000001
``````

Returned as 32 bytes because the modulus length was 32 bytes. The `ADJUSTED_EXPONENT_LENGTH` would be 255, and the gas cost would be `32**2 * 255 / 100 = 2611` gas (note that this is ~five thirds of the cost of using the EXP opcode to compute a 32-byte exponent). A 4096-bit RSA exponentiation would cost `512**2 * 4095 / 100 = 10734796` 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 2621 or 41943, 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 - 978`, 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 length_of_EXPONENT.

This input data:

``````0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000020
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe
fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd
``````

Would parse a base length of 0, a modulus length of 32, and an exponent length of `2**256 - 1`, where the base is empty, the modulus is `2**256 - 2` and the exponent 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 then there is no further data so it right pads it with 31 zeroes.