# What's two-adicity? posted 3 weeks ago

Some elliptic curves (related to zero-knowledge proofs I believe) have been claiming high 2-adicity. But for some reason, it seems a bit hard to find a definition of what this term means. And oddly, it's not a complicated thing to explain. So here's a short note about it.

You can see this being mentioned for example by the pasta curves:

They have the same 2-adicity, 32, unlike the Tweedle curves that had 2-adicity of 33 and 34. This simplifies implementations and may assist in square root performance (used for point decompression and internally to Halo 2) due to a new algorithm recently discovered; 32 is more convenient for this algorithm.

Looking at the definition of one of its field in Rust you can see that it is defined specifically for a trait related to FFTs:

```
impl FftParameters for FqParameters {
type BigInt = BigInteger;
const TWO_ADICITY: u32 = 32;
#[rustfmt::skip]
const TWO_ADIC_ROOT_OF_UNITY: BigInteger = BigInteger([
0x218077428c9942de, 0xcc49578921b60494, 0xac2e5d27b2efbee2, 0xb79fa897f2db056
]);
```

so what's that? Well, simply put, a two-adicity of 32 means that there's a multiplicative subgroup of size $2^{32}$ that exists in the field. And the code above also defines a generator $g$ for it, such that $g^{2^{32}} = 1$ and $g^i \neq 1$ for $i \in [[1, 2^{32}-1]]$ (so it's a **primitive** $2^{32}$-th root of unity).

Lagrange's theorem tells us that if we have a group of order $n$, then we'll have subgroups with orders dividing $n$. So in our case, we have subgroups with all the powers of 2, up to the 32-th power of 2.

To find any of these groups, it is pretty straight forward as well. Notice that:

- let $h = g^2$, then $h^{2^{31}} = g^{2^{32}} = 1$ and so $h$ generates a subgroup of order 31
- let $h = g^{2^2}$, then $h^{2^{30}} = g^{2^{32}} = 1$ and so $h$ generates a subgroup of order 30
- and so on...

In arkworks you can see how this is implemented:

```
let size = n.next_power_of_two() as u64;
let log_size_of_group = ark_std::log2(usize::try_from(size).expect("too large"));
omega = Self::TWO_ADIC_ROOT_OF_UNITY;
for _ in log_size_of_group..Self::TWO_ADICITY {
omega.square_in_place();
}
```

this allows you to easily find subgroups of different sizes of powers of 2, which is useful in zero-knowledge proof systems as FFT optimizations apply well on domains that are powers of 2. You can read more about that in the mina book.

## Comments

## DD

You write "g^i \neq 1 for i \in [[1, 2^{32}]]"

I'm not sure about your notation, but i must be strictly smaller than 2^{32}, so either exclude it from the interval by using ")" as bracket or subtract 1 of the upper limit.

Other than that: nice post, very informative!

## david

Ah good point!

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