Batch Inverse

Reduction of the number of field inverse operations when computing batch operations.

What is Batch Inverse?

Batch Inverse is a technique used to efficiently compute the inverses of multiple values $(v_1, \dots, v_n)$ at once. Normally, calculating each inverse separately requires $n \times \mathsf{INV}$ (inverse operations), but with Batch Inversion, you can compute all the inverses using just a single inverse operation.

As inverse operations are significantly more expensive than multiplications, this method is especially useful in performance-sensitive contexts—such as in ZK provers.

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How Does It Work?

Let’s walk through an example of computing the inverses of four values: $(a, b, c, d)$

1. Compute Prefix Products

Multiply the values one by one, keeping track of intermediate results:

• $P_1 = a$

• $P_2 = ab$

• $P_3 = abc$

• $P_4 = abcd$

So we get:

$$P = (P_1, P_2, P_3, P_4)$$

2. Compute Inverse of the Final Product

Compute the inverse of the final product:

$$t = P_4^{-1} = (abcd)^{-1}$$

This is the only inverse operation required.

3. Compute Each Inverse in Reverse

Now, traverse the values in reverse to compute individual inverses:

• $d^{-1} = P_3 \times t$, then update $t = t \times d = (abc)^{-1}$

• $c^{-1} = P_2 \times t$, then update $t = t \times c = (ab)^{-1}$

• $b^{-1} = P_1 \times t$, then update $t = t \times b = (a)^{-1}$

• $a^{-1} = t$

This phase involves only multiplication operations.

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Computational Complexity

If the vector has length $n$, the total computational cost is:

1. Prefix Product Computation: $(n - 1) \times \mathsf{MUL}$

2. One Inverse Operation: $1 \times \mathsf{INV}$

3. Reverse Traversal: $2(n - 1) \times \mathsf{MUL}$

Total:

$$3(n - 1) \times \mathsf{MUL} + 1 \times \mathsf{INV}$$

Written by Ryan Kim of Fractalyze