In a binary Merkle tree, we define a "generalized index" of a node as `2**depth + index`. Visually, this looks as follows:
```
1
2 3
4 5 6 7
...
```
Note that the generalized index has the convenient property that the two children of node `k` are `2k` and `2k+1`, and also that it equals the position of a node in the linear representation of the Merkle tree that's computed by this function:
```python
def merkle_tree(leaves):
o = [0] * len(leaves) + leaves
for i in range(len(leaves)-1, 0, -1):
o[i] = hash(o[i*2] + o[i*2+1])
return o
```
We will define Merkle proofs in terms of generalized indices.
### SSZ object to index
We can describe the hash tree of any SSZ object, rooted in `hash_tree_root(object)`, as a binary Merkle tree whose depth may vary. For example, an object `{x: bytes32, y: List[uint64]}` would look as follows:
We can now define a concept of a "path", a way of describing a function that takes as input an SSZ object and outputs some specific (possibly deeply nested) member. For example, `foo -> foo.x` is a path, as are `foo -> len(foo.y)` and `foo -> foo.y[5].w`. We'll describe paths as lists, which can have two representations. In "human-readable form", they are `["x"]`, `["y", "__len__"]` and `["y", 5, "w"]` respectively. In "encoded form", they are lists of `uint64` values, in these cases (assuming the fields of `foo` in order are `x` then `y`, and `w` is the first field of `y[i]`) `[0]`, `[1, 2**64-1]`, `[1, 5, 0]`.
We can now define a function `get_generalized_indices(object: Any, path: List[int], root=1: int) -> int` that converts an object and a path to a set of generalized indices (note that for constant-sized objects, there is only one generalized index and it only depends on the path, but for dynamically sized objects the indices may depend on the object itself too). For dynamically-sized objects, the set of indices will have more than one member because of the need to access an array's length to determine the correct generalized index for some array access.
We define a Merkle multiproof as a minimal subset of nodes in a Merkle tree needed to fully authenticate that a set of nodes actually are part of a Merkle tree with some specified root, at a particular set of generalized indices. For example, here is the Merkle multiproof for positions 0, 1, 6 in an 8-node Merkle tree (ie. generalized indices 8, 9, 14):
```
.
. .
. * * .
x x . . . . x *
```
. are unused nodes, * are used nodes, x are the values we are trying to prove. Notice how despite being a multiproof for 3 values, it requires only 3 auxiliary nodes, only one node more than would be required to prove a single value. Normally the efficiency gains are not quite that extreme, but the savings relative to individual Merkle proofs are still significant. As a rule of thumb, a multiproof for k nodes at the same level of an n-node tree has size `k * (n/k + log(n/k))`.
Here is code for creating and verifying a multiproof. First a helper:
```python
def log2(x):
return 0 if x == 1 else 1 + log2(x//2)
```
First, a method for computing the generalized indices of the auxiliary tree nodes that a proof of a given set of generalized indices will require:
# Get indices that cannot be recalculated from earlier indices
redundant_indices = set({})
proof = []
for index in maximal_indices:
if index not in redundant_indices:
proof.append(index)
while index > 1:
redundant_indices.add(index)
if (index ^ 1) not in redundant_indices:
break
index //= 2
return [i for i in proof if i not in tree_indices]
````
Generating a proof is simply a matter of taking the node of the SSZ hash tree with the union of the given generalized indices for each index given by `get_proof_indices`, and outputting the list of nodes in the same order.
We define `MerklePartial(f, arg1, arg2..., focus=0)` as being a `MerklePartial` object wrapping a Merkle multiproof of the set of nodes in the hash tree of the SSZ object `arg[focus]` that is needed to authenticate the parts of the object needed to compute `f(arg1, arg2...)`.
