🚧#8 Circom: Other Reference Circuits

- EscalarProduct - Decoder - Multiplexer

EscalarProduct

Verifies the inner/scalar product of two provided arrays

Like in1[w] . in2[w] === out.

• aux[n] collects the pairwise multiplication of elements in both input arrays.

• lc is a collector var, which basically returns the sum of elements in aux array.

out is constrained and assigned to lc, therefore, the circuit verifies that the sum of pairwise multiplication - inner product - of 2 arrays is correct.

Decoder

Decoder creates a mask based on the supplied index and size of array. Creates a mask of some array of defined size, by zero-ing out all other elements except the index we specified as inp value

    Decoder(3)
        input: inp = 1
        output: [0,1,0]

Creates a mask [0,1,0] that can be overlaid on an array of 3 elements to extract the 2nd element. Decoder returns array out:[0,1,0] and success: 1.

Returns a bitmask where the 1 is in the position of inp, if inp is less than w.

• if inp == i, assign 1 into out[i]; else assign 0.

• constraint out[i] to be either 0 or inp == i.

---

• lc sums all the elements of out. sum should be either 0 or 1

• success is constrained to be either 0 or 1.

• Therefore, lc should have a max value of 1.

This means out is essentially the binary representation of some figure [0,0,1,0], where there can only be a single 1.

• If success/lc is 0, the implication is that inp is larger than the array, since i never takes its value.

• decoder() returns a bitmask where the 1 is at the inp; if inp is less than w.

    If the index we were looking for was 4, in the same array:
    Consider the array [1,2,3]
        Decoder(3)    
            input: inp = 4
            output: [0,0,0] and success = 0
    
    since the input array only goes to index 2.

Multiplexer

Let's reduce this to 1 dimension first

Multiplexer allows us to retrieve the value/element of a specified array.

• Given an array inp[], return the element in said array for a specified selector.

• If we input [13,19,23,29] and set sel = 2, we will obtain out as 23.

• This is the element at index = 2.

input: inp = [13,19,23,29] and sel = 2
 
mask:  [0,0,1,0]
dot product:   [0,0,1,0] . [13,19,23,29] = 23
output: out = 23, success = 1

Multiplex lets us select the element in the place of index = 2, which is 23.

2-D Multiplex

Takes in an input of nested arrays, and lets us return the nested array of index value 'sel'

Example

// nested arrays
[
    [1,2,3,1],
    [3,4,5,2],
    [6,7,8,3]
]

Multiplex(3, 4):
    sel = 2

out returns us [6,7,8,3]