Xor Distance

We describe how to compute a XOR distance (as known as an Hamming weight distance) in Concrete. This can be useful in particular for biometry use-cases, where obviously, private is a very interesting feature.

The full code can be done here. Execution times of the different functions are given in the final section.

The Goal

Goal here is to compute

def dist_in_clear(x, y):
    return numpy.sum(hw(x ^ y))

function in FHE. This function XOR to values, and compute hw of the result, which is the Hamming weight, and is defined as the number of set bits.

For example, for x = 0xf3 = 0b11110011 and y = 0xbc = 0x10111100, we would have x ^ y = 0b01001111 = 0x4f, and so dist_in_clear(x, y) = 5 since there are 5 set bits in 0x4f. Or, in other words, dist_in_clear(x, y) = 5 means that bits of x and bits of y differ in 5 different locations:

0b11110011
0x10111100
   ^  ^^^^

This is a distance function, which can be used for various purpose, including measuring how two vectors are close to each other. In the context of biometry (or others), it may be very interesting to compute this function over encrypted x and y vectors.

First Implementation

In the full code, we use a first implementation, which is

def dist_in_fhe_directly_from_cp(x, y):
    return np.sum(hw(x ^ y))

Here, it's a pure copy of the code in Concrete, and it compiles directly into FHE code!

Second Implementation with `fhe.bits`

In the full code, we use a second implementation, which is

def dist_in_fhe_with_bits_1b(x, y):
    z = x + y
    zx = fhe.bits(z)[0]
    return np.sum(zx)

This function only works for bit-vectors x and y (as opposed to other functions). Here, we use fhe.bits operator to extract the least-significant bit of the addition x+y: indeed, this least signification bit is exactly x ^ y.

Third Implementation with Concatenation

In the full code, we use a third implementation, which is

def dist_in_fhe_with_xor_internal(x, y, bitsize_w):
    power = 2**bitsize_w
    table = fhe.LookupTable([hw((i % power) ^ (i // power)) for i in range(power**2)])

    z = x + power * y
    zx = table[z]

    return np.sum(zx)

Here, we concatenate the elements of x and y (which are of bitsize bitsize_w) into a 2 * bitsize_w input, and use a 2 * bitsize_w-bit programmable bootstrapping.

Fourth Implementation with `fhe.multivariate`

In the full code, we use a fourth implementation, which is

def dist_in_fhe_with_multivariate_internal(x, y):
    zx = fhe.multivariate(lambda x, y: hw(x ^ y))(x, y)
    return np.sum(zx)

Here, we use fhe.multivariate, which is a function which takes the two inputs x and y. Under the hood, it's going to be replaced by a 2 * bitsize_w-bit programmable bootstrapping.

Execution Time

All of the following timings were measured on an hpc7a machine, with Concrete 2.5.1.

If one executes the code for 120-bit vectors (of whatever shape), execution times should be:

    dist_in_fhe_with_multivariate_tables on 2 bits:  0.07 secunds
    dist_in_fhe_with_multivariate_tables on 1 bits:  0.09 secunds
             dist_in_fhe_with_xor_tables on 2 bits:  0.09 secunds
            dist_in_fhe_directly_from_cp on 2 bits:  0.10 secunds
             dist_in_fhe_with_xor_tables on 1 bits:  0.11 secunds
            dist_in_fhe_directly_from_cp on 1 bits:  0.12 secunds
                dist_in_fhe_with_bits_1b on 1 bits:  0.15 secunds
    dist_in_fhe_with_multivariate_tables on 3 bits:  0.27 secunds
             dist_in_fhe_with_xor_tables on 3 bits:  0.29 secunds
            dist_in_fhe_directly_from_cp on 3 bits:  0.31 secunds
            dist_in_fhe_directly_from_cp on 4 bits:  1.17 secunds
             dist_in_fhe_with_xor_tables on 4 bits:  2.18 secunds
    dist_in_fhe_with_multivariate_tables on 4 bits:  2.24 secunds

For 1200-bit vectors (obtained with python hamming_distance.py --nb_bits 1200), execution times should be:

    dist_in_fhe_with_multivariate_tables on 2 bits:  0.22 secunds
             dist_in_fhe_with_xor_tables on 2 bits:  0.29 secunds
            dist_in_fhe_directly_from_cp on 2 bits:  0.32 secunds
            dist_in_fhe_directly_from_cp on 1 bits:  0.36 secunds
             dist_in_fhe_with_xor_tables on 1 bits:  0.39 secunds
                dist_in_fhe_with_bits_1b on 1 bits:  0.44 secunds
    dist_in_fhe_with_multivariate_tables on 1 bits:  0.48 secunds
            dist_in_fhe_directly_from_cp on 3 bits:  0.73 secunds
    dist_in_fhe_with_multivariate_tables on 3 bits:  0.85 secunds
             dist_in_fhe_with_xor_tables on 3 bits:  1.14 secunds
            dist_in_fhe_directly_from_cp on 4 bits:  5.99 secunds
    dist_in_fhe_with_multivariate_tables on 4 bits:  7.17 secunds
             dist_in_fhe_with_xor_tables on 4 bits:  8.20 secunds

And finally, for 12804-bit vectors, execution times should be:

             dist_in_fhe_with_xor_tables on 2 bits:  2.53 secunds
    dist_in_fhe_with_multivariate_tables on 2 bits:  2.66 secunds
            dist_in_fhe_directly_from_cp on 2 bits:  3.64 secunds
            dist_in_fhe_directly_from_cp on 1 bits:  4.25 secunds
                dist_in_fhe_with_bits_1b on 1 bits:  4.40 secunds
             dist_in_fhe_with_xor_tables on 1 bits:  4.53 secunds
    dist_in_fhe_with_multivariate_tables on 1 bits:  4.71 secunds
            dist_in_fhe_directly_from_cp on 3 bits:  6.76 secunds
    dist_in_fhe_with_multivariate_tables on 3 bits:  7.93 secunds
             dist_in_fhe_with_xor_tables on 3 bits:  8.43 secunds
            dist_in_fhe_directly_from_cp on 4 bits: 23.72 secunds
             dist_in_fhe_with_xor_tables on 4 bits: 39.27 secunds
    dist_in_fhe_with_multivariate_tables on 4 bits: 40.89 secunds

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