# Table Lookups

In this tutorial, we will review the ways to perform direct table lookups in Concrete.

## Direct table lookup

Concrete provides a LookupTable class to create your own tables and apply them in your circuits.

LookupTables can have any number of elements. Let's call them N. As long as the lookup variable is in range [-N, N), Table Lookup is valid.

If you go out of bounds of this range, you will get the following error:

IndexError: index 10 is out of bounds for axis 0 with size 6

### With scalars.

You can create the lookup table using a list of integers and apply it using indexing:

from concrete import fhe

table = fhe.LookupTable([2, -1, 3, 0])

@fhe.compiler({"x": "encrypted"})
def f(x):
return table[x]

inputset = range(4)
circuit = f.compile(inputset)

assert circuit.encrypt_run_decrypt(0) == table[0] == 2
assert circuit.encrypt_run_decrypt(1) == table[1] == -1
assert circuit.encrypt_run_decrypt(2) == table[2] == 3
assert circuit.encrypt_run_decrypt(3) == table[3] == 0

### With tensors.

When you apply the table lookup to a tensor, you apply the scalar table lookup to each element of the tensor:

from concrete import fhe
import numpy as np

table = fhe.LookupTable([2, -1, 3, 0])

@fhe.compiler({"x": "encrypted"})
def f(x):
return table[x]

inputset = [np.random.randint(0, 4, size=(2, 3)) for _ in range(10)]
circuit = f.compile(inputset)

sample = [
[0, 1, 3],
[2, 3, 1],
]
expected_output = [
[2, -1, 0],
[3, 0, -1],
]
actual_output = circuit.encrypt_run_decrypt(np.array(sample))

for i in range(2):
for j in range(3):
assert actual_output[i][j] == expected_output[i][j] == table[sample[i][j]]

### With negative values.

LookupTable mimics array indexing in Python, which means if the lookup variable is negative, the table is looked up from the back:

from concrete import fhe

table = fhe.LookupTable([2, -1, 3, 0])

@fhe.compiler({"x": "encrypted"})
def f(x):
return table[-x]

inputset = range(1, 5)
circuit = f.compile(inputset)

assert circuit.encrypt_run_decrypt(1) == table[-1] == 0
assert circuit.encrypt_run_decrypt(2) == table[-2] == 3
assert circuit.encrypt_run_decrypt(3) == table[-3] == -1
assert circuit.encrypt_run_decrypt(4) == table[-4] == 2

## Direct multi-table lookup

In case you want to apply a different lookup table to each element of a tensor, you can have a LookupTable of LookupTables:

from concrete import fhe
import numpy as np

squared = fhe.LookupTable([i ** 2 for i in range(4)])
cubed = fhe.LookupTable([i ** 3 for i in range(4)])

table = fhe.LookupTable([
[squared, cubed],
[squared, cubed],
[squared, cubed],
])

@fhe.compiler({"x": "encrypted"})
def f(x):
return table[x]

inputset = [np.random.randint(0, 4, size=(3, 2)) for _ in range(10)]
circuit = f.compile(inputset)

sample = [
[0, 1],
[2, 3],
[3, 0],
]
expected_output = [
[0, 1],
[4, 27],
[9, 0]
]
actual_output = circuit.encrypt_run_decrypt(np.array(sample))

for i in range(3):
for j in range(2):
if j == 0:
assert actual_output[i][j] == expected_output[i][j] == squared[sample[i][j]]
else:
assert actual_output[i][j] == expected_output[i][j] == cubed[sample[i][j]]

In this example, we applied a squared table to the first column and a cubed table to the second one.

## Fused table lookup

Concrete tries to fuse some operations into table lookups automatically, so you don't need to create the lookup tables manually:

from concrete import fhe
import numpy as np

@fhe.compiler({"x": "encrypted"})
def f(x):
return (42 * np.sin(x)).astype(np.int64) // 10

inputset = range(8)
circuit = f.compile(inputset)

for x in range(8):
assert circuit.encrypt_run_decrypt(x) == f(x)

All lookup tables need to be from integers to integers. So, without .astype(np.int64), Concrete will not be able to fuse.

The function is first traced into:

Concrete then fuses appropriate nodes:

Fusing makes the code more readable and easier to modify, so try to utilize it over manual LookupTables as much as possible.

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