# Table lookup

In this tutorial, we are going to go over the ways to perform direct table lookups in **Concrete Numpy**. Please read [Compiling and Executing](../basics/compiling\_and\_executing.md) before reading further to see how you can compile the functions below.

## Direct table lookup

**Concrete Numpy** provides a special class to allow direct table lookups. Here is how to use it:

```python
import concrete.numpy as cnp

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

def f(x):
    return table[x]
```

where

* `x = "encrypted"` scalar

results in

<!--pytest-codeblocks:skip-->
```python
circuit.encrypt_run_decrypt(0) == 2
circuit.encrypt_run_decrypt(1) == -1
circuit.encrypt_run_decrypt(2) == 3
circuit.encrypt_run_decrypt(3) == 0
```

Moreover, direct lookup tables can be used with tensors where the same table lookup is applied to each value in the tensor, so

* `x = "encrypted"` tensor of shape `(2, 3)`

results in

<!--pytest-codeblocks:skip-->
```python
input = np.array([[0, 1, 3], [2, 3, 1]], dtype=np.uint8)
circuit.encrypt_run_decrypt(input) == [[2, 1, 0], [3, 0, 1]]
```

Direct table lookups behaves like array indexing in python. Which means, if the lookup variable is negative, table is looked up from the back.

```python
import concrete.numpy as cnp

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

def f(x):
    return table[-x]
```

where

* `x = "encrypted"` scalar

results in

<!--pytest-codeblocks:skip-->
```python
circuit.encrypt_run_decrypt(0) == 2
circuit.encrypt_run_decrypt(1) == 0
circuit.encrypt_run_decrypt(2) == 3
circuit.encrypt_run_decrypt(3) == 1
circuit.encrypt_run_decrypt(4) == 2
```

Lastly, a `LookupTable` can have any number of elements, let's call it **N**, as long as the lookup variable is in range \[-**N**, **N**). 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
```

Note that, number of elements in the lookup table doesn't affect the performance in any way.

## Direct multi table lookup

Sometimes you may want to apply a different lookup table to each value in a tensor. That's where direct multi lookup table becomes handy. Here is how to use it:

<!--pytest-codeblocks:skip-->
```python
import concrete.numpy as cnp

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

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

def f(x):
    return table[x]
```

where

* `x = "encrypted"` tensor of shape `(3, 2)`

results in

<!--pytest-codeblocks:skip-->
```python
input = np.array([[2, 3], [1, 2], [3, 0]], dtype=np.uint8)
circuit.encrypt_run_decrypt(input) == [[4, 27], [1, 8], [9, 0]]
```

Basically, we applied `squared` table to the first column and `cubed` to the second one.

## Fused table lookup

Direct tables are tedious to prepare by hand. When possible, **Concrete Numpy** fuses the floating point operations into table lookups automatically. There are some limitations on fusing operations, which you can learn more about on the next tutorial, [Working With Floating Points](working\_with\_floating\_points.md).

Here is an example function that results in fused table lookup:

<!--pytest-codeblocks:skip-->
```python
def f(x):
    return 127 - (50 * (np.sin(x) + 1)).astype(np.int64) # astype is to go back to integer world
```

where

* `x = "encrypted"` scalar

results in

<!--pytest-codeblocks:skip-->
```python
circuit.encrypt_run_decrypt(0) == 77
circuit.encrypt_run_decrypt(1) == 35
circuit.encrypt_run_decrypt(2) == 32
circuit.encrypt_run_decrypt(3) == 70
circuit.encrypt_run_decrypt(4) == 115
circuit.encrypt_run_decrypt(5) == 125
circuit.encrypt_run_decrypt(6) == 91
circuit.encrypt_run_decrypt(7) == 45
```

Initially, the function is converted to this operation graph

![](../\_static/tutorials/table-lookup/1.initial.graph.png)

and after floating point operations are fused, we get the following operation graph

![](../\_static/tutorials/table-lookup/3.final.graph.png)

Internally, it uses the following lookup table

<!--pytest-codeblocks:skip-->
```python
table = cnp.LookupTable([50, 92, 95, 57, 12, 2, 36, 82])
```

which is calculated by:

<!--pytest-codeblocks:skip-->
```python
[(50 * (np.sin(x) + 1)).astype(np.int64) for x in range(2 ** 3)]
```