doc: write arithmetic operations tutorial

docs/user/tutorial/ARITHMETIC_OPERATIONS.md

@@ -1,3 +1,215 @@
 # Arithmetic Operations
 
-Umut to do: #312
+In this tutorial, we are going to go over all arithmetic operations available in `concretefhe`. Please read [Compiling and Executing](../howto/COMPILING_AND_EXECUTING.md) before reading further to see how you can compile the functions below.
+
+## Addition
+
+### Static ClearScalar and EncryptedScalar
+
+```python
+def f(x):
+    return x + 42
+```
+
+or
+
+```python
+def f(x):
+    return 42 + x
+```
+
+where
+
+- `x = EncryptedScalar(UnsignedInteger(bits))`
+
+results in
+
+```python
+engine.run(3) == 45
+engine.run(0) == 42
+```
+
+### Dynamic ClearScalar and EncryptedScalar
+
+```python
+def f(x, y):
+    return x + y
+```
+
+or
+
+```python
+def f(x, y):
+    return y + x
+```
+
+results in
+
+```python
+engine.run(6, 4) == 10
+engine.run(1, 1) == 2
+```
+
+where
+
+- `x = EncryptedScalar(UnsignedInteger(bits))`
+- `y = ClearScalar(UnsignedInteger(bits))`
+
+### EncryptedScalar and EncryptedScalar
+
+```python
+def f(x, y):
+    return x + y
+```
+
+where
+
+- `x = EncryptedScalar(UnsignedInteger(bits))`
+- `y = EncryptedScalar(UnsignedInteger(bits))`
+
+results in
+
+```python
+engine.run(7, 7) == 14
+engine.run(3, 4) == 7
+```
+
+## Subtraction
+
+### Static ClearScalar and EncryptedScalar 
+
+```python
+def f(x):
+    return 3 - x
+```
+
+where
+
+- `x = EncryptedScalar(UnsignedInteger(bits))`
+
+results in
+
+```python
+engine.run(2) == 1
+engine.run(3) == 0
+```
+
+### Dynamic ClearScalar and EncryptedScalar
+
+```python
+def f(x, y):
+    return y - x
+```
+
+where
+
+- `x = EncryptedScalar(UnsignedInteger(bits))`
+- `y = ClearScalar(UnsignedInteger(bits))`
+
+results in
+
+```python
+engine.run(2, 4) == 2
+engine.run(1, 7) == 6
+```
+
+## Multiplication
+
+### Static ClearScalar and EncryptedScalar
+
+```python
+def f(x):
+    return x * 2
+```
+
+or
+
+```python
+def f(x):
+    return 2 * x
+```
+
+where
+
+- `x = EncryptedScalar(UnsignedInteger(bits))`
+
+results in
+
+```python
+engine.run(2) == 4
+engine.run(5) == 10
+```
+
+### Dynamic ClearScalar and EncryptedScalar
+
+```python
+def f(x, y):
+    return x * y
+```
+
+or
+
+```python
+def f(x, y):
+    return y * x
+```
+
+where
+
+- `x = EncryptedScalar(UnsignedInteger(bits))`
+- `y = ClearScalar(UnsignedInteger(bits))`
+
+results in
+
+```python
+engine.run(2, 3) == 6
+engine.run(1, 7) == 7
+```
+
+## Dot Product
+
+### Dynamic ClearTensor and EncryptedTensor
+
+```python
+def f(x, y):
+    return np.dot(x, y)
+```
+
+or
+
+```python
+def f(x, y):
+    return np.dot(y, x)
+```
+
+where
+
+- `x = EncryptedTensor(UnsignedInteger(bits), shape=(2,))`
+- `y = ClearTensor(UnsignedInteger(bits), shape=(2,))`
+
+results in
+
+```python
+engine.run([1, 1], [2, 3]) == 5
+engine.run([2, 3], [2, 3]) == 13
+```
+
+## Combining all together
+
+```python
+def f(x, y, z):
+    return 100 - (2 * (np.dot(x, y) + z))
+```
+
+where
+
+- `x = EncryptedTensor(UnsignedInteger(bits), shape=(2,))`
+- `y = ClearTensor(UnsignedInteger(bits), shape=(2,))`
+- `z = EncryptedScalar(UnsignedInteger(bits))`
+
+results in
+
+```python
+engine.run([1, 2], [4, 3], 10) == 60
+engine.run([2, 3], [3, 2], 5) == 66
+```