> For the complete documentation index, see [llms.txt](https://docs.zama.org/tfhe-rs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.zama.org/tfhe-rs/tfhe-rs/0.3-1/getting-started/operations.md).

# Operations

The table below contains an overview of the available operations in `TFHE-rs`. More details, and further examples, are given in the following sections.

| name                  | symbol      | FheUint/FheUint      | FheUint/Uint               | Uint/FheUint               |
| --------------------- | ----------- | -------------------- | -------------------------- | -------------------------- |
| Neg                   | `-`         | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Add                   | `+`         | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Sub                   | `-`         | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Mul                   | `*`         | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Div                   | `/`         | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Rem                   | `%`         | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Not                   | `!`         | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| BitAnd                | `&`         | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| BitOr                 | `\|`        | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| BitXor                | `^`         | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Shr                   | `>>`        | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Shl                   | `<<`        | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Min                   | `min`       | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Max                   | `max`       | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Greater than          | `gt`        | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Greater or equal than | `ge`        | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Lower than            | `lt`        | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Lower or equal than   | `le`        | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Equal                 | `eq`        | :heavy\_check\_mark: | :heavy\_check\_mark:       | :heavy\_check\_mark:       |
| Cast (into dest type) | `cast_into` | :heavy\_check\_mark: | :heavy\_multiplication\_x: | :heavy\_multiplication\_x: |
| Cast (from src type)  | `cast_from` | :heavy\_check\_mark: | :heavy\_multiplication\_x: | :heavy\_multiplication\_x: |

## Boolean Operations

Native homomorphic Booleans support common Boolean operations.

The list of supported operations is:

| name                                                          | symbol | type   |
| ------------------------------------------------------------- | ------ | ------ |
| [BitAnd](https://doc.rust-lang.org/std/ops/trait.BitAnd.html) | `&`    | Binary |
| [BitOr](https://doc.rust-lang.org/std/ops/trait.BitOr.html)   | `\|`   | Binary |
| [BitXor](https://doc.rust-lang.org/std/ops/trait.BitXor.html) | `^`    | Binary |
| [Not](https://doc.rust-lang.org/std/ops/trait.Not.html)       | `!`    | Unary  |

## ShortInt Operations

Native small homomorphic integer types (e.g., FheUint3 or FheUint4) easily compute various operations. In general, computing over encrypted data is as easy as computing over clear data, since the same operation symbol is used. The addition between two ciphertexts is done using the symbol `+` between two FheUint values. Many operations can be computed between a clear value (i.e. a scalar) and a ciphertext.

In Rust, operations on native types are modular. For example, computations on `u8` are carried out modulo $$2^{8}$$. A similar idea applies for FheUintX, where operations are done modulo $$2^{X}$$. For FheUint3, operations are done modulo $$8 = 2^{3}$$.

### Arithmetic operations.

Small homomorphic integer types support all common arithmetic operations, meaning `+`, `-`, `x`, `/`, `mod`.

The division operation implements a subtlety: since data is encrypted, it is possible to compute a division by 0. In this case, the division is tweaked so that dividing by 0 returns the max possible value for the message.

The list of supported operations is:

| name                                                    | symbol | type   |
| ------------------------------------------------------- | ------ | ------ |
| [Add](https://doc.rust-lang.org/std/ops/trait.Add.html) | `+`    | Binary |
| [Sub](https://doc.rust-lang.org/std/ops/trait.Sub.html) | `-`    | Binary |
| [Mul](https://doc.rust-lang.org/std/ops/trait.Mul.html) | `*`    | Binary |
| [Div](https://doc.rust-lang.org/std/ops/trait.Div.html) | `/`    | Binary |
| [Rem](https://doc.rust-lang.org/std/ops/trait.Rem.html) | `%`    | Binary |
| [Neg](https://doc.rust-lang.org/std/ops/trait.Neg.html) | `-`    | Unary  |

A simple example on how to use these operations:

```rust
use tfhe::prelude::*;
use tfhe::{generate_keys, set_server_key, ConfigBuilder, FheUint3};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let config = ConfigBuilder::all_disabled().enable_default_uint3().build();
    let (keys, server_keys) = generate_keys(config);
    set_server_key(server_keys);

    let clear_a = 7;
    let clear_b = 3;
    let clear_c = 2;

    let mut a = FheUint3::try_encrypt(clear_a, &keys)?;
    let mut b = FheUint3::try_encrypt(clear_b, &keys)?;
    let mut c = FheUint3::try_encrypt(clear_c, &keys)?;


    a = a * &b;  // Clear equivalent computations: 7 * 3 mod 8 = 5
    b = &b + &c; // Clear equivalent computations: 3 + 2 mod 8 = 5
    b = b - 5;   // Clear equivalent computations: 5 - 5 mod 8 = 0

    let dec_a = a.decrypt(&keys);
    let dec_b = b.decrypt(&keys);

    // We homomorphically swapped values using bitwise operations
    assert_eq!(dec_a, (clear_a * clear_b) % 8);
    assert_eq!(dec_b, ((clear_b + clear_c) - 5) % 8);

    Ok(())
}
```

### Bitwise operations.

Small homomorphic integer types support some bitwise operations.

The list of supported operations is:

| name                                                                                 | symbol         | type   |
| ------------------------------------------------------------------------------------ | -------------- | ------ |
| [Not](https://doc.rust-lang.org/std/ops/trait.Not.html)                              | `!`            | Unary  |
| [BitAnd](https://doc.rust-lang.org/std/ops/trait.BitAnd.html)                        | `&`            | Binary |
| [BitOr](https://doc.rust-lang.org/std/ops/trait.BitOr.html)                          | `\|`           | Binary |
| [BitXor](https://doc.rust-lang.org/std/ops/trait.BitXor.html)                        | `^`            | Binary |
| [Shr](https://doc.rust-lang.org/std/ops/trait.Shr.html)                              | `>>`           | Binary |
| [Shl](https://doc.rust-lang.org/std/ops/trait.Shl.html)                              | `<<`           | Binary |
| [Rotate Right](https://doc.rust-lang.org/std/primitive.u32.html#method.rotate_right) | `rotate_right` | Binary |
| [Rotate Left](https://doc.rust-lang.org/std/primitive.u32.html#method.rotate_left)   | `rotate_left`  | Binary |

A simple example on how to use these operations:

```rust
use tfhe::prelude::*;
use tfhe::{generate_keys, set_server_key, ConfigBuilder, FheUint3};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let config = ConfigBuilder::all_disabled().enable_default_uint3().build();
    let (keys, server_keys) = generate_keys(config);
    set_server_key(server_keys);

    let clear_a = 7;
    let clear_b = 3;

    let mut a = FheUint3::try_encrypt(clear_a, &keys)?;
    let mut b = FheUint3::try_encrypt(clear_b, &keys)?;

    a = a ^ &b;
    b = b ^ &a;
    a = a ^ &b;

    let dec_a = a.decrypt(&keys);
    let dec_b = b.decrypt(&keys);

    // We homomorphically swapped values using bitwise operations
    assert_eq!(dec_a, clear_b);
    assert_eq!(dec_b, clear_a);

    Ok(())
}
```

### Comparisons.

Small homomorphic integer types support comparison operations.

Due to some Rust limitations, it is not possible to overload the comparison symbols because of the inner definition of the operations. Rust expects to have a Boolean as an output, whereas a ciphertext is returned when using homomorphic types.

You will need to use different methods instead of using symbols for the comparisons. These methods follow the same naming conventions as the two standard Rust traits:

* [PartialOrd](https://doc.rust-lang.org/std/cmp/trait.PartialOrd.html)
* [PartialEq](https://doc.rust-lang.org/std/cmp/trait.PartialEq.html)

The list of supported operations is:

| name                                                                        | symbol | type   |
| --------------------------------------------------------------------------- | ------ | ------ |
| [Equal](https://doc.rust-lang.org/std/cmp/trait.PartialEq.html)             | `eq`   | Binary |
| [Not Equal](https://doc.rust-lang.org/std/cmp/trait.PartialEq.html)         | `ne`   | Binary |
| [Greater Than](https://doc.rust-lang.org/std/cmp/trait.PartialOrd.html)     | `gt`   | Binary |
| [Greater or Equal](https://doc.rust-lang.org/std/cmp/trait.PartialOrd.html) | `ge`   | Binary |
| [Lower](https://doc.rust-lang.org/std/cmp/trait.PartialOrd.html)            | `lt`   | Binary |
| [Lower or Equal](https://doc.rust-lang.org/std/cmp/trait.PartialOrd.html)   | `le`   | Binary |

A simple example on how to use these operations:

```rust
use tfhe::prelude::*;
use tfhe::{generate_keys, set_server_key, ConfigBuilder, FheUint3};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let config = ConfigBuilder::all_disabled().enable_default_uint3().build();
    let (keys, server_keys) = generate_keys(config);
    set_server_key(server_keys);

    let clear_a = 7;
    let clear_b = 3;

    let mut a = FheUint3::try_encrypt(clear_a, &keys)?;
    let mut b = FheUint3::try_encrypt(clear_b, &keys)?;

    assert_eq!(a.gt(&b).decrypt(&keys) != 0, true);
    assert_eq!(b.le(&a).decrypt(&keys) != 0, true);

    Ok(())
}
```

### Univariate function evaluation.

The shortint type also supports the computation of univariate functions, which make use of TFHE's *programmable bootstrapping*.

A simple example on how to use these operations:

```rust
use tfhe::prelude::*;
use tfhe::{generate_keys, set_server_key, ConfigBuilder, FheUint4};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let config = ConfigBuilder::all_disabled().enable_default_uint4().build();
    let (keys, server_keys) = generate_keys(config);
    set_server_key(server_keys);

    let pow_5 = |value: u64| {
        value.pow(5) % FheUint4::MODULUS as u64
    };

    let clear_a = 12;
    let a = FheUint4::try_encrypt(12, &keys)?;

    let c = a.map(pow_5);
    let decrypted = c.decrypt(&keys);
    assert_eq!(decrypted, pow_5(clear_a) as u8);

    Ok(())
}
```

### Bivariate function evaluations.

Using the shortint type allows you to evaluate bivariate functions (i.e., functions that take two ciphertexts as input).

A simple code example:

```rust
use tfhe::prelude::*;
use tfhe::{generate_keys, set_server_key, ConfigBuilder, FheUint2};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let config = ConfigBuilder::all_disabled().enable_default_uint2().build();
    let (keys, server_keys) = generate_keys(config);
    set_server_key(server_keys);

    let clear_a = 1;
    let clear_b = 3;
    let a = FheUint2::try_encrypt(clear_a, &keys)?;
    let b = FheUint2::try_encrypt(clear_b, &keys)?;


    let c = a.bivariate_function(&b, std::cmp::max);
    let decrypted = c.decrypt(&keys);
    assert_eq!(decrypted, std::cmp::max(clear_a, clear_b) as u8);

    Ok(())
}
```

## Integer

In TFHE-rs, integers are used to encrypt any messages larger than 4 bits. All supported operations are listed below.

### Arithmetic operations.

Homomorphic integer types support arithmetic operations.

The list of supported operations is:

| name                                                      | symbol | type   |
| --------------------------------------------------------- | ------ | ------ |
| [Neg](https://doc.rust-lang.org/std/ops/trait.Neg.html)   | `-`    | Unary  |
| [Add](https://doc.rust-lang.org/std/ops/trait.Add.html)   | `+`    | Binary |
| [Sub](https://doc.rust-lang.org/std/ops/trait.Sub.html)   | `-`    | Binary |
| [Mul](https://doc.rust-lang.org/std/ops/trait.Mul.html)   | `*`    | Binary |
| [Div](https://doc.rust-lang.org/std/ops/trait.Div.html)\* | `/`    | Binary |
| [Rem](https://doc.rust-lang.org/std/ops/trait.Rem.html)\* | `%`    | Binary |

For the division operator, the convention is to return the modulus - 1. For instance, for FheUint8, the modulus is $$2^8=256$$, so a division by 0 will return an encryption of 255. For the remainder operator, the convention is to return the first input without any modification. For instance, for ct1 = FheUint8(63) and ct2 = FheUint8(0), then ct1 % ct2 will return FheUint8(63).

A simple example on how to use these operations:

```rust
use tfhe::prelude::*;
use tfhe::{generate_keys, set_server_key, ConfigBuilder, FheUint8};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let config = ConfigBuilder::all_disabled().enable_default_integers().build();
    let (keys, server_keys) = generate_keys(config);
    set_server_key(server_keys);

    let clear_a = 15_u64;
    let clear_b = 27_u64;
    let clear_c = 43_u64;

    let mut a = FheUint8::try_encrypt(clear_a, &keys)?;
    let mut b = FheUint8::try_encrypt(clear_b, &keys)?;
    let mut c = FheUint8::try_encrypt(clear_c, &keys)?;


    a = a * &b;  // Clear equivalent computations: 15 * 27 mod 256 = 149
    b = &b + &c; // Clear equivalent computations: 27 + 43 mod 256 = 70
    b = b - 76u8;   // Clear equivalent computations: 70 - 76 mod 256 = 250

    let dec_a: u8 = a.decrypt(&keys);
    let dec_b: u8 = b.decrypt(&keys);

    assert_eq!(dec_a, ((clear_a * clear_b) % 256_u64) as u8);
    assert_eq!(dec_b, (((clear_b  + clear_c).wrapping_sub(76_u64)) % 256_u64) as u8);

    Ok(())
}
```

### Bitwise operations.

Homomorphic integer types support some bitwise operations.

The list of supported operations is:

| name                                                                                 | symbol         | type   |
| ------------------------------------------------------------------------------------ | -------------- | ------ |
| [Not](https://doc.rust-lang.org/std/ops/trait.Not.html)                              | `!`            | Unary  |
| [BitAnd](https://doc.rust-lang.org/std/ops/trait.BitAnd.html)                        | `&`            | Binary |
| [BitOr](https://doc.rust-lang.org/std/ops/trait.BitOr.html)                          | `\|`           | Binary |
| [BitXor](https://doc.rust-lang.org/std/ops/trait.BitXor.html)                        | `^`            | Binary |
| [Shr](https://doc.rust-lang.org/std/ops/trait.Shr.html)                              | `>>`           | Binary |
| [Shl](https://doc.rust-lang.org/std/ops/trait.Shl.html)                              | `<<`           | Binary |
| [Rotate Right](https://doc.rust-lang.org/std/primitive.u32.html#method.rotate_right) | `rotate_right` | Binary |
| [Rotate Left](https://doc.rust-lang.org/std/primitive.u32.html#method.rotate_left)   | `rotate_left`  | Binary |

A simple example on how to use these operations:

```rust
use tfhe::prelude::*;
use tfhe::{generate_keys, set_server_key, ConfigBuilder, FheUint8};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let config = ConfigBuilder::all_disabled().enable_default_integers().build();
    let (keys, server_keys) = generate_keys(config);
    set_server_key(server_keys);

    let clear_a = 164;
    let clear_b = 212;

    let mut a = FheUint8::try_encrypt(clear_a, &keys)?;
    let mut b = FheUint8::try_encrypt(clear_b, &keys)?;

    a = a ^ &b;
    b = b ^ &a;
    a = a ^ &b;

    let dec_a: u8 = a.decrypt(&keys);
    let dec_b: u8 = b.decrypt(&keys);

    // We homomorphically swapped values using bitwise operations
    assert_eq!(dec_a, clear_b);
    assert_eq!(dec_b, clear_a);

    Ok(())
}
```

### Comparisons.

Homomorphic integers support comparison operations. Since Rust does not allow the overloading of these operations, a simple function has been associated to each one.

The list of supported operations is:

| name                                                                        | symbol | type   |
| --------------------------------------------------------------------------- | ------ | ------ |
| [Equal](https://doc.rust-lang.org/std/cmp/trait.PartialEq.html)             | `eq`   | Binary |
| [Not Equal](https://doc.rust-lang.org/std/cmp/trait.PartialEq.html)         | `ne`   | Binary |
| [Greater Than](https://doc.rust-lang.org/std/cmp/trait.PartialOrd.html)     | `gt`   | Binary |
| [Greater or Equal](https://doc.rust-lang.org/std/cmp/trait.PartialOrd.html) | `ge`   | Binary |
| [Lower](https://doc.rust-lang.org/std/cmp/trait.PartialOrd.html)            | `lt`   | Binary |
| [Lower or Equal](https://doc.rust-lang.org/std/cmp/trait.PartialOrd.html)   | `le`   | Binary |

A simple example on how to use these operations:

```rust
use tfhe::prelude::*;
use tfhe::{generate_keys, set_server_key, ConfigBuilder, FheUint8};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let config = ConfigBuilder::all_disabled().enable_default_integers().build();
    let (keys, server_keys) = generate_keys(config);
    set_server_key(server_keys);

    let clear_a:u8 = 164;
    let clear_b:u8 = 212;

    let mut a = FheUint8::try_encrypt(clear_a, &keys)?;
    let mut b = FheUint8::try_encrypt(clear_b, &keys)?;

    let greater = a.gt(&b);
    let greater_or_equal = a.ge(&b);
    let lower = a.lt(&b);
    let lower_or_equal = a.le(&b);
    let equal = a.eq(&b);

    let dec_gt : u8 = greater.decrypt(&keys);
    let dec_ge : u8 = greater_or_equal.decrypt(&keys);
    let dec_lt : u8 = lower.decrypt(&keys);
    let dec_le : u8 = lower_or_equal.decrypt(&keys);
    let dec_eq : u8 = equal.decrypt(&keys);

    // We homomorphically swapped values using bitwise operations
    assert_eq!(dec_gt, (clear_a > clear_b ) as u8);
    assert_eq!(dec_ge, (clear_a >= clear_b) as u8);
    assert_eq!(dec_lt, (clear_a < clear_b ) as u8);
    assert_eq!(dec_le, (clear_a <= clear_b) as u8);
    assert_eq!(dec_eq, (clear_a == clear_b) as u8);

    Ok(())
}
```

### Min/Max.

Homomorphic integers support the min/max operations.

| name | symbol | type   |
| ---- | ------ | ------ |
| Min  | `min`  | Binary |
| Max  | `max`  | Binary |

A simple example on how to use these operations:

```rust
use tfhe::prelude::*;
use tfhe::{generate_keys, set_server_key, ConfigBuilder, FheUint8};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let config = ConfigBuilder::all_disabled().enable_default_integers().build();
    let (keys, server_keys) = generate_keys(config);
    set_server_key(server_keys);

    let clear_a:u8 = 164;
    let clear_b:u8 = 212;

    let mut a = FheUint8::try_encrypt(clear_a, &keys)?;
    let mut b = FheUint8::try_encrypt(clear_b, &keys)?;

    let min = a.min(&b);
    let max = a.max(&b);

    let dec_min : u8 = min.decrypt(&keys);
    let dec_max : u8 = max.decrypt(&keys);

    // We homomorphically swapped values using bitwise operations
    assert_eq!(dec_min, u8::min(clear_a, clear_b));
    assert_eq!(dec_max, u8::max(clear_a, clear_b));

    Ok(())
}
```

### Casting.

Casting between integer types is possible via the `cast_from` associated function or the `cast_into` method.

```rust
use tfhe::prelude::*;
use tfhe::{generate_keys, set_server_key, ConfigBuilder, FheUint8, FheUint32, FheUint16};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let config = ConfigBuilder::all_disabled()
        .enable_default_integers()
        .build();
    let (client_key, server_key) = generate_keys(config);

    // Casting requires server_key to set
    // (encryptions/decryptions do not need server_key to be set)
    set_server_key(server_key);

    {
        let clear = 12_837u16;
        let a = FheUint16::encrypt(clear, &client_key);

        // Downcasting
        let a: FheUint8 = a.cast_into();
        let da: u8 = a.decrypt(&client_key);
        assert_eq!(da, clear as u8);

        // Upcasting
        let a: FheUint32 = a.cast_into();
        let da: u32 = a.decrypt(&client_key);
        assert_eq!(da, (clear as u8) as u32);
    }

    {
        let clear = 12_837u16;
        let a = FheUint16::encrypt(clear, &client_key);

        // Upcasting
        let a = FheUint32::cast_from(a);
        let da: u32 = a.decrypt(&client_key);
        assert_eq!(da, clear as u32);

        // Downcasting
        let a = FheUint8::cast_from(a);
        let da: u8 = a.decrypt(&client_key);
        assert_eq!(da, (clear as u32) as u8);
    }

    Ok(())
}
```


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