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# Types & Operations

## Types

TFHE-rs includes two main types to represent encrypted data:
• FheUint: this is the homomorphic equivalent of Rust unsigned integers u8, u16, ...
• FheInt: this is the homomorphic equivalent of Rust (signed) integers i8, i16, ...
In the same manner as many programming languages, the number of bits used to represent the data must be chosen when declaring a variable. For instance:
// let clear_a: u64 = 7;
let mut a = FheUint64::try_encrypt(clear_a, &keys)?;
// let clear_b: i8 = 3;
let mut b = FheInt8::try_encrypt(clear_b, &keys)?;
// let clear_c: u128 = 2;
let mut c = FheUint128::try_encrypt(clear_c, &keys)?;

## Operation list

The table below contains an overview of the available operations in TFHE-rs. The notation Enc (for Encypted) either refers to FheInt or FheUint, for any size between 1 and 256-bits.
More details, and further examples, are given in the following sections.
 name symbol Enc/Enc Enc/ Int Neg - ​✔​ ​✔​ Add + ​✔​ ​✔​ Sub - ​✔​ ​✔​ Mul * ​✔​ ​✔​ Div / ​✔​ ​✔​ Rem % ​✔​ ​✔​ Not ! ​✔​ ​✔​ BitAnd & ​✔​ ​✔​ BitOr | ​✔​ ​✔​ BitXor ^ ​✔​ ​✔​ Shr >> ​✔​ ​✔​ Shl << ​✔​ ​✔​ Min min ​✔​ ​✔​ Max max ​✔​ ​✔​ Greater than gt ​✔​ ​✔​ Greater or equal than ge ​✔​ ​✔​ Lower than lt ​✔​ ​✔​ Lower or equal than le ​✔​ ​✔​ Equal eq ​✔​ ​✔​ Cast (into dest type) cast_into ​✔​ ​✖​ Cast (from src type) cast_from ​✔​ ​✖​ Ternary operator if_then_else ​✔​ ​✖​

## Integer

In TFHE-rs, integers are used to encrypt all messages which are 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
-
Unary
+
Binary
Sub
-
Binary
Mul
*
Binary
Div*
/
Binary
Rem*
%
Binary
For division by 0, the convention is to return 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, if ct1 = FheUint8(63) and ct2 = FheUint8(0) then ct1 % ct2 will return FheUint8(63).
A simple example of how to use these operations:
use tfhe::prelude::*;
use tfhe::{generate_keys, set_server_key, ConfigBuilder, FheInt8, 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 clear_d = -87_i64;
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)?;
let mut d = FheInt8::try_encrypt(clear_d, &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
d = d - 13i8; // Clear equivalent computations: -87 - 13 = 100 in [-128, 128[
let dec_a: u8 = a.decrypt(&keys);
let dec_b: u8 = b.decrypt(&keys);
let dec_d: i8 = d.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);
assert_eq!(dec_d, (clear_d - 13) as i8);
Ok(())
}

### Bitwise operations.

Homomorphic integer types support some bitwise operations.
The list of supported operations is:
name
symbol
type
Not
!
Unary
BitAnd
&
Binary
BitOr
|
Binary
BitXor
^
Binary
Shr
>>
Binary
Shl
<<
Binary
rotate_right
Binary
rotate_left
Binary
A simple example of how to use these operations:
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.
Due to some Rust limitations, it is not possible to overload the comparison symbols because of the inner definition of the operations. This is because 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:
The list of supported operations is:
name
symbol
type
Equal
eq
Binary
Not Equal
ne
Binary
gt
Binary
ge
Binary
Lower
lt
Binary
le
Binary
A simple example of how to use these operations:
use tfhe::prelude::*;
use tfhe::{generate_keys, set_server_key, ConfigBuilder, FheInt8};
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: i8 = -121;
let clear_b: i8 = 87;
let mut a = FheInt8::try_encrypt(clear_a, &keys)?;
let mut b = FheInt8::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: i8 = greater.decrypt(&keys);
let dec_ge: i8 = greater_or_equal.decrypt(&keys);
let dec_lt: i8 = lower.decrypt(&keys);
let dec_le: i8 = lower_or_equal.decrypt(&keys);
let dec_eq: i8 = equal.decrypt(&keys);
// We homomorphically swapped values using bitwise operations
assert_eq!(dec_gt, (clear_a > clear_b ) as i8);
assert_eq!(dec_ge, (clear_a >= clear_b) as i8);
assert_eq!(dec_lt, (clear_a < clear_b ) as i8);
assert_eq!(dec_le, (clear_a <= clear_b) as i8);
assert_eq!(dec_eq, (clear_a == clear_b) as i8);
Ok(())
}

### Min/Max.

Homomorphic integers support the min/max operations.
name
symbol
type
Min
min
Binary
Max
max
Binary
A simple example of how to use these operations:
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(())
}

### Ternary conditional operator.

The ternary conditional operator allows computing conditional instructions of the form if cond { choice_if } else { choice_else }.
name
symbol
type
Ternary operator
if_then_else
Ternary
The syntax is encrypted_condition.if_then_else(encrypted_choice_if, encrypted_choice_else). The encrypted_condition should be an encryption of 0 or 1 in order to be valid.
use tfhe::prelude::*;
use tfhe::{generate_keys, set_server_key, ConfigBuilder, FheInt32};
fn main() -> Result<(), Box<dyn std::error::Error>> {
// Basic configuration to use homomorphic integers
let config = ConfigBuilder::all_disabled()
.enable_default_integers()
.build();
// Key generation
let (client_key, server_keys) = generate_keys(config);
let clear_a = 32i32;
let clear_b = -45i32;
// Encrypting the input data using the (private) client_key
// FheInt32: Encrypted equivalent to i32
let encrypted_a = FheInt32::try_encrypt(clear_a, &client_key)?;
let encrypted_b = FheInt32::try_encrypt(clear_b, &client_key)?;
// On the server side:
set_server_key(server_keys);
// Clear equivalent computations: 32 > -45
let encrypted_comp = &encrypted_a.gt(&encrypted_b);
let clear_res: i32 = encrypted_comp.decrypt(&client_key);
assert_eq!(clear_res, (clear_a > clear_b) as i32);
// encrypted_comp contains the result of the comparison, i.e.,
// a boolean value. This acts as a condition on which the
// if_then_else function can be applied on.
// Clear equivalent computations:
// if 32 > -45 {result = 32} else {result = -45}
let encrypted_res = &encrypted_comp.if_then_else(&encrypted_a, &encrypted_b);
let clear_res: i32 = encrypted_res.decrypt(&client_key);
assert_eq!(clear_res, clear_a);
Ok(())
}

### Casting.

Casting between integer types is possible via the cast_from associated function or the cast_into method.
use tfhe::prelude::*;
use tfhe::{generate_keys, set_server_key, ConfigBuilder, FheInt16, 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);
}
{
let clear = 12_837i16;
let a = FheInt16::encrypt(clear, &client_key);
// Casting from FheInt16 to FheUint16
let a = FheUint16::cast_from(a);
let da: u16 = a.decrypt(&client_key);
assert_eq!(da, clear as u16);
}
Ok(())
}

## ShortInt Operations

Native small homomorphic integer types (e.g., FheUint3 or FheUint4) can easily compute various operations. In general, computing over encrypted data in TFHE-rs 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 presents 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 maximum possible value for the message.
The list of supported operations is:
name
symbol
type
+
Binary
Sub
-
Binary
Mul
*
Binary
Div
/
Binary
Rem
%
Binary
Neg
-
Unary
A simple example of how to use these operations:
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
!
Unary
BitAnd
&
Binary
BitOr
|
Binary
BitXor
^
Binary
Shr
>>
Binary
Shl
<<
Binary
rotate_right
Binary
rotate_left
Binary
A simple example of how to use these operations:
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:
The list of supported operations is:
name
symbol
type
Equal
eq
Binary
Not Equal
ne
Binary
gt
Binary
ge
Binary
Lower
lt
Binary
le
Binary
A simple example of how to use these operations:
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, &