#fhe #homomorphic #arithmetics #parallel


PARMESAN: Parallel ARithMEticS over ENcrypted data

9 releases

0.1.0 Apr 20, 2023
0.1.0-alpha Apr 19, 2023
0.0.20-alpha-4 Sep 2, 2022
0.0.20-alpha-3 Aug 26, 2022
0.0.2 Jul 31, 2021

#105 in Math


3.5K SLoC

Rust 3K SLoC // 0.3% comments Ruby 355 SLoC // 0.2% comments


Parallel ARithMEticS over ENcrypted data

Parmesan implements selected parallel algorithms for multi-digit arithmetics over TFHE ciphertexts. Namely:

  • addition/subtraction,
  • scalar multiplication (i.e., multiplication by a known integer),
  • multiplication,
  • squaring,
  • signum,
  • maximum of two numbers,
  • rounding, and
  • evaluation of a simple neural network.

Disclaimer: Parmesan is currently an experimental library, which serves as a proof-of-concept for parallelizable arithmetics over encrypted data. Hence neither correct functionality nor any level of code quality is guaranteed.

The Short Story

In the standard integer representation, parallel addition is not possible due to the carry, which can propagate all the way from the LSB to the MSB. However, using, e.g., an alphabet {-1,0,1} for base-2 integer representation, a parallel addition algorithm does exist. Other operations like (scalar) multiplication and squaring benefit from the fast addition, too.

The Long Story

See our full paper on eprint and our preceeding study that compares various approaches for parallel addition (also on eprint).

Use parmesan

Add a dependency to your Cargo.toml file in your Rust project.

parmesan = { version = "^0.0.20-alpha", features = ["measure"] }
colored = "^2.0.0"

For the best performance, compile with

$ RUSTFLAGS="-C target-cpu=native" cargo build --release


use std::error::Error;

use colored::Colorize;

use parmesan::params;
use parmesan::userovo::*;
use parmesan::ParmesanUserovo;
use parmesan::ParmesanCloudovo;
use parmesan::arithmetics::ParmArithmetics;

pub fn main() -> Result<(), Box<dyn Error>> {

    // =================================
    //  Initialization
    // ---------------------------------
    //  Global Scope
    let par = &params::PAR_TFHE_V0_2__M4_C0;

    // ---------------------------------
    //  Userovo Scope
    let pu = ParmesanUserovo::new(par)?;
    let pub_k = pu.export_pub_keys();
    // ---------------------------------
    //  Cloudovo Scope
    let pc = ParmesanCloudovo::new(par, &pub_k);

    // =================================
    //  U: Encryption
    let a: Vec<i32> = vec![1,0,1,-1,-1,0,-1,1,1,-1,1,1,1,-1,-1,0,0,1,1,0,0,0,0,-1,0,0,0,0,0,-1,0,0,];
    let b: Vec<i32> = vec![-1,0,0,-1,1,1,-1,1,-1,0,0,1,0,1,1,0,0,0,-1,0,0,1,0,0,-1,0,-1,-1,-1,1,1,0,];
    let ca = pu.encrypt_vec(&a)?;
    let cb = pu.encrypt_vec(&b)?;
    // convert to actual numbers
    let a_val = encryption::convert_from_vec(&a)?;
    let b_val = encryption::convert_from_vec(&b)?;
    // print plain inputs
    println!("a   = {:12}", a_val);
    println!("b   = {:12}", b_val);

    // =================================
    //  C: Evaluation
    let c_add_a_b = ParmArithmetics::add(&pc, &ca, &cb);

    // =================================
    //  U: Decryption
    let add_a_b = pu.decrypt(&c_add_a_b)?;

    let mut summary_text = format!("\nResults:\n");
    summary_text = format!("{}\nAddition:", summary_text);
    summary_text = format!("{}\na + b         = {:12} :: {} (exp. {})", summary_text,
                            if add_a_b == a_val + b_val {String::from("PASS").bold().green()} else {String::from("FAIL").bold().red()},
                            a_val + b_val

    println!("{}", summary_text);

    println!("\nDemo END\n");



For benchmarks, we implement an experimental tool bench-parmesan, which also compares Parmesan and Concrete v0.2. Results can be found in the whitepaper.

Processor Load

In particular, we measure the processor load; in the graph below, find our results for 32-bit maximum that were measured on a machine with AMD EPYC 7543.

proc load max

An extra graph that shows the processor load during 32-bit multiplication can be found here.

Version Log

  • v0.1.0-alpha: rewritten using tfhe v0.2 instead of concrete_core.
  • v0.0.20-alpha-4: algorithms for whitepaper.


Parmesan is licensed under AGPLv3.


This work was supported by the MESRI-BMBF French-German joint project UPCARE (ANR-20-CYAL-0003-01), granted to EURECOM.


Computational resources were supplied by the project e-Infrastruktura CZ (e-INFRA CZ LM2018140) supported by the Ministry of Education, Youth and Sports of the Czech Republic.

e-infra logo


~342K SLoC