created_at2019-03-13 23:32:17.858464
updated_at2021-01-31 16:29:15.034388
descriptionFull Domain Hash (FDH) for extending the size of a hash digest to an arbitrary length
Todd McGee (tpmcgee)



Full Domain Hash

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A Full Domain Hash (FDH) is a useful cryptographic construction that limits the domain of the digest of a hash function (for example ensuring the digest is less than modulus n in RSA). Secondarily, it can also be used to extend the size of a hash digest to an arbitrary length, turning a regular hash function into an XOF hash function.

We construct an FDH by computing a number of cycles where:

cycles=(target length)/(digest length) + 1

We then compute:

FDH(M) = HASH(M||0) || HASH(M||1) || ... || HASH(M||cycles−1)

Where HASH is any hash function, M is the message, || denotes concatenation, and numerical values are single-byte u8.

FDHs are usually used with an RSA signature scheme where the target length is the size of the key, and the domain is less than modulus n. See https://en.wikipedia.org/wiki/Full_Domain_Hash

This crate makes extensive use of the digest crate's cryptograhic hash traits, so most useful methods are implemented as part of digest traits. These traits are re-exported for convenience. See https://github.com/RustCrypto/hashes for a list of compatible hashes.

It should be noted that FDH is not constant-time in relation to the message. While the variable-time natue of a FDH cannot be used to recover the message (except in pathological cases), it can be used to eliminate certain values from the set of all possible values for the message.


  use sha2::Sha256;
  use fdh::{FullDomainHash, VariableOutput, Input};

  // Expand SHA256 from 256 bits to 1024 bits.
  let output_bits = 1024;
  let output_bytes = 1024 / 8;
  let mut hasher = FullDomainHash::<Sha256>::new(output_bytes)?;
  hasher.input(b"ATTACK AT DAWN");
  let result = hasher.vec_result();


This crate also supports no_std, so it can be used in embedded or other settings with no allocation.

use sha2::Sha256;
use fdh::{FullDomainHash, Input, ExtendableOutput, XofReader};

// Expand SHA256 from 256 bits to 512 bits (and beyond!), reading it in 16 byte chunks.
let mut hasher = FullDomainHash::<Sha256>::default();
hasher.input(b"ATTACK AT DAWN");
let mut reader = hasher.xof_result();
let mut read_buf = <[u8; 16]>::default();

// Read the first 16 bytes into read_buf
reader.read(&mut read_buf);

// Read the second 16 bytes into read_buf
reader.read(&mut read_buf);

// If we want, we can just keep going, reading as many bits as we want indefinitely.
reader.read(&mut read_buf);
reader.read(&mut read_buf);

Restricted Domain

This crate also supports getting a digest that is within a specific domain. It follows an algorithim like so:

fn digest_in_domain(message, iv):
    digest = fdh(message, iv)
    while not in_domain(digest):
        digest = fdh(message, iv)
    return digest, iv

The method results_in_domain() is provided to accomplish this. The helper methods results_between(), results_lt(), results_gt() are provided for the common case where the digest must be in a certain range.

An example that produces a digest that is odd:

use sha2::Sha512;
use fdh::{FullDomainHash, Input, VariableOutput};
use num_bigint::BigUint;
use num_integer::Integer;

// Get a full domain hash that is odd
let mut hasher = FullDomainHash::<Sha512>::new(64).unwrap();

fn digest_is_odd(digest: &[u8]) -> bool {
let iv = 0;

let (digest, iv) = hasher.results_in_domain(iv, digest_is_odd).unwrap();


  1. Patrick Hayes (linkedin) (github) - Available for hire.
Commit count: 112

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