diff --git a/.github/workflows/x448.yml b/.github/workflows/x448.yml new file mode 100644 index 000000000..be86aeee4 --- /dev/null +++ b/.github/workflows/x448.yml @@ -0,0 +1,111 @@ +name: x448 + +on: + pull_request: + paths: + - ".github/workflows/x448.yml" + - "ed448-goldilocks/**" + - "x448/**" + - "Cargo.*" + push: + branches: master + +defaults: + run: + working-directory: x448 + +env: + CARGO_INCREMENTAL: 0 + RUSTFLAGS: "-Dwarnings" + RUSTDOCFLAGS: "-Dwarnings" + +jobs: + build: + runs-on: ubuntu-latest + strategy: + matrix: + rust: + - 1.85.0 # MSRV + - stable + target: + - thumbv7em-none-eabi + - wasm32-unknown-unknown + steps: + - uses: actions/checkout@v4 + - uses: dtolnay/rust-toolchain@master + with: + toolchain: ${{ matrix.rust }} + targets: ${{ matrix.target }} + - run: cargo build --target ${{ matrix.target }} --release + + test: + runs-on: ubuntu-latest + strategy: + matrix: + include: + # 32-bit Linux + - target: i686-unknown-linux-gnu + rust: 1.85.0 # MSRV + deps: sudo apt update && sudo apt install gcc-multilib + - target: i686-unknown-linux-gnu + rust: stable + deps: sudo apt update && sudo apt install gcc-multilib + + # 64-bit Linux + - target: x86_64-unknown-linux-gnu + rust: 1.85.0 # MSRV + - target: x86_64-unknown-linux-gnu + rust: stable + + steps: + - uses: actions/checkout@v4 + - uses: dtolnay/rust-toolchain@master + with: + toolchain: ${{ matrix.rust }} + targets: ${{ matrix.target }} + - uses: RustCrypto/actions/cargo-hack-install@master + - run: ${{ matrix.deps }} + - run: cargo test --release --target ${{ matrix.target }} + + cross: + strategy: + matrix: + include: + # ARM32 + - target: armv7-unknown-linux-gnueabihf + rust: 1.85.0 # MSRV (cross) + - target: armv7-unknown-linux-gnueabihf + rust: stable + + # ARM64 + - target: aarch64-unknown-linux-gnu + rust: 1.85.0 # MSRV (cross) + - target: aarch64-unknown-linux-gnu + rust: stable + + # PPC32 + - target: powerpc-unknown-linux-gnu + rust: 1.85.0 # MSRV (cross) + - target: powerpc-unknown-linux-gnu + rust: stable + + runs-on: ubuntu-latest + steps: + - uses: actions/checkout@v4 + - run: ${{ matrix.deps }} + - uses: dtolnay/rust-toolchain@master + with: + toolchain: ${{ matrix.rust }} + targets: ${{ matrix.target }} + - uses: RustCrypto/actions/cross-install@master + - run: cross test --release --target ${{ matrix.target }} + + doc: + runs-on: ubuntu-latest + steps: + - uses: actions/checkout@v4 + - uses: RustCrypto/actions/cargo-cache@master + - uses: dtolnay/rust-toolchain@master + with: + toolchain: stable + - run: cargo doc diff --git a/Cargo.lock b/Cargo.lock index 455653ad0..db2f9d2da 100644 --- a/Cargo.lock +++ b/Cargo.lock @@ -1192,6 +1192,7 @@ dependencies = [ name = "sm2" version = "0.14.0-pre" dependencies = [ + "der", "elliptic-curve", "hex-literal", "primefield", @@ -1567,6 +1568,16 @@ dependencies = [ "tap", ] +[[package]] +name = "x448" +version = "0.7.0-pre" +dependencies = [ + "ed448-goldilocks", + "rand", + "rand_core", + "zeroize", +] + [[package]] name = "zerocopy" version = "0.8.26" @@ -1592,3 +1603,17 @@ name = "zeroize" version = "1.8.1" source = "registry+https://github.com/rust-lang/crates.io-index" checksum = "ced3678a2879b30306d323f4542626697a464a97c0a07c9aebf7ebca65cd4dde" +dependencies = [ + "zeroize_derive", +] + +[[package]] +name = "zeroize_derive" +version = "1.4.2" +source = "registry+https://github.com/rust-lang/crates.io-index" +checksum = "ce36e65b0d2999d2aafac989fb249189a141aee1f53c612c1f37d72631959f69" +dependencies = [ + "proc-macro2", + "quote", + "syn 2.0.104", +] diff --git a/Cargo.toml b/Cargo.toml index 8085bee70..281cafed9 100644 --- a/Cargo.toml +++ b/Cargo.toml @@ -14,7 +14,8 @@ members = [ "p521", "primefield", "primeorder", - "sm2" + "sm2", + "x448" ] [profile.dev] @@ -24,6 +25,7 @@ opt-level = 2 ecdsa = { git = "https://github.com/RustCrypto/signatures.git" } elliptic-curve = { git = "https://github.com/RustCrypto/traits.git" } +ed448-goldilocks = { path = "ed448-goldilocks" } hash2curve = { path = "hash2curve" } primefield = { path = "primefield" } primeorder = { path = "primeorder" } diff --git a/x448/Cargo.toml b/x448/Cargo.toml new file mode 100644 index 000000000..50ea6b92e --- /dev/null +++ b/x448/Cargo.toml @@ -0,0 +1,26 @@ +[package] +name = "x448" +version = "0.7.0-pre" +authors = ["RustCrypto Developers"] +categories = ["cryptography"] +keywords = ["cryptography", "crypto", "x448", "diffie-hellman", "curve448",] +homepage = "https://docs.rs/x448/" +repository = "https://github.com/RustCrypto/elliptic-curves/tree/master/x448" +documentation = "https://docs.rs/ed448-goldilocks" +license = "Apache-2.0 OR MIT" +edition = "2024" +rust-version = "1.85" +readme = "README.md" +description = "A pure-Rust implementation of X448." + +[dependencies] +ed448-goldilocks = { version = "0.14.0-pre.0", default-features = false } +rand_core = { version = "0.9", default-features = false } + +[dependencies.zeroize] +version = "1" +default-features = false +features = ["zeroize_derive"] + +[dev-dependencies] +rand = "0.9" diff --git a/x448/src/lib.rs b/x448/src/lib.rs new file mode 100644 index 000000000..5d62031a3 --- /dev/null +++ b/x448/src/lib.rs @@ -0,0 +1,407 @@ +#![no_std] + +use ed448_goldilocks::{ + MontgomeryPoint, + elliptic_curve::{bigint::U448, scalar::FromUintUnchecked}, +}; +use rand_core::{CryptoRng, RngCore}; +use zeroize::Zeroize; + +type MontgomeryScalar = ed448_goldilocks::Scalar; + +/// Computes a Scalar according to RFC7748 +/// given a byte array of length 56 +impl From<[u8; 56]> for Secret { + fn from(arr: [u8; 56]) -> Secret { + let mut secret = Secret(arr); + secret.clamp(); + secret + } +} + +/// Given a Secret Key, compute the corresponding public key +/// using the generator specified in RFC7748 +/// XXX: Waiting for upstream PR to use pre-computation +impl From<&Secret> for PublicKey { + fn from(secret: &Secret) -> PublicKey { + let secret = secret.as_scalar(); + let point = &MontgomeryPoint::GENERATOR * &secret; + PublicKey(point) + } +} + +/// A PublicKey is a point on Curve448. +#[derive(Debug, PartialEq, Eq, Copy, Clone)] +pub struct PublicKey(MontgomeryPoint); + +/// A Secret is a Scalar on Curve448. +#[derive(Clone, Zeroize)] +#[zeroize(drop)] +pub struct Secret([u8; 56]); + +/// A SharedSecret is a point on Curve448. +/// This point is the result of a Diffie-Hellman key exchange. +#[derive(Zeroize)] +#[zeroize(drop)] +pub struct SharedSecret(MontgomeryPoint); + +impl PublicKey { + /// Converts a bytes slice into a Public key + /// Returns None if: + /// - The length of the slice is not 56 + /// - The point is a low order point + pub fn from_bytes(bytes: &[u8]) -> Option { + let public_key = PublicKey::from_bytes_unchecked(bytes)?; + if public_key.0.is_low_order() { + return None; + } + Some(public_key) + } + /// Converts a bytes slice into a Public key + /// Returns None if: + /// - The length of the slice is not 56 + pub fn from_bytes_unchecked(bytes: &[u8]) -> Option { + // First check if we have 56 bytes + if bytes.len() != 56 { + return None; + } + + // Check if the point has low order + let arr = slice_to_array(bytes); + let point = MontgomeryPoint(arr); + + Some(PublicKey(point)) + } + + /// Converts a public key into a byte slice + pub fn as_bytes(&self) -> &[u8; 56] { + self.0.as_bytes() + } +} + +impl SharedSecret { + /// Converts a shared secret into a byte slice + pub fn as_bytes(&self) -> &[u8; 56] { + self.0.as_bytes() + } +} + +impl Secret { + /// Generate a x448 `Secret` key. + // Taken from dalek-x25519 + pub fn new(csprng: &mut T) -> Self + where + T: RngCore + CryptoRng + ?Sized, + { + let mut bytes = [0u8; 56]; + + csprng.fill_bytes(&mut bytes); + + Secret::from(bytes) + } + + /// Clamps the secret key according to RFC7748 + fn clamp(&mut self) { + self.0[0] &= 252; + self.0[55] |= 128; + } + + /// Views a Secret as a Scalar + fn as_scalar(&self) -> MontgomeryScalar { + let secret = U448::from_le_slice(&self.0); + MontgomeryScalar::from_uint_unchecked(secret) + } + + /// Performs a Diffie-hellman key exchange between the secret key and an external public key + pub fn as_diffie_hellman(&self, public_key: &PublicKey) -> Option { + // Check if the point is one of the low order points + if public_key.0.is_low_order() { + return None; + } + let shared_key = &public_key.0 * &self.as_scalar(); + Some(SharedSecret(shared_key)) + } + + /// Performs a Diffie-hellman key exchange once between the secret key and an external public key + pub fn to_diffie_hellman(self, public_key: &PublicKey) -> Option { + self.as_diffie_hellman(public_key) + } + + /// Converts a byte slice into a secret and clamp + pub fn from_bytes(bytes: &[u8]) -> Option { + // First check if we have 56 bytes + if bytes.len() != 56 { + return None; + } + + let secret = Secret::from(slice_to_array(bytes)); + Some(secret) + } + + /// Converts a secret into a byte array + pub fn as_bytes(&self) -> &[u8; 56] { + &self.0 + } +} + +fn slice_to_array(bytes: &[u8]) -> [u8; 56] { + let mut array: [u8; 56] = [0; 56]; + array.copy_from_slice(bytes); + array +} + +/// A safe version of the x448 function defined in RFC448. +/// Currently, the only reason I can think of for using the raw function is FFI. +/// Option is FFI safe[1]. So we can still maintain that the invariant that +/// we do not return a low order point. +/// +/// [1]: https://github.com/rust-lang/nomicon/issues/59 +pub fn x448(scalar_bytes: [u8; 56], point_bytes: [u8; 56]) -> Option<[u8; 56]> { + let point = PublicKey::from_bytes(&point_bytes)?; + let scalar = Secret::from(scalar_bytes).as_scalar(); + Some((&point.0 * &scalar).0) +} +/// An unchecked version of the x448 function defined in RFC448 +/// No checks are made on the points. +pub fn x448_unchecked(scalar_bytes: [u8; 56], point_bytes: [u8; 56]) -> [u8; 56] { + let point = MontgomeryPoint(point_bytes); + let scalar = Secret::from(scalar_bytes).as_scalar(); + (&point * &scalar).0 +} + +pub const X448_BASEPOINT_BYTES: [u8; 56] = MontgomeryPoint::GENERATOR.0; + +#[cfg(test)] +mod test { + extern crate alloc; + + use super::*; + use alloc::vec; + + #[test] + fn test_low_order() { + // Notice, that this is the only way to add low order points into the system + // and this is not exposed to the user. The user will use `from_bytes` which will check for low order points. + let bad_key_a = PublicKey(MontgomeryPoint::LOW_A); + let checked_bad_key_a = PublicKey::from_bytes(&MontgomeryPoint::LOW_A.0); + assert!(checked_bad_key_a.is_none()); + + let bad_key_b = PublicKey(MontgomeryPoint::LOW_B); + let checked_bad_key_b = PublicKey::from_bytes(&MontgomeryPoint::LOW_B.0); + assert!(checked_bad_key_b.is_none()); + + let bad_key_c = PublicKey(MontgomeryPoint::LOW_C); + let checked_bad_key_c = PublicKey::from_bytes(&MontgomeryPoint::LOW_C.0); + assert!(checked_bad_key_c.is_none()); + + let mut rng = rand::rng(); + let bob_priv = Secret::new(&mut rng); + + // If for some reason, these low order points are added to the system + // The Diffie-Hellman key exchange for the honest party will return None. + let shared_bob = bob_priv.as_diffie_hellman(&bad_key_a); + assert!(shared_bob.is_none()); + + let shared_bob = bob_priv.as_diffie_hellman(&bad_key_b); + assert!(shared_bob.is_none()); + + let shared_bob = bob_priv.as_diffie_hellman(&bad_key_c); + assert!(shared_bob.is_none()); + } + + #[test] + fn test_random_dh() { + let mut rng = rand::rng(); + let alice_priv = Secret::new(&mut rng); + let alice_pub = PublicKey::from(&alice_priv); + + let bob_priv = Secret::new(&mut rng); + let bob_pub = PublicKey::from(&bob_priv); + + // Since Alice and Bob are both using the API correctly + // If by chance, a low order point is generated, the clamping function will + // remove it. + let low_order = alice_pub.0.is_low_order() || bob_pub.0.is_low_order(); + assert!(low_order == false); + + // Both Alice and Bob perform the DH key exchange. + // As mentioned above, we unwrap because both Parties are using the API correctly. + let shared_alice = alice_priv.as_diffie_hellman(&bob_pub).unwrap(); + let shared_bob = bob_priv.as_diffie_hellman(&alice_pub).unwrap(); + + assert_eq!(shared_alice.as_bytes()[..], shared_bob.as_bytes()[..]); + } + + #[test] + fn test_rfc_test_vectors_alice_bob() { + let alice_priv = Secret::from_bytes(&[ + 0x9a, 0x8f, 0x49, 0x25, 0xd1, 0x51, 0x9f, 0x57, 0x75, 0xcf, 0x46, 0xb0, 0x4b, 0x58, + 0x0, 0xd4, 0xee, 0x9e, 0xe8, 0xba, 0xe8, 0xbc, 0x55, 0x65, 0xd4, 0x98, 0xc2, 0x8d, + 0xd9, 0xc9, 0xba, 0xf5, 0x74, 0xa9, 0x41, 0x97, 0x44, 0x89, 0x73, 0x91, 0x0, 0x63, + 0x82, 0xa6, 0xf1, 0x27, 0xab, 0x1d, 0x9a, 0xc2, 0xd8, 0xc0, 0xa5, 0x98, 0x72, 0x6b, + ]) + .unwrap(); + let got_alice_pub = PublicKey::from(&alice_priv); + + let expected_alice_pub = [ + 0x9b, 0x8, 0xf7, 0xcc, 0x31, 0xb7, 0xe3, 0xe6, 0x7d, 0x22, 0xd5, 0xae, 0xa1, 0x21, 0x7, + 0x4a, 0x27, 0x3b, 0xd2, 0xb8, 0x3d, 0xe0, 0x9c, 0x63, 0xfa, 0xa7, 0x3d, 0x2c, 0x22, + 0xc5, 0xd9, 0xbb, 0xc8, 0x36, 0x64, 0x72, 0x41, 0xd9, 0x53, 0xd4, 0xc, 0x5b, 0x12, + 0xda, 0x88, 0x12, 0xd, 0x53, 0x17, 0x7f, 0x80, 0xe5, 0x32, 0xc4, 0x1f, 0xa0, + ]; + assert_eq!(got_alice_pub.as_bytes()[..], expected_alice_pub[..]); + + let bob_priv = Secret::from_bytes(&[ + 0x1c, 0x30, 0x6a, 0x7a, 0xc2, 0xa0, 0xe2, 0xe0, 0x99, 0xb, 0x29, 0x44, 0x70, 0xcb, + 0xa3, 0x39, 0xe6, 0x45, 0x37, 0x72, 0xb0, 0x75, 0x81, 0x1d, 0x8f, 0xad, 0xd, 0x1d, + 0x69, 0x27, 0xc1, 0x20, 0xbb, 0x5e, 0xe8, 0x97, 0x2b, 0xd, 0x3e, 0x21, 0x37, 0x4c, + 0x9c, 0x92, 0x1b, 0x9, 0xd1, 0xb0, 0x36, 0x6f, 0x10, 0xb6, 0x51, 0x73, 0x99, 0x2d, + ]) + .unwrap(); + let got_bob_pub = PublicKey::from(&bob_priv); + + let expected_bob_pub = [ + 0x3e, 0xb7, 0xa8, 0x29, 0xb0, 0xcd, 0x20, 0xf5, 0xbc, 0xfc, 0xb, 0x59, 0x9b, 0x6f, + 0xec, 0xcf, 0x6d, 0xa4, 0x62, 0x71, 0x7, 0xbd, 0xb0, 0xd4, 0xf3, 0x45, 0xb4, 0x30, + 0x27, 0xd8, 0xb9, 0x72, 0xfc, 0x3e, 0x34, 0xfb, 0x42, 0x32, 0xa1, 0x3c, 0xa7, 0x6, + 0xdc, 0xb5, 0x7a, 0xec, 0x3d, 0xae, 0x7, 0xbd, 0xc1, 0xc6, 0x7b, 0xf3, 0x36, 0x9, + ]; + assert_eq!(got_bob_pub.as_bytes()[..], expected_bob_pub[..]); + + let bob_shared = bob_priv.to_diffie_hellman(&got_alice_pub).unwrap(); + let alice_shared = alice_priv.to_diffie_hellman(&got_bob_pub).unwrap(); + assert_eq!(bob_shared.as_bytes()[..], alice_shared.as_bytes()[..]); + + let expected_shared = [ + 0x7, 0xff, 0xf4, 0x18, 0x1a, 0xc6, 0xcc, 0x95, 0xec, 0x1c, 0x16, 0xa9, 0x4a, 0xf, 0x74, + 0xd1, 0x2d, 0xa2, 0x32, 0xce, 0x40, 0xa7, 0x75, 0x52, 0x28, 0x1d, 0x28, 0x2b, 0xb6, + 0xc, 0xb, 0x56, 0xfd, 0x24, 0x64, 0xc3, 0x35, 0x54, 0x39, 0x36, 0x52, 0x1c, 0x24, 0x40, + 0x30, 0x85, 0xd5, 0x9a, 0x44, 0x9a, 0x50, 0x37, 0x51, 0x4a, 0x87, 0x9d, + ]; + + assert_eq!(bob_shared.as_bytes()[..], expected_shared[..]); + } + + #[test] + fn test_rfc_test_vectors_fixed() { + struct Test { + secret: [u8; 56], + point: [u8; 56], + expected: [u8; 56], + } + + let test_vectors = vec![ + Test { + secret: [ + 0x3d, 0x26, 0x2f, 0xdd, 0xf9, 0xec, 0x8e, 0x88, 0x49, 0x52, 0x66, 0xfe, 0xa1, + 0x9a, 0x34, 0xd2, 0x88, 0x82, 0xac, 0xef, 0x4, 0x51, 0x4, 0xd0, 0xd1, 0xaa, + 0xe1, 0x21, 0x70, 0xa, 0x77, 0x9c, 0x98, 0x4c, 0x24, 0xf8, 0xcd, 0xd7, 0x8f, + 0xbf, 0xf4, 0x49, 0x43, 0xeb, 0xa3, 0x68, 0xf5, 0x4b, 0x29, 0x25, 0x9a, 0x4f, + 0x1c, 0x60, 0xa, 0xd3, + ], + point: [ + 0x6, 0xfc, 0xe6, 0x40, 0xfa, 0x34, 0x87, 0xbf, 0xda, 0x5f, 0x6c, 0xf2, 0xd5, + 0x26, 0x3f, 0x8a, 0xad, 0x88, 0x33, 0x4c, 0xbd, 0x7, 0x43, 0x7f, 0x2, 0xf, 0x8, + 0xf9, 0x81, 0x4d, 0xc0, 0x31, 0xdd, 0xbd, 0xc3, 0x8c, 0x19, 0xc6, 0xda, 0x25, + 0x83, 0xfa, 0x54, 0x29, 0xdb, 0x94, 0xad, 0xa1, 0x8a, 0xa7, 0xa7, 0xfb, 0x4e, + 0xf8, 0xa0, 0x86, + ], + expected: [ + 0xce, 0x3e, 0x4f, 0xf9, 0x5a, 0x60, 0xdc, 0x66, 0x97, 0xda, 0x1d, 0xb1, 0xd8, + 0x5e, 0x6a, 0xfb, 0xdf, 0x79, 0xb5, 0xa, 0x24, 0x12, 0xd7, 0x54, 0x6d, 0x5f, + 0x23, 0x9f, 0xe1, 0x4f, 0xba, 0xad, 0xeb, 0x44, 0x5f, 0xc6, 0x6a, 0x1, 0xb0, + 0x77, 0x9d, 0x98, 0x22, 0x39, 0x61, 0x11, 0x1e, 0x21, 0x76, 0x62, 0x82, 0xf7, + 0x3d, 0xd9, 0x6b, 0x6f, + ], + }, + Test { + secret: [ + 0x20, 0x3d, 0x49, 0x44, 0x28, 0xb8, 0x39, 0x93, 0x52, 0x66, 0x5d, 0xdc, 0xa4, + 0x2f, 0x9d, 0xe8, 0xfe, 0xf6, 0x0, 0x90, 0x8e, 0xd, 0x46, 0x1c, 0xb0, 0x21, + 0xf8, 0xc5, 0x38, 0x34, 0x5d, 0xd7, 0x7c, 0x3e, 0x48, 0x6, 0xe2, 0x5f, 0x46, + 0xd3, 0x31, 0x5c, 0x44, 0xe0, 0xa5, 0xb4, 0x37, 0x12, 0x82, 0xdd, 0x2c, 0x8d, + 0x5b, 0xe3, 0x9, 0x5f, + ], + point: [ + 0xf, 0xbc, 0xc2, 0xf9, 0x93, 0xcd, 0x56, 0xd3, 0x30, 0x5b, 0xb, 0x7d, 0x9e, + 0x55, 0xd4, 0xc1, 0xa8, 0xfb, 0x5d, 0xbb, 0x52, 0xf8, 0xe9, 0xa1, 0xe9, 0xb6, + 0x20, 0x1b, 0x16, 0x5d, 0x1, 0x58, 0x94, 0xe5, 0x6c, 0x4d, 0x35, 0x70, 0xbe, + 0xe5, 0x2f, 0xe2, 0x5, 0xe2, 0x8a, 0x78, 0xb9, 0x1c, 0xdf, 0xbd, 0xe7, 0x1c, + 0xe8, 0xd1, 0x57, 0xdb, + ], + expected: [ + 0x88, 0x4a, 0x2, 0x57, 0x62, 0x39, 0xff, 0x7a, 0x2f, 0x2f, 0x63, 0xb2, 0xdb, + 0x6a, 0x9f, 0xf3, 0x70, 0x47, 0xac, 0x13, 0x56, 0x8e, 0x1e, 0x30, 0xfe, 0x63, + 0xc4, 0xa7, 0xad, 0x1b, 0x3e, 0xe3, 0xa5, 0x70, 0xd, 0xf3, 0x43, 0x21, 0xd6, + 0x20, 0x77, 0xe6, 0x36, 0x33, 0xc5, 0x75, 0xc1, 0xc9, 0x54, 0x51, 0x4e, 0x99, + 0xda, 0x7c, 0x17, 0x9d, + ], + }, + ]; + + for vector in test_vectors { + let public_key = PublicKey::from_bytes(&vector.point).unwrap(); + let secret = Secret::from_bytes(&vector.secret).unwrap(); + + let got = secret.as_diffie_hellman(&public_key).unwrap(); + + assert_eq!(got.as_bytes()[..], vector.expected[..]) + } + } + + // This function is needed for the second set of test vectors in RFC7748 + fn swap(secret: &mut [u8; 56], public_key: &mut [u8; 56], result: &[u8; 56]) { + // set point to be the secret + *public_key = *secret; + // set the secret to be the result + *secret = *result; + } + + #[test] + #[ignore] + fn test_rfc_test_vectors_iteration() { + let one_iter = [ + 0x3f, 0x48, 0x2c, 0x8a, 0x9f, 0x19, 0xb0, 0x1e, 0x6c, 0x46, 0xee, 0x97, 0x11, 0xd9, + 0xdc, 0x14, 0xfd, 0x4b, 0xf6, 0x7a, 0xf3, 0x7, 0x65, 0xc2, 0xae, 0x2b, 0x84, 0x6a, + 0x4d, 0x23, 0xa8, 0xcd, 0xd, 0xb8, 0x97, 0x8, 0x62, 0x39, 0x49, 0x2c, 0xaf, 0x35, 0xb, + 0x51, 0xf8, 0x33, 0x86, 0x8b, 0x9b, 0xc2, 0xb3, 0xbc, 0xa9, 0xcf, 0x41, 0x13, + ]; + let one_k_iter = [ + 0xaa, 0x3b, 0x47, 0x49, 0xd5, 0x5b, 0x9d, 0xaf, 0x1e, 0x5b, 0x0, 0x28, 0x88, 0x26, + 0xc4, 0x67, 0x27, 0x4c, 0xe3, 0xeb, 0xbd, 0xd5, 0xc1, 0x7b, 0x97, 0x5e, 0x9, 0xd4, + 0xaf, 0x6c, 0x67, 0xcf, 0x10, 0xd0, 0x87, 0x20, 0x2d, 0xb8, 0x82, 0x86, 0xe2, 0xb7, + 0x9f, 0xce, 0xea, 0x3e, 0xc3, 0x53, 0xef, 0x54, 0xfa, 0xa2, 0x6e, 0x21, 0x9f, 0x38, + ]; + let one_mil_iter = [ + 0x7, 0x7f, 0x45, 0x36, 0x81, 0xca, 0xca, 0x36, 0x93, 0x19, 0x84, 0x20, 0xbb, 0xe5, + 0x15, 0xca, 0xe0, 0x0, 0x24, 0x72, 0x51, 0x9b, 0x3e, 0x67, 0x66, 0x1a, 0x7e, 0x89, + 0xca, 0xb9, 0x46, 0x95, 0xc8, 0xf4, 0xbc, 0xd6, 0x6e, 0x61, 0xb9, 0xb9, 0xc9, 0x46, + 0xda, 0x8d, 0x52, 0x4d, 0xe3, 0xd6, 0x9b, 0xd9, 0xd9, 0xd6, 0x6b, 0x99, 0x7e, 0x37, + ]; + + let mut point = MontgomeryPoint::GENERATOR.0; + let mut scalar = MontgomeryPoint::GENERATOR.0; + let mut result = [0u8; 56]; + + // Iterate 1 time then check value on 1st iteration + for _ in 1..=1 { + result = x448(scalar, point).unwrap(); + swap(&mut scalar, &mut point, &result); + } + assert_eq!(&result[..], &one_iter[..]); + + // Iterate 999 times then check value on 1_000th iteration + for _ in 1..=999 { + result = x448(scalar, point).unwrap(); + swap(&mut scalar, &mut point, &result); + } + assert_eq!(&result[..], &one_k_iter[..]); + + // Iterate 999_000 times then check value on 1_000_000th iteration + for _ in 1..=999_000 { + result = x448(scalar, point).unwrap(); + swap(&mut scalar, &mut point, &result); + } + assert_eq!(&result[..], &one_mil_iter[..]); + } +}