// If the "u128-support" feature is enabled, this library can use
// more efficient arithmetic. Only available in the nightly compiler.
#![cfg_attr(feature = "u128-support", feature(i128_type))]
// `clippy` is a code linting tool for improving code quality by catching
// common mistakes or strange code patterns. If the `clippy` feature is
// provided, it is enabled and all compiler warnings are prohibited.
#![cfg_attr(feature = "clippy", deny(warnings))]
#![cfg_attr(feature = "clippy", feature(plugin))]
#![cfg_attr(feature = "clippy", plugin(clippy))]
#![cfg_attr(feature = "clippy", allow(inline_always))]
#![cfg_attr(feature = "clippy", allow(too_many_arguments))]
#![cfg_attr(feature = "clippy", allow(unreadable_literal))]
#![cfg_attr(feature = "clippy", allow(new_without_default_derive))]
// Force public structures to implement Debug
#![deny(missing_debug_implementations)]
extern crate rand;
extern crate byteorder;
#[cfg(test)]
pub mod tests;
pub mod bls12_381;
mod wnaf;
pub use self::wnaf::Wnaf;
use std::fmt;
use std::error::Error;
use std::io::{self, Read, Write};
/// An "engine" is a collection of types (fields, elliptic curve groups, etc.)
/// with well-defined relationships. In particular, the G1/G2 curve groups are
/// of prime order `r`, and are equipped with a bilinear pairing function.
pub trait Engine: Sized + 'static + Clone
{
/// This is the scalar field of the G1/G2 groups.
type Fr: PrimeField + SqrtField;
/// The projective representation of an element in G1.
type G1: CurveProjective<Engine=Self, Base=Self::Fq, Scalar=Self::Fr, Affine=Self::G1Affine> + From<Self::G1Affine>;
/// The affine representation of an element in G1.
type G1Affine: CurveAffine<Engine=Self, Base=Self::Fq, Scalar=Self::Fr, Projective=Self::G1, Pair=Self::G2Affine, PairingResult=Self::Fqk> + From<Self::G1>;
/// The projective representation of an element in G2.
type G2: CurveProjective<Engine=Self, Base=Self::Fqe, Scalar=Self::Fr, Affine=Self::G2Affine> + From<Self::G2Affine>;
/// The affine representation of an element in G2.
type G2Affine: CurveAffine<Engine=Self, Base=Self::Fqe, Scalar=Self::Fr, Projective=Self::G2, Pair=Self::G1Affine, PairingResult=Self::Fqk> + From<Self::G2>;
/// The base field that hosts G1.
type Fq: PrimeField + SqrtField;
/// The extension field that hosts G2.
type Fqe: SqrtField;
/// The extension field that hosts the target group of the pairing.
type Fqk: Field;
/// Perform a miller loop with some number of (G1, G2) pairs.
fn miller_loop<'a, I>(i: I) -> Self::Fqk
where I: IntoIterator<Item=&'a (
&'a <Self::G1Affine as CurveAffine>::Prepared,
&'a <Self::G2Affine as CurveAffine>::Prepared
)>;
/// Perform final exponentiation of the result of a miller loop.
fn final_exponentiation(&Self::Fqk) -> Option<Self::Fqk>;
/// Performs a complete pairing operation `(p, q)`.
fn pairing<G1, G2>(p: G1, q: G2) -> Self::Fqk
where G1: Into<Self::G1Affine>,
G2: Into<Self::G2Affine>
{
Self::final_exponentiation(&Self::miller_loop(
[(
&(p.into().prepare()),
&(q.into().prepare())
)].into_iter()
)).unwrap()
}
}
/// Projective representation of an elliptic curve point guaranteed to be
/// in the correct prime order subgroup.
pub trait CurveProjective: PartialEq +
Eq +
Sized +
Copy +
Clone +
Send +
Sync +
fmt::Debug +
fmt::Display +
rand::Rand +
'static
{
type Engine: Engine<Fr=Self::Scalar>;
type Scalar: PrimeField + SqrtField;
type Base: SqrtField;
type Affine: CurveAffine<Projective=Self, Scalar=Self::Scalar>;
/// Returns the additive identity.
fn zero() -> Self;
/// Returns a fixed generator of unknown exponent.
fn one() -> Self;
/// Determines if this point is the point at infinity.
fn is_zero(&self) -> bool;
/// Normalizes a slice of projective elements so that
/// conversion to affine is cheap.
fn batch_normalization(v: &mut [Self]);
/// Checks if the point is already "normalized" so that
/// cheap affine conversion is possible.
fn is_normalized(&self) -> bool;
/// Doubles this element.
fn double(&mut self);
/// Adds another element to this element.
fn add_assign(&mut self, other: &Self);
/// Subtracts another element from this element.
fn sub_assign(&mut self, other: &Self) {
let mut tmp = *other;
tmp.negate();
self.add_assign(&tmp);
}
/// Adds an affine element to this element.
fn add_assign_mixed(&mut self, other: &Self::Affine);
/// Negates this element.
fn negate(&mut self);
/// Performs scalar multiplication of this element.
fn mul_assign<S: Into<<Self::Scalar as PrimeField>::Repr>>(&mut self, other: S);
/// Converts this element into its affine representation.
fn into_affine(&self) -> Self::Affine;
/// Recommends a wNAF window table size given a scalar. Always returns a number
/// between 2 and 22, inclusive.
fn recommended_wnaf_for_scalar(scalar: <Self::Scalar as PrimeField>::Repr) -> usize;
/// Recommends a wNAF window size given the number of scalars you intend to multiply
/// a base by. Always returns a number between 2 and 22, inclusive.
fn recommended_wnaf_for_num_scalars(num_scalars: usize) -> usize;
}
/// Affine representation of an elliptic curve point guaranteed to be
/// in the correct prime order subgroup.
pub trait CurveAffine: Copy +
Clone +
Sized +
Send +
Sync +
fmt::Debug +
fmt::Display +
PartialEq +
Eq +
'static
{
type Engine: Engine<Fr=Self::Scalar>;
type Scalar: PrimeField + SqrtField;
type Base: SqrtField;
type Projective: CurveProjective<Affine=Self, Scalar=Self::Scalar>;
type Prepared: Clone + Send + Sync + 'static;
type Uncompressed: EncodedPoint<Affine=Self>;
type Compressed: EncodedPoint<Affine=Self>;
type Pair: CurveAffine<Pair=Self>;
type PairingResult: Field;
/// Returns the additive identity.
fn zero() -> Self;
/// Returns a fixed generator of unknown exponent.
fn one() -> Self;
/// Determines if this point represents the point at infinity; the
/// additive identity.
fn is_zero(&self) -> bool;
/// Negates this element.
fn negate(&mut self);
/// Performs scalar multiplication of this element with mixed addition.
fn mul<S: Into<<Self::Scalar as PrimeField>::Repr>>(&self, other: S) -> Self::Projective;
/// Prepares this element for pairing purposes.
fn prepare(&self) -> Self::Prepared;
/// Perform a pairing
fn pairing_with(&self, other: &Self::Pair) -> Self::PairingResult;
/// Converts this element into its affine representation.
fn into_projective(&self) -> Self::Projective;
/// Converts this element into its compressed encoding, so long as it's not
/// the point at infinity.
fn into_compressed(&self) -> Self::Compressed {
<Self::Compressed as EncodedPoint>::from_affine(*self)
}
/// Converts this element into its uncompressed encoding, so long as it's not
/// the point at infinity.
fn into_uncompressed(&self) -> Self::Uncompressed {
<Self::Uncompressed as EncodedPoint>::from_affine(*self)
}
}
/// An encoded elliptic curve point, which should essentially wrap a `[u8; N]`.
pub trait EncodedPoint: Sized +
Send +
Sync +
AsRef<[u8]> +
AsMut<[u8]> +
Clone +
Copy +
'static
{
type Affine: CurveAffine;
/// Creates an empty representation.
fn empty() -> Self;
/// Returns the number of bytes consumed by this representation.
fn size() -> usize;
/// Converts an `EncodedPoint` into a `CurveAffine` element,
/// if the encoding represents a valid element.
fn into_affine(&self) -> Result<Self::Affine, GroupDecodingError>;
/// Converts an `EncodedPoint` into a `CurveAffine` element,
/// without guaranteeing that the encoding represents a valid
/// element. This is useful when the caller knows the encoding is
/// valid already.
///
/// If the encoding is invalid, this can break API invariants,
/// so caution is strongly encouraged.
fn into_affine_unchecked(&self) -> Result<Self::Affine, GroupDecodingError>;
/// Creates an `EncodedPoint` from an affine point, as long as the
/// point is not the point at infinity.
fn from_affine(affine: Self::Affine) -> Self;
}
/// This trait represents an element of a field.
pub trait Field: Sized +
Eq +
Copy +
Clone +
Send +
Sync +
fmt::Debug +
fmt::Display +
'static +
rand::Rand
{
/// Returns the zero element of the field, the additive identity.
fn zero() -> Self;
/// Returns the one element of the field, the multiplicative identity.
fn one() -> Self;
/// Returns true iff this element is zero.
fn is_zero(&self) -> bool;
/// Squares this element.
fn square(&mut self);
/// Doubles this element.
fn double(&mut self);
/// Negates this element.
fn negate(&mut self);
/// Adds another element to this element.
fn add_assign(&mut self, other: &Self);
/// Subtracts another element from this element.
fn sub_assign(&mut self, other: &Self);
/// Multiplies another element by this element.
fn mul_assign(&mut self, other: &Self);
/// Computes the multiplicative inverse of this element, if nonzero.
fn inverse(&self) -> Option<Self>;
/// Exponentiates this element by a power of the base prime modulus via
/// the Frobenius automorphism.
fn frobenius_map(&mut self, power: usize);
/// Exponentiates this element by a number represented with `u64` limbs,
/// least significant digit first.
fn pow<S: AsRef<[u64]>>(&self, exp: S) -> Self
{
let mut res = Self::one();
let mut found_one = false;
for i in BitIterator::new(exp) {
if found_one {
res.square();
} else {
found_one = i;
}
if i {
res.mul_assign(self);
}
}
res
}
}
/// This trait represents an element of a field that has a square root operation described for it.
pub trait SqrtField: Field
{
/// Returns the Legendre symbol of the field element.
fn legendre(&self) -> LegendreSymbol;
/// Returns the square root of the field element, if it is
/// quadratic residue.
fn sqrt(&self) -> Option<Self>;
}
/// This trait represents a wrapper around a biginteger which can encode any element of a particular
/// prime field. It is a smart wrapper around a sequence of `u64` limbs, least-significant digit
/// first.
pub trait PrimeFieldRepr: Sized +
Copy +
Clone +
Eq +
Ord +
Send +
Sync +
Default +
fmt::Debug +
fmt::Display +
'static +
rand::Rand +
AsRef<[u64]> +
AsMut<[u64]> +
From<u64>
{
/// Subtract another represetation from this one, returning the borrow bit.
fn sub_noborrow(&mut self, other: &Self) -> bool;
/// Add another representation to this one, returning the carry bit.
fn add_nocarry(&mut self, other: &Self) -> bool;
/// Compute the number of bits needed to encode this number. Always a
/// multiple of 64.
fn num_bits(&self) -> u32;
/// Returns true iff this number is zero.
fn is_zero(&self) -> bool;
/// Returns true iff this number is odd.
fn is_odd(&self) -> bool;
/// Returns true iff this number is even.
fn is_even(&self) -> bool;
/// Performs a rightwise bitshift of this number, effectively dividing
/// it by 2.
fn div2(&mut self);
/// Performs a rightwise bitshift of this number by some amount.
fn divn(&mut self, amt: u32);
/// Performs a leftwise bitshift of this number, effectively multiplying
/// it by 2. Overflow is ignored.
fn mul2(&mut self);
/// Performs a leftwise bitshift of this number by some amount.
fn muln(&mut self, amt: u32);
/// Writes this `PrimeFieldRepr` as a big endian integer. Always writes
/// `(num_bits` / 8) bytes.
fn write_be<W: Write>(&self, mut writer: W) -> io::Result<()> {
use byteorder::{WriteBytesExt, BigEndian};
for digit in self.as_ref().iter().rev() {
writer.write_u64::<BigEndian>(*digit)?;
}
Ok(())
}
/// Reads a big endian integer occupying (`num_bits` / 8) bytes into this
/// representation.
fn read_be<R: Read>(&mut self, mut reader: R) -> io::Result<()> {
use byteorder::{ReadBytesExt, BigEndian};
for digit in self.as_mut().iter_mut().rev() {
*digit = reader.read_u64::<BigEndian>()?;
}
Ok(())
}
}
#[derive(Debug, PartialEq)]
pub enum LegendreSymbol {
Zero = 0,
QuadraticResidue = 1,
QuadraticNonResidue = -1
}
/// An error that may occur when trying to interpret a `PrimeFieldRepr` as a
/// `PrimeField` element.
#[derive(Debug)]
pub enum PrimeFieldDecodingError {
/// The encoded value is not in the field
NotInField(String)
}
impl Error for PrimeFieldDecodingError {
fn description(&self) -> &str {
match *self {
PrimeFieldDecodingError::NotInField(..) => "not an element of the field"
}
}
}
impl fmt::Display for PrimeFieldDecodingError {
fn fmt(&self, f: &mut fmt::Formatter) -> Result<(), fmt::Error> {
match *self {
PrimeFieldDecodingError::NotInField(ref repr) => {
write!(f, "{} is not an element of the field", repr)
}
}
}
}
/// An error that may occur when trying to decode an `EncodedPoint`.
#[derive(Debug)]
pub enum GroupDecodingError {
/// The coordinate(s) do not lie on the curve.
NotOnCurve,
/// The element is not part of the r-order subgroup.
NotInSubgroup,
/// One of the coordinates could not be decoded
CoordinateDecodingError(&'static str, PrimeFieldDecodingError),
/// The compression mode of the encoded element was not as expected
UnexpectedCompressionMode,
/// The encoding contained bits that should not have been set
UnexpectedInformation
}
impl Error for GroupDecodingError {
fn description(&self) -> &str {
match *self {
GroupDecodingError::NotOnCurve => "coordinate(s) do not lie on the curve",
GroupDecodingError::NotInSubgroup => "the element is not part of an r-order subgroup",
GroupDecodingError::CoordinateDecodingError(..) => "coordinate(s) could not be decoded",
GroupDecodingError::UnexpectedCompressionMode => "encoding has unexpected compression mode",
GroupDecodingError::UnexpectedInformation => "encoding has unexpected information"
}
}
}
impl fmt::Display for GroupDecodingError {
fn fmt(&self, f: &mut fmt::Formatter) -> Result<(), fmt::Error> {
match *self {
GroupDecodingError::CoordinateDecodingError(description, ref err) => {
write!(f, "{} decoding error: {}", description, err)
},
_ => {
write!(f, "{}", self.description())
}
}
}
}
/// This represents an element of a prime field.
pub trait PrimeField: Field
{
/// The prime field can be converted back and forth into this biginteger
/// representation.
type Repr: PrimeFieldRepr + From<Self>;
/// Interpret a string of numbers as a (congruent) prime field element.
/// Does not accept unnecessary leading zeroes or a blank string.
fn from_str(s: &str) -> Option<Self> {
if s.is_empty() {
return None;
}
if s == "0" {
return Some(Self::zero());
}
let mut res = Self::zero();
let ten = Self::from_repr(Self::Repr::from(10)).unwrap();
let mut first_digit = true;
for c in s.chars() {
match c.to_digit(10) {
Some(c) => {
if first_digit {
if c == 0 {
return None;
}
first_digit = false;
}
res.mul_assign(&ten);
res.add_assign(&Self::from_repr(Self::Repr::from(u64::from(c))).unwrap());
},
None => {
return None;
}
}
}
Some(res)
}
/// Convert this prime field element into a biginteger representation.
fn from_repr(Self::Repr) -> Result<Self, PrimeFieldDecodingError>;
/// Convert a biginteger representation into a prime field element, if
/// the number is an element of the field.
fn into_repr(&self) -> Self::Repr;
/// Returns the field characteristic; the modulus.
fn char() -> Self::Repr;
/// How many bits are needed to represent an element of this field.
const NUM_BITS: u32;
/// How many bits of information can be reliably stored in the field element.
const CAPACITY: u32;
/// Returns the multiplicative generator of `char()` - 1 order. This element
/// must also be quadratic nonresidue.
fn multiplicative_generator() -> Self;
/// 2^s * t = `char()` - 1 with t odd.
const S: u32;
/// Returns the 2^s root of unity computed by exponentiating the `multiplicative_generator()`
/// by t.
fn root_of_unity() -> Self;
}
#[derive(Debug)]
pub struct BitIterator<E> {
t: E,
n: usize
}
impl<E: AsRef<[u64]>> BitIterator<E> {
pub fn new(t: E) -> Self {
let n = t.as_ref().len() * 64;
BitIterator {
t: t,
n: n
}
}
}
impl<E: AsRef<[u64]>> Iterator for BitIterator<E> {
type Item = bool;
fn next(&mut self) -> Option<bool> {
if self.n == 0 {
None
} else {
self.n -= 1;
let part = self.n / 64;
let bit = self.n - (64 * part);
Some(self.t.as_ref()[part] & (1 << bit) > 0)
}
}
}
#[test]
fn test_bit_iterator() {
let mut a = BitIterator::new([0xa953d79b83f6ab59, 0x6dea2059e200bd39]);
let expected = "01101101111010100010000001011001111000100000000010111101001110011010100101010011110101111001101110000011111101101010101101011001";
for e in expected.chars() {
assert!(a.next().unwrap() == (e == '1'));
}
assert!(a.next().is_none());
let expected = "1010010101111110101010000101101011101000011101110101001000011001100100100011011010001011011011010001011011101100110100111011010010110001000011110100110001100110011101101000101100011100100100100100001010011101010111110011101011000011101000111011011101011001";
let mut a = BitIterator::new([0x429d5f3ac3a3b759, 0xb10f4c66768b1c92, 0x92368b6d16ecd3b4, 0xa57ea85ae8775219]);
for e in expected.chars() {
assert!(a.next().unwrap() == (e == '1'));
}
assert!(a.next().is_none());
}
use self::arith::*;
#[cfg(feature = "u128-support")]
mod arith {
/// Calculate a - b - borrow, returning the result and modifying
/// the borrow value.
#[inline(always)]
pub(crate) fn sbb(a: u64, b: u64, borrow: &mut u64) -> u64 {
let tmp = (1u128 << 64) + u128::from(a) - u128::from(b) - u128::from(*borrow);
*borrow = if tmp >> 64 == 0 { 1 } else { 0 };
tmp as u64
}
/// Calculate a + b + carry, returning the sum and modifying the
/// carry value.
#[inline(always)]
pub(crate) fn adc(a: u64, b: u64, carry: &mut u64) -> u64 {
let tmp = u128::from(a) + u128::from(b) + u128::from(*carry);
*carry = (tmp >> 64) as u64;
tmp as u64
}
/// Calculate a + (b * c) + carry, returning the least significant digit
/// and setting carry to the most significant digit.
#[inline(always)]
pub(crate) fn mac_with_carry(a: u64, b: u64, c: u64, carry: &mut u64) -> u64 {
let tmp = (u128::from(a)) + u128::from(b) * u128::from(c) + u128::from(*carry);
*carry = (tmp >> 64) as u64;
tmp as u64
}
}
#[cfg(not(feature = "u128-support"))]
mod arith {
#[inline(always)]
fn split_u64(i: u64) -> (u64, u64) {
(i >> 32, i & 0xFFFFFFFF)
}
#[inline(always)]
fn combine_u64(hi: u64, lo: u64) -> u64 {
(hi << 32) | lo
}
#[inline(always)]
pub(crate) fn sbb(a: u64, b: u64, borrow: &mut u64) -> u64 {
let (a_hi, a_lo) = split_u64(a);
let (b_hi, b_lo) = split_u64(b);
let (b, r0) = split_u64((1 << 32) + a_lo - b_lo - *borrow);
let (b, r1) = split_u64((1 << 32) + a_hi - b_hi - ((b == 0) as u64));
*borrow = (b == 0) as u64;
combine_u64(r1, r0)
}
#[inline(always)]
pub(crate) fn adc(a: u64, b: u64, carry: &mut u64) -> u64 {
let (a_hi, a_lo) = split_u64(a);
let (b_hi, b_lo) = split_u64(b);
let (carry_hi, carry_lo) = split_u64(*carry);
let (t, r0) = split_u64(a_lo + b_lo + carry_lo);
let (t, r1) = split_u64(t + a_hi + b_hi + carry_hi);
*carry = t;
combine_u64(r1, r0)
}
#[inline(always)]
pub(crate) fn mac_with_carry(a: u64, b: u64, c: u64, carry: &mut u64) -> u64 {
/*
[ b_hi | b_lo ]
[ c_hi | c_lo ] *
-------------------------------------------
[ b_lo * c_lo ] <-- w
[ b_hi * c_lo ] <-- x
[ b_lo * c_hi ] <-- y
[ b_hi * c_lo ] <-- z
[ a_hi | a_lo ]
[ C_hi | C_lo ]
*/
let (a_hi, a_lo) = split_u64(a);
let (b_hi, b_lo) = split_u64(b);
let (c_hi, c_lo) = split_u64(c);
let (carry_hi, carry_lo) = split_u64(*carry);
let (w_hi, w_lo) = split_u64(b_lo * c_lo);
let (x_hi, x_lo) = split_u64(b_hi * c_lo);
let (y_hi, y_lo) = split_u64(b_lo * c_hi);
let (z_hi, z_lo) = split_u64(b_hi * c_hi);
let (t, r0) = split_u64(w_lo + a_lo + carry_lo);
let (t, r1) = split_u64(t + w_hi + x_lo + y_lo + a_hi + carry_hi);
let (t, r2) = split_u64(t + x_hi + y_hi + z_lo);
let (_, r3) = split_u64(t + z_hi);
*carry = combine_u64(r3, r2);
combine_u64(r1, r0)
}
}