diff --git a/src/edwards.rs b/src/edwards.rs index 998af8d..40b2b5e 100644 --- a/src/edwards.rs +++ b/src/edwards.rs @@ -117,11 +117,15 @@ use backend::serial::curve_models::CompletedPoint; use backend::serial::curve_models::ProjectiveNielsPoint; use backend::serial::curve_models::ProjectivePoint; -use window::LookupTable; +use window::LookupTableRadix16; +use window::LookupTableRadix64; +use window::LookupTableRadix128; +use window::LookupTableRadix256; #[allow(unused_imports)] use prelude::*; +use traits::BasepointTable; use traits::ValidityCheck; use traits::{Identity, IsIdentity}; @@ -743,56 +747,116 @@ impl EdwardsPoint { } } +macro_rules! impl_basepoint_table { + (Name = $name:ident, LookupTable = $table:ident, Point = $point:ty, Radix = $radix:expr, Additions = $adds:expr) => { + /// A precomputed table of multiples of a basepoint, for accelerating /// fixed-base scalar multiplication. One table, for the Ed25519 /// basepoint, is provided in the `constants` module. /// -/// The basepoint tables are reasonably large (30KB), so they should -/// probably be boxed. +/// The basepoint tables are reasonably large, so they should probably be boxed. +/// +/// The sizes for the tables and the number of additions required for one scalar +/// multiplication are as follows: +/// +/// * [`EdwardsBasepointTableRadix16`]: 30KB, 64A +/// (this is the default size, and is used for [`ED25519_BASEPOINT_TABLE`]) +/// * [`EdwardsBasepointTableRadix64`]: 120KB, 43A +/// * [`EdwardsBasepointTableRadix128`]: 240KB, 37A +/// * [`EdwardsBasepointTableRadix256`]: 480KB, 33A +/// +/// # Why 33 additions for radix-256? +/// +/// Normally, the radix-256 tables would allow for only 32 additions per scalar +/// multiplication. However, due to the fact that standardised definitions of +/// legacy protocols—such as x25519—require allowing unreduced 255-bit scalar +/// invariants, when converting such an unreduced scalar's representation to +/// radix-\\(2^{8}\\), we cannot guarantee the carry bit will fit in the last +/// coefficient (the coefficients are `i8`s). When, \\(w\\), the power-of-2 of +/// the radix, is \\(w < 8\\), we can fold the final carry onto the last +/// coefficient, \\(d\\), because \\(d < 2^{w/2}\\), so +/// $$ +/// d + carry \cdot 2^{w} = d + 1 \cdot 2^{w} < 2^{w+1} < 2^{8} +/// $$ +/// When \\(w = 8\\), we can't fit \\(carry \cdot 2^{w}\\) into an `i8`, so we +/// add the carry bit onto an additional coefficient. #[derive(Clone)] -pub struct EdwardsBasepointTable(pub(crate) [LookupTable; 32]); +pub struct $name(pub(crate) [$table; 32]); -impl EdwardsBasepointTable { - /// The computation uses Pippeneger's algorithm, as described on - /// page 13 of the Ed25519 paper. Write the scalar \\(a\\) in radix \\(16\\) with - /// coefficients in \\([-8,8)\\), i.e., +impl BasepointTable for $name { + type Point = $point; + + /// Create a table of precomputed multiples of `basepoint`. + fn create(basepoint: &$point) -> $name { + // XXX use init_with + let mut table = $name([$table::default(); 32]); + let mut P = *basepoint; + for i in 0..32 { + // P = (2w)^i * B + table.0[i] = $table::from(&P); + P = P.mul_by_pow_2($radix + $radix); + } + table + } + + /// Get the basepoint for this table as an `EdwardsPoint`. + fn basepoint(&self) -> $point { + // self.0[0].select(1) = 1*(16^2)^0*B + // but as an `AffineNielsPoint`, so add identity to convert to extended. + (&<$point>::identity() + &self.0[0].select(1)).to_extended() + } + + /// The computation uses Pippeneger's algorithm, as described for the + /// specific case of radix-16 on page 13 of the Ed25519 paper. + /// + /// # Piggenger's Algorithm Generalised + /// + /// Write the scalar \\(a\\) in radix-\\(w\\), where \\(w\\) is a power of + /// 2, with coefficients in \\([\frac{-w}{2},\frac{w}{2})\\), i.e., /// $$ - /// a = a\_0 + a\_1 16\^1 + \cdots + a\_{63} 16\^{63}, + /// a = a\_0 + a\_1 w\^1 + \cdots + a\_{x} w\^{x}, /// $$ - /// with \\(-8 \leq a_i < 8\\), \\(-8 \leq a\_{63} \leq 8\\). Then + /// with /// $$ - /// a B = a\_0 B + a\_1 16\^1 B + \cdots + a\_{63} 16\^{63} B. + /// \frac{-w}{2} \leq a_i < \frac{w}{2}, \cdots, \frac{-w}{2} \leq a\_{x} \leq \frac{w}{2} + /// $$ + /// and the number of additions, \\(x\\), is given by \\(x = \lceil \frac{256}{w} \rceil\\). + /// Then + /// $$ + /// a B = a\_0 B + a\_1 w\^1 B + \cdots + a\_{x-1} w\^{x-1} B. /// $$ /// Grouping even and odd coefficients gives /// $$ /// \begin{aligned} - /// a B = \quad a\_0 16\^0 B +& a\_2 16\^2 B + \cdots + a\_{62} 16\^{62} B \\\\ - /// + a\_1 16\^1 B +& a\_3 16\^3 B + \cdots + a\_{63} 16\^{63} B \\\\ - /// = \quad(a\_0 16\^0 B +& a\_2 16\^2 B + \cdots + a\_{62} 16\^{62} B) \\\\ - /// + 16(a\_1 16\^0 B +& a\_3 16\^2 B + \cdots + a\_{63} 16\^{62} B). \\\\ + /// a B = \quad a\_0 w\^0 B +& a\_2 w\^2 B + \cdots + a\_{x-2} w\^{x-2} B \\\\ + /// + a\_1 w\^1 B +& a\_3 w\^3 B + \cdots + a\_{x-1} w\^{x-1} B \\\\ + /// = \quad(a\_0 w\^0 B +& a\_2 w\^2 B + \cdots + a\_{x-2} w\^{x-2} B) \\\\ + /// + w(a\_1 w\^0 B +& a\_3 w\^2 B + \cdots + a\_{x-1} w\^{x-2} B). \\\\ /// \end{aligned} /// $$ /// For each \\(i = 0 \ldots 31\\), we create a lookup table of /// $$ - /// [16\^{2i} B, \ldots, 8\cdot16\^{2i} B], + /// [w\^{2i} B, \ldots, \frac{w}{2}\cdotw\^{2i} B], /// $$ - /// and use it to select \\( x \cdot 16\^{2i} \cdot B \\) in constant time. + /// and use it to select \\( y \cdot w\^{2i} \cdot B \\) in constant time. /// - /// The radix-\\(16\\) representation requires that the scalar is bounded + /// The radix-\\(w\\) representation requires that the scalar is bounded /// by \\(2\^{255}\\), which is always the case. - fn basepoint_mul(&self, scalar: &Scalar) -> EdwardsPoint { - let a = scalar.to_radix_16(); + /// + /// The above algorithm is trivially generalised to other powers-of-2 radices. + fn basepoint_mul(&self, scalar: &Scalar) -> $point { + let a = scalar.to_radix_2w($radix); let tables = &self.0; - let mut P = EdwardsPoint::identity(); + let mut P = <$point>::identity(); - for i in (0..64).filter(|x| x % 2 == 1) { + for i in (0..$adds).filter(|x| x % 2 == 1) { P = (&P + &tables[i/2].select(a[i])).to_extended(); } - P = P.mul_by_pow_2(4); + P = P.mul_by_pow_2($radix); - for i in (0..64).filter(|x| x % 2 == 0) { + for i in (0..$adds).filter(|x| x % 2 == 0) { P = (&P + &tables[i/2].select(a[i])).to_extended(); } @@ -800,49 +864,53 @@ impl EdwardsBasepointTable { } } -impl<'a, 'b> Mul<&'b Scalar> for &'a EdwardsBasepointTable { - type Output = EdwardsPoint; +impl<'a, 'b> Mul<&'b Scalar> for &'a $name { + type Output = $point; /// Construct an `EdwardsPoint` from a `Scalar` \\(a\\) by /// computing the multiple \\(aB\\) of this basepoint \\(B\\). - fn mul(self, scalar: &'b Scalar) -> EdwardsPoint { + fn mul(self, scalar: &'b Scalar) -> $point { // delegate to a private function so that its documentation appears in internal docs self.basepoint_mul(scalar) } } -impl<'a, 'b> Mul<&'a EdwardsBasepointTable> for &'b Scalar { - type Output = EdwardsPoint; +impl<'a, 'b> Mul<&'a $name> for &'b Scalar { + type Output = $point; /// Construct an `EdwardsPoint` from a `Scalar` \\(a\\) by /// computing the multiple \\(aB\\) of this basepoint \\(B\\). - fn mul(self, basepoint_table: &'a EdwardsBasepointTable) -> EdwardsPoint { + fn mul(self, basepoint_table: &'a $name) -> $point { basepoint_table * self } } -impl EdwardsBasepointTable { - /// Create a table of precomputed multiples of `basepoint`. - pub fn create(basepoint: &EdwardsPoint) -> EdwardsBasepointTable { - // XXX use init_with - let mut table = EdwardsBasepointTable([LookupTable::default(); 32]); - let mut P = *basepoint; +impl Debug for $name { + fn fmt(&self, f: &mut ::core::fmt::Formatter) -> ::core::fmt::Result { + write!(f, "{:?}([\n", stringify!($name))?; for i in 0..32 { - // P = (16^2)^i * B - table.0[i] = LookupTable::from(&P); - P = P.mul_by_pow_2(8); + write!(f, "\t{:?},\n", &self.0[i])?; } - table - } - - /// Get the basepoint for this table as an `EdwardsPoint`. - pub fn basepoint(&self) -> EdwardsPoint { - // self.0[0].select(1) = 1*(16^2)^0*B - // but as an `AffineNielsPoint`, so add identity to convert to extended. - (&EdwardsPoint::identity() + &self.0[0].select(1)).to_extended() + write!(f, "])") } } +}} // End macro_rules! impl_basepoint_table + +// The number of additions required is ceil(256/w) where w is the radix representation. +impl_basepoint_table! {Name = EdwardsBasepointTable, LookupTable = LookupTableRadix16, Point = EdwardsPoint, Radix = 4, Additions = 64} +impl_basepoint_table! {Name = EdwardsBasepointTableRadix64, LookupTable = LookupTableRadix64, Point = EdwardsPoint, Radix = 6, Additions = 43} +impl_basepoint_table! {Name = EdwardsBasepointTableRadix128, LookupTable = LookupTableRadix128, Point = EdwardsPoint, Radix = 7, Additions = 37} +impl_basepoint_table! {Name = EdwardsBasepointTableRadix256, LookupTable = LookupTableRadix256, Point = EdwardsPoint, Radix = 8, Additions = 33} + +/// A type-alias for [`EdwardsBasepointTable`] because the latter is +/// used as a constructor in the `constants` module. +// +// Same as for `LookupTableRadix16`, we have to define `EdwardsBasepointTable` +// first, because it's used as a constructor, and then provide a type alias for +// it. +pub type EdwardsBasepointTableRadix16 = EdwardsBasepointTable; + impl EdwardsPoint { /// Multiply by the cofactor: return \\([8]P\\). pub fn mul_by_cofactor(&self) -> EdwardsPoint { @@ -930,16 +998,6 @@ impl Debug for EdwardsPoint { } } -impl Debug for EdwardsBasepointTable { - fn fmt(&self, f: &mut ::core::fmt::Formatter) -> ::core::fmt::Result { - write!(f, "EdwardsBasepointTable([\n")?; - for i in 0..32 { - write!(f, "\t{:?},\n", &self.0[i])?; - } - write!(f, "])") - } -} - // ------------------------------------------------------------------------ // Tests // ------------------------------------------------------------------------ @@ -1148,6 +1206,29 @@ mod test { assert_eq!(bp2.compress(), BASE2_CMPRSSD); } + /// Test that all the basepoint table types compute the same results. + #[test] + fn basepoint_tables() { + let P = &constants::ED25519_BASEPOINT_POINT; + let a = A_SCALAR; + + let table_radix16 = EdwardsBasepointTableRadix16::create(&P); + let table_radix64 = EdwardsBasepointTableRadix64::create(&P); + let table_radix128 = EdwardsBasepointTableRadix128::create(&P); + let table_radix256 = EdwardsBasepointTableRadix256::create(&P); + + let aP = (&constants::ED25519_BASEPOINT_TABLE * &a).compress(); + let aP16 = (&table_radix16 * &a).compress(); + let aP64 = (&table_radix64 * &a).compress(); + let aP128 = (&table_radix128 * &a).compress(); + let aP256 = (&table_radix256 * &a).compress(); + + assert_eq!(aP, aP16); + assert_eq!(aP16, aP64); + assert_eq!(aP64, aP128); + assert_eq!(aP128, aP256); + } + /// Check that converting to projective and then back to extended round-trips. #[test] fn basepoint_projective_extended_round_trip() { diff --git a/src/ristretto.rs b/src/ristretto.rs index c4b6170..380df18 100644 --- a/src/ristretto.rs +++ b/src/ristretto.rs @@ -185,6 +185,7 @@ use prelude::*; use scalar::Scalar; +use traits::BasepointTable; use traits::Identity; #[cfg(any(feature = "alloc", feature = "std"))] use traits::{MultiscalarMul, VartimeMultiscalarMul, VartimePrecomputedMultiscalarMul}; diff --git a/src/scalar.rs b/src/scalar.rs index d365d25..3e70367 100644 --- a/src/scalar.rs +++ b/src/scalar.rs @@ -989,10 +989,11 @@ impl Scalar { /// Returns a size hint indicating how many entries of the return /// value of `to_radix_2w` are nonzero. pub(crate) fn to_radix_2w_size_hint(w: usize) -> usize { - debug_assert!(w >= 6); + debug_assert!(w == 4 || w >= 6); debug_assert!(w <= 8); let digits_count = match w { + 4 => (256 + w - 1)/w as usize, 6 => (256 + w - 1)/w as usize, 7 => (256 + w - 1)/w as usize, // See comment in to_radix_2w on handling the terminal carry. @@ -1000,7 +1001,7 @@ impl Scalar { _ => panic!("invalid radix parameter"), }; - debug_assert!(digits_count <= 43); + debug_assert!(digits_count <= 64); digits_count } @@ -1022,10 +1023,14 @@ impl Scalar { /// $$ /// with \\(-2\^w/2 \leq a_i < 2\^w/2\\) for \\(0 \leq i < (n-1)\\) and \\(-2\^w/2 \leq a_{n-1} \leq 2\^w/2\\). /// - pub(crate) fn to_radix_2w(&self, w: usize) -> [i8; 43] { - debug_assert!(w >= 6); + pub(crate) fn to_radix_2w(&self, w: usize) -> [i8; 64] { + debug_assert!(w == 4 || w >= 6); debug_assert!(w <= 8); + if w == 4 { + return self.to_radix_16(); + } + use byteorder::{ByteOrder, LittleEndian}; // Scalar formatted as four `u64`s with carry bit packed into the highest bit. @@ -1036,7 +1041,7 @@ impl Scalar { let window_mask: u64 = radix - 1; let mut carry = 0u64; - let mut digits = [0i8; 43]; + let mut digits = [0i8; 64]; let digits_count = (256 + w - 1)/w as usize; for i in 0..digits_count { // Construct a buffer of bits of the scalar, starting at `bit_offset`.