Implement larger sizes of basepoint tables.

This implements a macro for implementing the BasepointTable trait, and
uses the macro to create basepoint table types. The default table
still uses radix-16 representation and is ~30KB in size.  The new
table types, and their memory usage and additions required per
basepoint multiplication are:

 * `EdwardsBasepointTableRadix64`: ~120KB, 43 additions
 * `EdwardsBasepointTableRadix128`: ~240KB, 37 additions
 * `EdwardsBasepointTableRadix256`: ~480KB, 32 additions

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
+/// The basepoint tables are reasonably large, so they should probably be boxed.
-/// 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<AffineNielsPoint>; 32]);
+pub struct $name(pub(crate) [$table<AffineNielsPoint>; 32]);
 
-impl EdwardsBasepointTable {
+impl BasepointTable for $name {
-    /// The computation uses Pippeneger's algorithm, as described on
+    type Point = $point;
-    /// page 13 of the Ed25519 paper.  Write the scalar \\(a\\) in radix \\(16\\) with
+
-    /// coefficients in \\([-8,8)\\), i.e.,
+    /// Create a table of precomputed multiples of `basepoint`.
+    fn create(basepoint: &$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
     /// $$
     /// \begin{aligned}
-    ///     a B = \quad a\_0 16\^0 B +& a\_2 16\^2 B + \cdots + a\_{62} 16\^{62} B    \\\\
+    ///     a B = \quad a\_0 w\^0 B +& a\_2 w\^2 B + \cdots + a\_{x-2} w\^{x-2} B    \\\\
-    ///               + a\_1 16\^1 B +& a\_3 16\^3 B + \cdots + a\_{63} 16\^{63} B    \\\\
+    ///               + a\_1 w\^1 B +& a\_3 w\^3 B + \cdots + a\_{x-1} w\^{x-1} B    \\\\
-    ///         = \quad(a\_0 16\^0 B +& a\_2 16\^2 B + \cdots + a\_{62} 16\^{62} B)   \\\\
+    ///         = \quad(a\_0 w\^0 B +& a\_2 w\^2 B + \cdots + a\_{x-2} w\^{x-2} B)   \\\\
-    ///            + 16(a\_1 16\^0 B +& a\_3 16\^2 B + \cdots + a\_{63} 16\^{62} B).  \\\\
+    ///             + w(a\_1 w\^0 B +& a\_3 w\^2 B + \cdots + a\_{x-1} w\^{x-2} B).  \\\\
     /// \end{aligned}
     /// $$
     /// 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 {
+impl<'a, 'b> Mul<&'b Scalar> for &'a $name {
-    type Output = EdwardsPoint;
+    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 {
+impl<'a, 'b> Mul<&'a $name> for &'b Scalar {
-    type Output = EdwardsPoint;
+    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 {
+impl Debug for $name {
-    /// Create a table of precomputed multiples of `basepoint`.
+    fn fmt(&self, f: &mut ::core::fmt::Formatter) -> ::core::fmt::Result {
-    pub fn create(basepoint: &EdwardsPoint) -> EdwardsBasepointTable {
+        write!(f, "{:?}([\n", stringify!($name))?;
-        // XXX use init_with
-        let mut table = EdwardsBasepointTable([LookupTable::default(); 32]);
-        let mut P = *basepoint;
         for i in 0..32 {
-            // P = (16^2)^i * B
+            write!(f, "\t{:?},\n", &self.0[i])?;
-            table.0[i] = LookupTable::from(&P);
-            P = P.mul_by_pow_2(8);
         }
-        table
+        write!(f, "])")
-    }
-
-    /// 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()
     }
 }
 
+}} // 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() {

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};

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] {
+    pub(crate) fn to_radix_2w(&self, w: usize) -> [i8; 64] {
-        debug_assert!(w >= 6);
+        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`.