daelk-cryptography curve25519-dalek源码解析——之Field表示[通俗易懂]

https://github.com/dalek-cryptography/curve25519-dalek

1. Scalar结构

针对p<2²⁵⁵的域filed，采用scalar以little-endian的数组形式来表示：【对于Curve25519，其p值为 2²⁵⁵ – 19】

/// The `Scalar` struct holds an integer \\(s < 2\^{255} \\) which
/// represents an element of \\(\mathbb Z / \ell\\).
#[derive(Copy, Clone)]
pub struct Scalar {
    /// `bytes` is a little-endian byte encoding of an integer representing a scalar modulo the
    /// group order.
    ///
    /// # Invariant
    ///
    /// The integer representing this scalar must be bounded above by \\(2\^{255}\\), or
    /// equivalently the high bit of `bytes[31]` must be zero.
    ///
    /// This ensures that there is room for a carry bit when computing a NAF representation.
    //
    // XXX This is pub(crate) so we can write literal constants.  If const fns were stable, we could
    //     make the Scalar constructors const fns and use those instead.
    pub(crate) bytes: [u8; 32], 
}

Scalar类型中的bytes成员定义为pub(crate)，表示该成员可在本crate内public可见，但对除本crate外的其它crates中不可见。

因此，对于：
x = 2238329342913194256032495932344128051776374960164957527413114840482143558222

sage: hex(2238329342913194256032495932344128051776374960164957527413114840482143
....: 558222)
'4f2d979a8f449d44442cc1b1085a552527dc21b64b413598408475d34b45a4e'
sage: len('4f2d979a8f449d44442cc1b1085a552527dc21b64b413598408475d34b45a4e'
....: )
63  //对应的 the high bit of `bytes[31]` must be zero.

/// // x = 2238329342913194256032495932344128051776374960164957527413114840482143558222
    /// let X: Scalar = Scalar::from_bytes_mod_order([
    ///         0x4e, 0x5a, 0xb4, 0x34, 0x5d, 0x47, 0x08, 0x84,
    ///         0x59, 0x13, 0xb4, 0x64, 0x1b, 0xc2, 0x7d, 0x52,
    ///         0x52, 0xa5, 0x85, 0x10, 0x1b, 0xcc, 0x42, 0x44,
    ///         0xd4, 0x49, 0xf4, 0xa8, 0x79, 0xd9, 0xf2, 0x04,
    ///     ]);

2. UnpackedScalar结构

程序中默认采用的是u64_backend feature
UnpackedScalar用于代表GF(l)域，其中l=2^252 + 27742317777372353535851937790883648493

/// An `UnpackedScalar` represents an element of the field GF(l), optimized for speed.
///
/// This is a type alias for one of the scalar types in the `backend`
/// module.
#[cfg(feature = "u64_backend")]
type UnpackedScalar = backend::serial::u64::scalar::Scalar52;

/// An `UnpackedScalar` represents an element of the field GF(l), optimized for speed.
///
/// This is a type alias for one of the scalar types in the `backend`
/// module.
#[cfg(feature = "u32_backend")]
type UnpackedScalar = backend::serial::u32::scalar::Scalar29;

参照libsnark中的格式const mp_size_t alt_bn128_r_limbs = (alt_bn128_r_bitcount+GMP_NUMB_BITS-1)/GMP_NUMB_BITS;，即bitcount=255：

对于64位系统，数组大小的计算公式为n=roundup[(bitcount+64-1)/64]=5，为了减少计算复杂度（无需考虑所有64位，仅需关注libm位的计算操作），libm仅需用满足libm*n略大于等于bitcount，此时libm本可以取值51（51*5=255），但考虑到Montgomery multiplication reduce的需要，libm取值52。
对于32位系统，数组大小的计算公式为n=roundup[(bitcount+32-1)/32]=9，同理此时libm取值29。

/// The `Scalar52` struct represents an element in
/// \\(\mathbb Z / \ell \mathbb Z\\) as 5 \\(52\\)-bit limbs.
#[derive(Copy,Clone)]
pub struct Scalar52(pub [u64; 5]);

3. Scalar与UnpackedScalar转换

	/// let inv_X: Scalar = X.invert();
    /// assert!(XINV == inv_X);
    /// let should_be_one: Scalar = &inv_X * &X;
    /// assert!(should_be_one == Scalar::one());
    /// ```
    pub fn invert(&self) -> Scalar {
        self.unpack().invert().pack()
    }

	/// Unpack this `Scalar` to an `UnpackedScalar` for faster arithmetic.
    pub(crate) fn unpack(&self) -> UnpackedScalar {
        UnpackedScalar::from_bytes(&self.bytes)
    }

3.1 Scalar转换为UnpackedScalar

Scalar转换为UnpackedScalar的代码细节为：

	/// Unpack a 32 byte / 256 bit scalar into 5 52-bit limbs.
    pub fn from_bytes(bytes: &[u8; 32]) -> Scalar52 {
        let mut words = [0u64; 4];
        for i in 0..4 {
            for j in 0..8 {
                words[i] |= (bytes[(i * 8) + j] as u64) << (j * 8);
            }
        }

        let mask = (1u64 << 52) - 1; //仅取52bit
        let top_mask = (1u64 << 48) - 1; //仅取48bit
        let mut s = Scalar52::zero();
		// 一共仅保留256bit，words数组中是将scalar值按64bit为单位分别存储
		// 以下是要以52bit单位分别存储到s数组中，需要对words中的内容进行移位及mask处理，s数组内一共存储256bit有效位数。
        s[ 0] =   words[0]                            & mask; 
        s[ 1] = ((words[0] >> 52) | (words[1] << 12)) & mask;
        s[ 2] = ((words[1] >> 40) | (words[2] << 24)) & mask;
        s[ 3] = ((words[2] >> 28) | (words[3] << 36)) & mask;
        s[ 4] =  (words[3] >> 16)                     & top_mask;

        s
    }

3.2 invert()操作

有限域内的乘法具有以下特征：

x^(p-2) * x = x^(p-1) = 1 (mod p)

由此可推测出，求有限域的x值的倒数可转换为求x^(p-2)的值。

程序中，对Scalar值求倒数，是先通过unpack()函数将Scalar转换为UnpackedScalar，然后对UnpackedScalar求倒数，最后通过pack()函数将UnpackedScalar转换为Scalar值。

impl Scalar {
	/// let inv_X: Scalar = X.invert();
    /// assert!(XINV == inv_X);
    /// let should_be_one: Scalar = &inv_X * &X;
    /// assert!(should_be_one == Scalar::one());
    /// ```
    pub fn invert(&self) -> Scalar {
        self.unpack().invert().pack()
    }
	/// Unpack this `Scalar` to an `UnpackedScalar` for faster arithmetic.
    pub(crate) fn unpack(&self) -> UnpackedScalar {
        UnpackedScalar::from_bytes(&self.bytes)
    }
}

impl UnpackedScalar {
	/// Inverts an UnpackedScalar not in Montgomery form.
    pub fn invert(&self) -> UnpackedScalar {
        self.to_montgomery().montgomery_invert().from_montgomery()
    }
	/// Pack the limbs of this `UnpackedScalar` into a `Scalar`.
    fn pack(&self) -> Scalar {
        Scalar{ bytes: self.to_bytes() }
    }
}

对于u64_backend feature, 有 type UnpackedScalar = backend::serial::u64::scalar::Scalar52;，所以对于
to_montgomery()的具体实现如下：

impl Scalar52 {
	/// Puts a Scalar52 in to Montgomery form, i.e. computes `a*R (mod l)`
    #[inline(never)]
    pub fn to_montgomery(&self) -> Scalar52 {
        Scalar52::montgomery_mul(self, &constants::RR) //将数组中52*5=260，260bit所有位数都用上。pub struct Scalar52(pub [u64; 5]);
    }
    
	/// Compute `(a * b) / R` (mod l), where R is the Montgomery modulus 2^260
    #[inline(never)]
    pub fn montgomery_mul(a: &Scalar52, b: &Scalar52) -> Scalar52 {
        Scalar52::montgomery_reduce(&Scalar52::mul_internal(a, b))
    }

	/// Compute `a * b`
    #[inline(always)]
    pub (crate) fn mul_internal(a: &Scalar52, b: &Scalar52) -> [u128; 9] {
        let mut z = [0u128; 9];

        z[0] = m(a[0],b[0]);
        z[1] = m(a[0],b[1]) + m(a[1],b[0]);
        z[2] = m(a[0],b[2]) + m(a[1],b[1]) + m(a[2],b[0]);
        z[3] = m(a[0],b[3]) + m(a[1],b[2]) + m(a[2],b[1]) + m(a[3],b[0]);
        z[4] = m(a[0],b[4]) + m(a[1],b[3]) + m(a[2],b[2]) + m(a[3],b[1]) + m(a[4],b[0]);
        z[5] =                m(a[1],b[4]) + m(a[2],b[3]) + m(a[3],b[2]) + m(a[4],b[1]);
        z[6] =                               m(a[2],b[4]) + m(a[3],b[3]) + m(a[4],b[2]);
        z[7] =                                              m(a[3],b[4]) + m(a[4],b[3]);
        z[8] =                                                             m(a[4],b[4]);

        z
    }
    
    /// u64 * u64 = u128 multiply helper
	#[inline(always)]
	fn m(x: u64, y: u64) -> u128 {
 	   (x as u128) * (y as u128)
	}
	
 	/// Compute `limbs/R` (mod l), where R is the Montgomery modulus 2^260
    #[inline(always)]
    pub (crate) fn montgomery_reduce(limbs: &[u128; 9]) -> Scalar52 {

        #[inline(always)]
        fn part1(sum: u128) -> (u128, u64) {
            let p = (sum as u64).wrapping_mul(constants::LFACTOR) & ((1u64 << 52) - 1);
            ((sum + m(p,constants::L[0])) >> 52, p)
        }

        #[inline(always)]
        fn part2(sum: u128) -> (u128, u64) {
            let w = (sum as u64) & ((1u64 << 52) - 1);
            (sum >> 52, w)
        }

        // note: l3 is zero, so its multiplies can be skipped
        let l = &constants::L;

        // the first half computes the Montgomery adjustment factor n, and begins adding n*l to make limbs divisible by R
        let (carry, n0) = part1(        limbs[0]);
        let (carry, n1) = part1(carry + limbs[1] + m(n0,l[1]));
        let (carry, n2) = part1(carry + limbs[2] + m(n0,l[2]) + m(n1,l[1]));
        let (carry, n3) = part1(carry + limbs[3]              + m(n1,l[2]) + m(n2,l[1]));
        let (carry, n4) = part1(carry + limbs[4] + m(n0,l[4])              + m(n2,l[2]) + m(n3,l[1]));

        // limbs is divisible by R now, so we can divide by R by simply storing the upper half as the result
        let (carry, r0) = part2(carry + limbs[5]              + m(n1,l[4])              + m(n3,l[2]) + m(n4,l[1]));
        let (carry, r1) = part2(carry + limbs[6]                           + m(n2,l[4])              + m(n4,l[2]));
        let (carry, r2) = part2(carry + limbs[7]                                        + m(n3,l[4])             );
        let (carry, r3) = part2(carry + limbs[8]                                                     + m(n4,l[4]));
        let         r4 = carry as u64;

        // result may be >= l, so attempt to subtract l
        Scalar52::sub(&Scalar52([r0,r1,r2,r3,r4]), l)
    }
}

4. constant.rs中常量值sage验证

/// constant.rs中有记录一些常量值。
/// `L` is the order of base point, i.e. 2^252 + 27742317777372353535851937790883648493
pub(crate) const L: Scalar52 = Scalar52([ 0x0002631a5cf5d3ed, 0x000dea2f79cd6581, 0x000000000014def9, 0x0000000000000000, 0x0000100000000000 ]);

/// 其实即为L[0]*LFACTOR = -1 (mod 2^52) = 2^52-1 (mod 2^52)
/// (L[i]<<52)*LFACTOR = 0 (mod 2^52) 其中 1 =< i <= 4
/// `L` * `LFACTOR` = -1 (mod 2^52)
pub(crate) const LFACTOR: u64 = 0x51da312547e1b;

/// `R` = R % L where R = 2^260
pub(crate) const R: Scalar52 = Scalar52([ 0x000f48bd6721e6ed, 0x0003bab5ac67e45a, 0x000fffffeb35e51b, 0x000fffffffffffff, 0x00000fffffffffff ]);

/// `RR` = (R^2) % L where R = 2^260
pub(crate) const RR: Scalar52 = Scalar52([ 0x0009d265e952d13b, 0x000d63c715bea69f, 0x0005be65cb687604, 0x0003dceec73d217f, 0x000009411b7c309a ]);

对应的sage验证为：

sage: 2^252 + 27742317777372353535851937790883648493
7237005577332262213973186563042994240857116359379907606001950938285454250989
sage: is_prime(72370055773322622139731865630429942408571163593799076060019509382
....: 85454250989)
True
sage: hex(7237005577332262213973186563042994240857116359379907606001950938285454
....: 250989)
'1000000000000000000000000000000014def9dea2f79cd65812631a5cf5d3ed'  //即pub(crate) const L: Scalar52为数组内每个元素只截取13个数字（52bit），按little-endian方式存储。

sage: L=2^252 + 27742317777372353535851937790883648493
sage: LFACTOR=0x51da312547e1b
sage: LFACTOR
1439961107955227
sage: mod(L*LFACTOR, 2^52)  //即`L` * `LFACTOR` = -1 (mod 2^52)
4503599627370495
sage: 2^52
4503599627370496

sage: R=2^260
sage: mod(R,L)
7237005577332262213973186563042994233755083008372585100823854863819240236781
sage: hex(7237005577332262213973186563042994233755083008372585100823854863819240
....: 236781)
'fffffffffffffffffffffffffffffeb35e51b3bab5ac67e45af48bd6721e6ed' //即pub(crate) const R: Scalar52为数组内每个元素只截取13个数字（52bit），按little-endian方式存储。

sage: mod(R^2, L)
4185850391763183796333492317919282507600454137915443218209456916606550724923
sage: hex(4185850391763183796333492317919282507600454137915443218209456916606550
....: 724923)
'9411b7c309a3dceec73d217f5be65cb687604d63c715bea69f9d265e952d13b'
sage:

sage: gcd(L,R) //符合Montgomery reduction定义的条件。可参见https://blog.csdn.net/mutourend/article/details/95613967 第2.4.1节内容
1