fe.go

     1  // Copyright (c) 2017 The Go Authors. All rights reserved.
     2  // Use of this source code is governed by a BSD-style
     3  // license that can be found in the LICENSE file.
     4  
     5  // Package field implements fast arithmetic modulo 2^255-19.
     6  package field
     7  
     8  import (
     9  	"crypto/subtle"
    10  	"encoding/binary"
    11  	"math/bits"
    12  )
    13  
    14  // Element represents an element of the field GF(2^255-19). Note that this
    15  // is not a cryptographically secure group, and should only be used to interact
    16  // with edwards25519.Point coordinates.
    17  //
    18  // This type works similarly to math/big.Int, and all arguments and receivers
    19  // are allowed to alias.
    20  //
    21  // The zero value is a valid zero element.
    22  type Element struct {
    23  	// An element t represents the integer
    24  	//     t.l0 + t.l1*2^51 + t.l2*2^102 + t.l3*2^153 + t.l4*2^204
    25  	//
    26  	// Between operations, all limbs are expected to be lower than 2^52.
    27  	l0 uint64
    28  	l1 uint64
    29  	l2 uint64
    30  	l3 uint64
    31  	l4 uint64
    32  }
    33  
    34  const maskLow51Bits uint64 = (1 << 51) - 1
    35  
    36  var feZero = &Element{0, 0, 0, 0, 0}
    37  
    38  // Zero sets v = 0, and returns v.
    39  func (v *Element) Zero() *Element {
    40  	*v = *feZero
    41  	return v
    42  }
    43  
    44  var feOne = &Element{1, 0, 0, 0, 0}
    45  
    46  // One sets v = 1, and returns v.
    47  func (v *Element) One() *Element {
    48  	*v = *feOne
    49  	return v
    50  }
    51  
    52  // reduce reduces v modulo 2^255 - 19 and returns it.
    53  func (v *Element) reduce() *Element {
    54  	v.carryPropagate()
    55  
    56  	// After the light reduction we now have a field element representation
    57  	// v < 2^255 + 2^13 * 19, but need v < 2^255 - 19.
    58  
    59  	// If v >= 2^255 - 19, then v + 19 >= 2^255, which would overflow 2^255 - 1,
    60  	// generating a carry. That is, c will be 0 if v < 2^255 - 19, and 1 otherwise.
    61  	c := (v.l0 + 19) >> 51
    62  	c = (v.l1 + c) >> 51
    63  	c = (v.l2 + c) >> 51
    64  	c = (v.l3 + c) >> 51
    65  	c = (v.l4 + c) >> 51
    66  
    67  	// If v < 2^255 - 19 and c = 0, this will be a no-op. Otherwise, it's
    68  	// effectively applying the reduction identity to the carry.
    69  	v.l0 += 19 * c
    70  
    71  	v.l1 += v.l0 >> 51
    72  	v.l0 = v.l0 & maskLow51Bits
    73  	v.l2 += v.l1 >> 51
    74  	v.l1 = v.l1 & maskLow51Bits
    75  	v.l3 += v.l2 >> 51
    76  	v.l2 = v.l2 & maskLow51Bits
    77  	v.l4 += v.l3 >> 51
    78  	v.l3 = v.l3 & maskLow51Bits
    79  	// no additional carry
    80  	v.l4 = v.l4 & maskLow51Bits
    81  
    82  	return v
    83  }
    84  
    85  // Add sets v = a + b, and returns v.
    86  func (v *Element) Add(a, b *Element) *Element {
    87  	v.l0 = a.l0 + b.l0
    88  	v.l1 = a.l1 + b.l1
    89  	v.l2 = a.l2 + b.l2
    90  	v.l3 = a.l3 + b.l3
    91  	v.l4 = a.l4 + b.l4
    92  	// Using the generic implementation here is actually faster than the
    93  	// assembly. Probably because the body of this function is so simple that
    94  	// the compiler can figure out better optimizations by inlining the carry
    95  	// propagation. TODO
    96  	return v.carryPropagateGeneric()
    97  }
    98  
    99  // Subtract sets v = a - b, and returns v.
   100  func (v *Element) Subtract(a, b *Element) *Element {
   101  	// We first add 2 * p, to guarantee the subtraction won't underflow, and
   102  	// then subtract b (which can be up to 2^255 + 2^13 * 19).
   103  	v.l0 = (a.l0 + 0xFFFFFFFFFFFDA) - b.l0
   104  	v.l1 = (a.l1 + 0xFFFFFFFFFFFFE) - b.l1
   105  	v.l2 = (a.l2 + 0xFFFFFFFFFFFFE) - b.l2
   106  	v.l3 = (a.l3 + 0xFFFFFFFFFFFFE) - b.l3
   107  	v.l4 = (a.l4 + 0xFFFFFFFFFFFFE) - b.l4
   108  	return v.carryPropagate()
   109  }
   110  
   111  // Negate sets v = -a, and returns v.
   112  func (v *Element) Negate(a *Element) *Element {
   113  	return v.Subtract(feZero, a)
   114  }
   115  
   116  // Invert sets v = 1/z mod p, and returns v.
   117  //
   118  // If z == 0, Invert returns v = 0.
   119  func (v *Element) Invert(z *Element) *Element {
   120  	// Inversion is implemented as exponentiation with exponent p − 2. It uses the
   121  	// same sequence of 255 squarings and 11 multiplications as [Curve25519].
   122  	var z2, z9, z11, z2_5_0, z2_10_0, z2_20_0, z2_50_0, z2_100_0, t Element
   123  
   124  	z2.Square(z)             // 2
   125  	t.Square(&z2)            // 4
   126  	t.Square(&t)             // 8
   127  	z9.Multiply(&t, z)       // 9
   128  	z11.Multiply(&z9, &z2)   // 11
   129  	t.Square(&z11)           // 22
   130  	z2_5_0.Multiply(&t, &z9) // 31 = 2^5 - 2^0
   131  
   132  	t.Square(&z2_5_0) // 2^6 - 2^1
   133  	for i := 0; i < 4; i++ {
   134  		t.Square(&t) // 2^10 - 2^5
   135  	}
   136  	z2_10_0.Multiply(&t, &z2_5_0) // 2^10 - 2^0
   137  
   138  	t.Square(&z2_10_0) // 2^11 - 2^1
   139  	for i := 0; i < 9; i++ {
   140  		t.Square(&t) // 2^20 - 2^10
   141  	}
   142  	z2_20_0.Multiply(&t, &z2_10_0) // 2^20 - 2^0
   143  
   144  	t.Square(&z2_20_0) // 2^21 - 2^1
   145  	for i := 0; i < 19; i++ {
   146  		t.Square(&t) // 2^40 - 2^20
   147  	}
   148  	t.Multiply(&t, &z2_20_0) // 2^40 - 2^0
   149  
   150  	t.Square(&t) // 2^41 - 2^1
   151  	for i := 0; i < 9; i++ {
   152  		t.Square(&t) // 2^50 - 2^10
   153  	}
   154  	z2_50_0.Multiply(&t, &z2_10_0) // 2^50 - 2^0
   155  
   156  	t.Square(&z2_50_0) // 2^51 - 2^1
   157  	for i := 0; i < 49; i++ {
   158  		t.Square(&t) // 2^100 - 2^50
   159  	}
   160  	z2_100_0.Multiply(&t, &z2_50_0) // 2^100 - 2^0
   161  
   162  	t.Square(&z2_100_0) // 2^101 - 2^1
   163  	for i := 0; i < 99; i++ {
   164  		t.Square(&t) // 2^200 - 2^100
   165  	}
   166  	t.Multiply(&t, &z2_100_0) // 2^200 - 2^0
   167  
   168  	t.Square(&t) // 2^201 - 2^1
   169  	for i := 0; i < 49; i++ {
   170  		t.Square(&t) // 2^250 - 2^50
   171  	}
   172  	t.Multiply(&t, &z2_50_0) // 2^250 - 2^0
   173  
   174  	t.Square(&t) // 2^251 - 2^1
   175  	t.Square(&t) // 2^252 - 2^2
   176  	t.Square(&t) // 2^253 - 2^3
   177  	t.Square(&t) // 2^254 - 2^4
   178  	t.Square(&t) // 2^255 - 2^5
   179  
   180  	return v.Multiply(&t, &z11) // 2^255 - 21
   181  }
   182  
   183  // Set sets v = a, and returns v.
   184  func (v *Element) Set(a *Element) *Element {
   185  	*v = *a
   186  	return v
   187  }
   188  
   189  // SetBytes sets v to x, which must be a 32-byte little-endian encoding.
   190  //
   191  // Consistent with RFC 7748, the most significant bit (the high bit of the
   192  // last byte) is ignored, and non-canonical values (2^255-19 through 2^255-1)
   193  // are accepted. Note that this is laxer than specified by RFC 8032.
   194  func (v *Element) SetBytes(x []byte) *Element {
   195  	if len(x) != 32 {
   196  		panic("edwards25519: invalid field element input size")
   197  	}
   198  
   199  	// Bits 0:51 (bytes 0:8, bits 0:64, shift 0, mask 51).
   200  	v.l0 = binary.LittleEndian.Uint64(x[0:8])
   201  	v.l0 &= maskLow51Bits
   202  	// Bits 51:102 (bytes 6:14, bits 48:112, shift 3, mask 51).
   203  	v.l1 = binary.LittleEndian.Uint64(x[6:14]) >> 3
   204  	v.l1 &= maskLow51Bits
   205  	// Bits 102:153 (bytes 12:20, bits 96:160, shift 6, mask 51).
   206  	v.l2 = binary.LittleEndian.Uint64(x[12:20]) >> 6
   207  	v.l2 &= maskLow51Bits
   208  	// Bits 153:204 (bytes 19:27, bits 152:216, shift 1, mask 51).
   209  	v.l3 = binary.LittleEndian.Uint64(x[19:27]) >> 1
   210  	v.l3 &= maskLow51Bits
   211  	// Bits 204:251 (bytes 24:32, bits 192:256, shift 12, mask 51).
   212  	// Note: not bytes 25:33, shift 4, to avoid overread.
   213  	v.l4 = binary.LittleEndian.Uint64(x[24:32]) >> 12
   214  	v.l4 &= maskLow51Bits
   215  
   216  	return v
   217  }
   218  
   219  // Bytes returns the canonical 32-byte little-endian encoding of v.
   220  func (v *Element) Bytes() []byte {
   221  	// This function is outlined to make the allocations inline in the caller
   222  	// rather than happen on the heap.
   223  	var out [32]byte
   224  	return v.bytes(&out)
   225  }
   226  
   227  func (v *Element) bytes(out *[32]byte) []byte {
   228  	t := *v
   229  	t.reduce()
   230  
   231  	var buf [8]byte
   232  	for i, l := range [5]uint64{t.l0, t.l1, t.l2, t.l3, t.l4} {
   233  		bitsOffset := i * 51
   234  		binary.LittleEndian.PutUint64(buf[:], l<<uint(bitsOffset%8))
   235  		for i, bb := range buf {
   236  			off := bitsOffset/8 + i
   237  			if off >= len(out) {
   238  				break
   239  			}
   240  			out[off] |= bb
   241  		}
   242  	}
   243  
   244  	return out[:]
   245  }
   246  
   247  // Equal returns 1 if v and u are equal, and 0 otherwise.
   248  func (v *Element) Equal(u *Element) int {
   249  	sa, sv := u.Bytes(), v.Bytes()
   250  	return subtle.ConstantTimeCompare(sa, sv)
   251  }
   252  
   253  // mask64Bits returns 0xffffffff if cond is 1, and 0 otherwise.
   254  func mask64Bits(cond int) uint64 { return ^(uint64(cond) - 1) }
   255  
   256  // Select sets v to a if cond == 1, and to b if cond == 0.
   257  func (v *Element) Select(a, b *Element, cond int) *Element {
   258  	m := mask64Bits(cond)
   259  	v.l0 = (m & a.l0) | (^m & b.l0)
   260  	v.l1 = (m & a.l1) | (^m & b.l1)
   261  	v.l2 = (m & a.l2) | (^m & b.l2)
   262  	v.l3 = (m & a.l3) | (^m & b.l3)
   263  	v.l4 = (m & a.l4) | (^m & b.l4)
   264  	return v
   265  }
   266  
   267  // Swap swaps v and u if cond == 1 or leaves them unchanged if cond == 0, and returns v.
   268  func (v *Element) Swap(u *Element, cond int) {
   269  	m := mask64Bits(cond)
   270  	t := m & (v.l0 ^ u.l0)
   271  	v.l0 ^= t
   272  	u.l0 ^= t
   273  	t = m & (v.l1 ^ u.l1)
   274  	v.l1 ^= t
   275  	u.l1 ^= t
   276  	t = m & (v.l2 ^ u.l2)
   277  	v.l2 ^= t
   278  	u.l2 ^= t
   279  	t = m & (v.l3 ^ u.l3)
   280  	v.l3 ^= t
   281  	u.l3 ^= t
   282  	t = m & (v.l4 ^ u.l4)
   283  	v.l4 ^= t
   284  	u.l4 ^= t
   285  }
   286  
   287  // IsNegative returns 1 if v is negative, and 0 otherwise.
   288  func (v *Element) IsNegative() int {
   289  	return int(v.Bytes()[0] & 1)
   290  }
   291  
   292  // Absolute sets v to |u|, and returns v.
   293  func (v *Element) Absolute(u *Element) *Element {
   294  	return v.Select(new(Element).Negate(u), u, u.IsNegative())
   295  }
   296  
   297  // Multiply sets v = x * y, and returns v.
   298  func (v *Element) Multiply(x, y *Element) *Element {
   299  	feMul(v, x, y)
   300  	return v
   301  }
   302  
   303  // Square sets v = x * x, and returns v.
   304  func (v *Element) Square(x *Element) *Element {
   305  	feSquare(v, x)
   306  	return v
   307  }
   308  
   309  // Mult32 sets v = x * y, and returns v.
   310  func (v *Element) Mult32(x *Element, y uint32) *Element {
   311  	x0lo, x0hi := mul51(x.l0, y)
   312  	x1lo, x1hi := mul51(x.l1, y)
   313  	x2lo, x2hi := mul51(x.l2, y)
   314  	x3lo, x3hi := mul51(x.l3, y)
   315  	x4lo, x4hi := mul51(x.l4, y)
   316  	v.l0 = x0lo + 19*x4hi // carried over per the reduction identity
   317  	v.l1 = x1lo + x0hi
   318  	v.l2 = x2lo + x1hi
   319  	v.l3 = x3lo + x2hi
   320  	v.l4 = x4lo + x3hi
   321  	// The hi portions are going to be only 32 bits, plus any previous excess,
   322  	// so we can skip the carry propagation.
   323  	return v
   324  }
   325  
   326  // mul51 returns lo + hi * 2⁵¹ = a * b.
   327  func mul51(a uint64, b uint32) (lo uint64, hi uint64) {
   328  	mh, ml := bits.Mul64(a, uint64(b))
   329  	lo = ml & maskLow51Bits
   330  	hi = (mh << 13) | (ml >> 51)
   331  	return
   332  }
   333  
   334  // Pow22523 set v = x^((p-5)/8), and returns v. (p-5)/8 is 2^252-3.
   335  func (v *Element) Pow22523(x *Element) *Element {
   336  	var t0, t1, t2 Element
   337  
   338  	t0.Square(x)             // x^2
   339  	t1.Square(&t0)           // x^4
   340  	t1.Square(&t1)           // x^8
   341  	t1.Multiply(x, &t1)      // x^9
   342  	t0.Multiply(&t0, &t1)    // x^11
   343  	t0.Square(&t0)           // x^22
   344  	t0.Multiply(&t1, &t0)    // x^31
   345  	t1.Square(&t0)           // x^62
   346  	for i := 1; i < 5; i++ { // x^992
   347  		t1.Square(&t1)
   348  	}
   349  	t0.Multiply(&t1, &t0)     // x^1023 -> 1023 = 2^10 - 1
   350  	t1.Square(&t0)            // 2^11 - 2
   351  	for i := 1; i < 10; i++ { // 2^20 - 2^10
   352  		t1.Square(&t1)
   353  	}
   354  	t1.Multiply(&t1, &t0)     // 2^20 - 1
   355  	t2.Square(&t1)            // 2^21 - 2
   356  	for i := 1; i < 20; i++ { // 2^40 - 2^20
   357  		t2.Square(&t2)
   358  	}
   359  	t1.Multiply(&t2, &t1)     // 2^40 - 1
   360  	t1.Square(&t1)            // 2^41 - 2
   361  	for i := 1; i < 10; i++ { // 2^50 - 2^10
   362  		t1.Square(&t1)
   363  	}
   364  	t0.Multiply(&t1, &t0)     // 2^50 - 1
   365  	t1.Square(&t0)            // 2^51 - 2
   366  	for i := 1; i < 50; i++ { // 2^100 - 2^50
   367  		t1.Square(&t1)
   368  	}
   369  	t1.Multiply(&t1, &t0)      // 2^100 - 1
   370  	t2.Square(&t1)             // 2^101 - 2
   371  	for i := 1; i < 100; i++ { // 2^200 - 2^100
   372  		t2.Square(&t2)
   373  	}
   374  	t1.Multiply(&t2, &t1)     // 2^200 - 1
   375  	t1.Square(&t1)            // 2^201 - 2
   376  	for i := 1; i < 50; i++ { // 2^250 - 2^50
   377  		t1.Square(&t1)
   378  	}
   379  	t0.Multiply(&t1, &t0)     // 2^250 - 1
   380  	t0.Square(&t0)            // 2^251 - 2
   381  	t0.Square(&t0)            // 2^252 - 4
   382  	return v.Multiply(&t0, x) // 2^252 - 3 -> x^(2^252-3)
   383  }
   384  
   385  // sqrtM1 is 2^((p-1)/4), which squared is equal to -1 by Euler's Criterion.
   386  var sqrtM1 = &Element{1718705420411056, 234908883556509,
   387  	2233514472574048, 2117202627021982, 765476049583133}
   388  
   389  // SqrtRatio sets r to the non-negative square root of the ratio of u and v.
   390  //
   391  // If u/v is square, SqrtRatio returns r and 1. If u/v is not square, SqrtRatio
   392  // sets r according to Section 4.3 of draft-irtf-cfrg-ristretto255-decaf448-00,
   393  // and returns r and 0.
   394  func (r *Element) SqrtRatio(u, v *Element) (rr *Element, wasSquare int) {
   395  	var a, b Element
   396  
   397  	// r = (u * v3) * (u * v7)^((p-5)/8)
   398  	v2 := a.Square(v)
   399  	uv3 := b.Multiply(u, b.Multiply(v2, v))
   400  	uv7 := a.Multiply(uv3, a.Square(v2))
   401  	r.Multiply(uv3, r.Pow22523(uv7))
   402  
   403  	check := a.Multiply(v, a.Square(r)) // check = v * r^2
   404  
   405  	uNeg := b.Negate(u)
   406  	correctSignSqrt := check.Equal(u)
   407  	flippedSignSqrt := check.Equal(uNeg)
   408  	flippedSignSqrtI := check.Equal(uNeg.Multiply(uNeg, sqrtM1))
   409  
   410  	rPrime := b.Multiply(r, sqrtM1) // r_prime = SQRT_M1 * r
   411  	// r = CT_SELECT(r_prime IF flipped_sign_sqrt | flipped_sign_sqrt_i ELSE r)
   412  	r.Select(rPrime, r, flippedSignSqrt|flippedSignSqrtI)
   413  
   414  	r.Absolute(r) // Choose the nonnegative square root.
   415  	return r, correctSignSqrt | flippedSignSqrt
   416  }
   417
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