nat.go

     1  // Copyright 2009 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  // This file implements unsigned multi-precision integers (natural
     6  // numbers). They are the building blocks for the implementation
     7  // of signed integers, rationals, and floating-point numbers.
     8  //
     9  // Caution: This implementation relies on the function "alias"
    10  //          which assumes that (nat) slice capacities are never
    11  //          changed (no 3-operand slice expressions). If that
    12  //          changes, alias needs to be updated for correctness.
    13  
    14  package big
    15  
    16  import (
    17  	"internal/byteorder"
    18  	"math/bits"
    19  	"math/rand"
    20  	"slices"
    21  	"sync"
    22  )
    23  
    24  // An unsigned integer x of the form
    25  //
    26  //	x = x[n-1]*_B^(n-1) + x[n-2]*_B^(n-2) + ... + x[1]*_B + x[0]
    27  //
    28  // with 0 <= x[i] < _B and 0 <= i < n is stored in a slice of length n,
    29  // with the digits x[i] as the slice elements.
    30  //
    31  // A number is normalized if the slice contains no leading 0 digits.
    32  // During arithmetic operations, denormalized values may occur but are
    33  // always normalized before returning the final result. The normalized
    34  // representation of 0 is the empty or nil slice (length = 0).
    35  type nat []Word
    36  
    37  var (
    38  	natOne  = nat{1}
    39  	natTwo  = nat{2}
    40  	natFive = nat{5}
    41  	natTen  = nat{10}
    42  )
    43  
    44  func (z nat) String() string {
    45  	return "0x" + string(z.itoa(false, 16))
    46  }
    47  
    48  func (z nat) norm() nat {
    49  	i := len(z)
    50  	for i > 0 && z[i-1] == 0 {
    51  		i--
    52  	}
    53  	return z[0:i]
    54  }
    55  
    56  func (z nat) make(n int) nat {
    57  	if n <= cap(z) {
    58  		return z[:n] // reuse z
    59  	}
    60  	if n == 1 {
    61  		// Most nats start small and stay that way; don't over-allocate.
    62  		return make(nat, 1)
    63  	}
    64  	// Choosing a good value for e has significant performance impact
    65  	// because it increases the chance that a value can be reused.
    66  	const e = 4 // extra capacity
    67  	return make(nat, n, n+e)
    68  }
    69  
    70  func (z nat) setWord(x Word) nat {
    71  	if x == 0 {
    72  		return z[:0]
    73  	}
    74  	z = z.make(1)
    75  	z[0] = x
    76  	return z
    77  }
    78  
    79  func (z nat) setUint64(x uint64) nat {
    80  	// single-word value
    81  	if w := Word(x); uint64(w) == x {
    82  		return z.setWord(w)
    83  	}
    84  	// 2-word value
    85  	z = z.make(2)
    86  	z[1] = Word(x >> 32)
    87  	z[0] = Word(x)
    88  	return z
    89  }
    90  
    91  func (z nat) set(x nat) nat {
    92  	z = z.make(len(x))
    93  	copy(z, x)
    94  	return z
    95  }
    96  
    97  func (z nat) add(x, y nat) nat {
    98  	m := len(x)
    99  	n := len(y)
   100  
   101  	switch {
   102  	case m < n:
   103  		return z.add(y, x)
   104  	case m == 0:
   105  		// n == 0 because m >= n; result is 0
   106  		return z[:0]
   107  	case n == 0:
   108  		// result is x
   109  		return z.set(x)
   110  	}
   111  	// m > 0
   112  
   113  	z = z.make(m + 1)
   114  	c := addVV(z[0:n], x, y)
   115  	if m > n {
   116  		c = addVW(z[n:m], x[n:], c)
   117  	}
   118  	z[m] = c
   119  
   120  	return z.norm()
   121  }
   122  
   123  func (z nat) sub(x, y nat) nat {
   124  	m := len(x)
   125  	n := len(y)
   126  
   127  	switch {
   128  	case m < n:
   129  		panic("underflow")
   130  	case m == 0:
   131  		// n == 0 because m >= n; result is 0
   132  		return z[:0]
   133  	case n == 0:
   134  		// result is x
   135  		return z.set(x)
   136  	}
   137  	// m > 0
   138  
   139  	z = z.make(m)
   140  	c := subVV(z[0:n], x, y)
   141  	if m > n {
   142  		c = subVW(z[n:], x[n:], c)
   143  	}
   144  	if c != 0 {
   145  		panic("underflow")
   146  	}
   147  
   148  	return z.norm()
   149  }
   150  
   151  func (x nat) cmp(y nat) (r int) {
   152  	m := len(x)
   153  	n := len(y)
   154  	if m != n || m == 0 {
   155  		switch {
   156  		case m < n:
   157  			r = -1
   158  		case m > n:
   159  			r = 1
   160  		}
   161  		return
   162  	}
   163  
   164  	i := m - 1
   165  	for i > 0 && x[i] == y[i] {
   166  		i--
   167  	}
   168  
   169  	switch {
   170  	case x[i] < y[i]:
   171  		r = -1
   172  	case x[i] > y[i]:
   173  		r = 1
   174  	}
   175  	return
   176  }
   177  
   178  // montgomery computes z mod m = x*y*2**(-n*_W) mod m,
   179  // assuming k = -1/m mod 2**_W.
   180  // z is used for storing the result which is returned;
   181  // z must not alias x, y or m.
   182  // See Gueron, "Efficient Software Implementations of Modular Exponentiation".
   183  // https://eprint.iacr.org/2011/239.pdf
   184  // In the terminology of that paper, this is an "Almost Montgomery Multiplication":
   185  // x and y are required to satisfy 0 <= z < 2**(n*_W) and then the result
   186  // z is guaranteed to satisfy 0 <= z < 2**(n*_W), but it may not be < m.
   187  func (z nat) montgomery(x, y, m nat, k Word, n int) nat {
   188  	// This code assumes x, y, m are all the same length, n.
   189  	// (required by addMulVVW and the for loop).
   190  	// It also assumes that x, y are already reduced mod m,
   191  	// or else the result will not be properly reduced.
   192  	if len(x) != n || len(y) != n || len(m) != n {
   193  		panic("math/big: mismatched montgomery number lengths")
   194  	}
   195  	z = z.make(n * 2)
   196  	clear(z)
   197  	var c Word
   198  	for i := 0; i < n; i++ {
   199  		d := y[i]
   200  		c2 := addMulVVWW(z[i:n+i], z[i:n+i], x, d, 0)
   201  		t := z[i] * k
   202  		c3 := addMulVVWW(z[i:n+i], z[i:n+i], m, t, 0)
   203  		cx := c + c2
   204  		cy := cx + c3
   205  		z[n+i] = cy
   206  		if cx < c2 || cy < c3 {
   207  			c = 1
   208  		} else {
   209  			c = 0
   210  		}
   211  	}
   212  	if c != 0 {
   213  		subVV(z[:n], z[n:], m)
   214  	} else {
   215  		copy(z[:n], z[n:])
   216  	}
   217  	return z[:n]
   218  }
   219  
   220  // alias reports whether x and y share the same base array.
   221  //
   222  // Note: alias assumes that the capacity of underlying arrays
   223  // is never changed for nat values; i.e. that there are
   224  // no 3-operand slice expressions in this code (or worse,
   225  // reflect-based operations to the same effect).
   226  func alias(x, y nat) bool {
   227  	return cap(x) > 0 && cap(y) > 0 && &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1]
   228  }
   229  
   230  // addTo implements z += x; z must be long enough.
   231  // (we don't use nat.add because we need z to stay the same
   232  // slice, and we don't need to normalize z after each addition)
   233  func addTo(z, x nat) {
   234  	if n := len(x); n > 0 {
   235  		if c := addVV(z[:n], z, x); c != 0 {
   236  			if n < len(z) {
   237  				addVW(z[n:], z[n:], c)
   238  			}
   239  		}
   240  	}
   241  }
   242  
   243  // mulRange computes the product of all the unsigned integers in the
   244  // range [a, b] inclusively. If a > b (empty range), the result is 1.
   245  // The caller may pass stk == nil to request that mulRange obtain and release one itself.
   246  func (z nat) mulRange(stk *stack, a, b uint64) nat {
   247  	switch {
   248  	case a == 0:
   249  		// cut long ranges short (optimization)
   250  		return z.setUint64(0)
   251  	case a > b:
   252  		return z.setUint64(1)
   253  	case a == b:
   254  		return z.setUint64(a)
   255  	case a+1 == b:
   256  		return z.mul(stk, nat(nil).setUint64(a), nat(nil).setUint64(b))
   257  	}
   258  
   259  	if stk == nil {
   260  		stk = getStack()
   261  		defer stk.free()
   262  	}
   263  
   264  	m := a + (b-a)/2 // avoid overflow
   265  	return z.mul(stk, nat(nil).mulRange(stk, a, m), nat(nil).mulRange(stk, m+1, b))
   266  }
   267  
   268  // A stack provides temporary storage for complex calculations
   269  // such as multiplication and division.
   270  // The stack is a simple slice of words, extended as needed
   271  // to hold all the temporary storage for a calculation.
   272  // In general, if a function takes a *stack, it expects a non-nil *stack.
   273  // However, certain functions may allow passing a nil *stack instead,
   274  // so that they can handle trivial stack-free cases without forcing the
   275  // caller to obtain and free a stack that will be unused. These functions
   276  // document that they accept a nil *stack in their doc comments.
   277  type stack struct {
   278  	w []Word
   279  }
   280  
   281  var stackPool sync.Pool
   282  
   283  // getStack returns a temporary stack.
   284  // The caller must call [stack.free] to give up use of the stack when finished.
   285  func getStack() *stack {
   286  	s, _ := stackPool.Get().(*stack)
   287  	if s == nil {
   288  		s = new(stack)
   289  	}
   290  	return s
   291  }
   292  
   293  // free returns the stack for use by another calculation.
   294  func (s *stack) free() {
   295  	s.w = s.w[:0]
   296  	stackPool.Put(s)
   297  }
   298  
   299  // save returns the current stack pointer.
   300  // A future call to restore with the same value
   301  // frees any temporaries allocated on the stack after the call to save.
   302  func (s *stack) save() int {
   303  	return len(s.w)
   304  }
   305  
   306  // restore restores the stack pointer to n.
   307  // It is almost always invoked as
   308  //
   309  //	defer stk.restore(stk.save())
   310  //
   311  // which makes sure to pop any temporaries allocated in the current function
   312  // from the stack before returning.
   313  func (s *stack) restore(n int) {
   314  	s.w = s.w[:n]
   315  }
   316  
   317  // nat returns a nat of n words, allocated on the stack.
   318  func (s *stack) nat(n int) nat {
   319  	nr := (n + 3) &^ 3 // round up to multiple of 4
   320  	off := len(s.w)
   321  	s.w = slices.Grow(s.w, nr)
   322  	s.w = s.w[:off+nr]
   323  	x := s.w[off : off+n : off+n]
   324  	if n > 0 {
   325  		x[0] = 0xfedcb
   326  	}
   327  	return x
   328  }
   329  
   330  // bitLen returns the length of x in bits.
   331  // Unlike most methods, it works even if x is not normalized.
   332  func (x nat) bitLen() int {
   333  	// This function is used in cryptographic operations. It must not leak
   334  	// anything but the Int's sign and bit size through side-channels. Any
   335  	// changes must be reviewed by a security expert.
   336  	if i := len(x) - 1; i >= 0 {
   337  		// bits.Len uses a lookup table for the low-order bits on some
   338  		// architectures. Neutralize any input-dependent behavior by setting all
   339  		// bits after the first one bit.
   340  		top := uint(x[i])
   341  		top |= top >> 1
   342  		top |= top >> 2
   343  		top |= top >> 4
   344  		top |= top >> 8
   345  		top |= top >> 16
   346  		top |= top >> 16 >> 16 // ">> 32" doesn't compile on 32-bit architectures
   347  		return i*_W + bits.Len(top)
   348  	}
   349  	return 0
   350  }
   351  
   352  // trailingZeroBits returns the number of consecutive least significant zero
   353  // bits of x.
   354  func (x nat) trailingZeroBits() uint {
   355  	if len(x) == 0 {
   356  		return 0
   357  	}
   358  	var i uint
   359  	for x[i] == 0 {
   360  		i++
   361  	}
   362  	// x[i] != 0
   363  	return i*_W + uint(bits.TrailingZeros(uint(x[i])))
   364  }
   365  
   366  // isPow2 returns i, true when x == 2**i and 0, false otherwise.
   367  func (x nat) isPow2() (uint, bool) {
   368  	var i uint
   369  	for x[i] == 0 {
   370  		i++
   371  	}
   372  	if i == uint(len(x))-1 && x[i]&(x[i]-1) == 0 {
   373  		return i*_W + uint(bits.TrailingZeros(uint(x[i]))), true
   374  	}
   375  	return 0, false
   376  }
   377  
   378  func same(x, y nat) bool {
   379  	return len(x) == len(y) && len(x) > 0 && &x[0] == &y[0]
   380  }
   381  
   382  // z = x << s
   383  func (z nat) lsh(x nat, s uint) nat {
   384  	if s == 0 {
   385  		if same(z, x) {
   386  			return z
   387  		}
   388  		if !alias(z, x) {
   389  			return z.set(x)
   390  		}
   391  	}
   392  
   393  	m := len(x)
   394  	if m == 0 {
   395  		return z[:0]
   396  	}
   397  	// m > 0
   398  
   399  	n := m + int(s/_W)
   400  	z = z.make(n + 1)
   401  	if s %= _W; s == 0 {
   402  		copy(z[n-m:n], x)
   403  		z[n] = 0
   404  	} else {
   405  		z[n] = lshVU(z[n-m:n], x, s)
   406  	}
   407  	clear(z[0 : n-m])
   408  
   409  	return z.norm()
   410  }
   411  
   412  // z = x >> s
   413  func (z nat) rsh(x nat, s uint) nat {
   414  	if s == 0 {
   415  		if same(z, x) {
   416  			return z
   417  		}
   418  		if !alias(z, x) {
   419  			return z.set(x)
   420  		}
   421  	}
   422  
   423  	m := len(x)
   424  	n := m - int(s/_W)
   425  	if n <= 0 {
   426  		return z[:0]
   427  	}
   428  	// n > 0
   429  
   430  	z = z.make(n)
   431  	if s %= _W; s == 0 {
   432  		copy(z, x[m-n:])
   433  	} else {
   434  		rshVU(z, x[m-n:], s)
   435  	}
   436  
   437  	return z.norm()
   438  }
   439  
   440  func (z nat) setBit(x nat, i uint, b uint) nat {
   441  	j := int(i / _W)
   442  	m := Word(1) << (i % _W)
   443  	n := len(x)
   444  	switch b {
   445  	case 0:
   446  		z = z.make(n)
   447  		copy(z, x)
   448  		if j >= n {
   449  			// no need to grow
   450  			return z
   451  		}
   452  		z[j] &^= m
   453  		return z.norm()
   454  	case 1:
   455  		if j >= n {
   456  			z = z.make(j + 1)
   457  			clear(z[n:])
   458  		} else {
   459  			z = z.make(n)
   460  		}
   461  		copy(z, x)
   462  		z[j] |= m
   463  		// no need to normalize
   464  		return z
   465  	}
   466  	panic("set bit is not 0 or 1")
   467  }
   468  
   469  // bit returns the value of the i'th bit, with lsb == bit 0.
   470  func (x nat) bit(i uint) uint {
   471  	j := i / _W
   472  	if j >= uint(len(x)) {
   473  		return 0
   474  	}
   475  	// 0 <= j < len(x)
   476  	return uint(x[j] >> (i % _W) & 1)
   477  }
   478  
   479  // sticky returns 1 if there's a 1 bit within the
   480  // i least significant bits, otherwise it returns 0.
   481  func (x nat) sticky(i uint) uint {
   482  	j := i / _W
   483  	if j >= uint(len(x)) {
   484  		if len(x) == 0 {
   485  			return 0
   486  		}
   487  		return 1
   488  	}
   489  	// 0 <= j < len(x)
   490  	for _, x := range x[:j] {
   491  		if x != 0 {
   492  			return 1
   493  		}
   494  	}
   495  	if x[j]<<(_W-i%_W) != 0 {
   496  		return 1
   497  	}
   498  	return 0
   499  }
   500  
   501  func (z nat) and(x, y nat) nat {
   502  	m := len(x)
   503  	n := len(y)
   504  	if m > n {
   505  		m = n
   506  	}
   507  	// m <= n
   508  
   509  	z = z.make(m)
   510  	for i := 0; i < m; i++ {
   511  		z[i] = x[i] & y[i]
   512  	}
   513  
   514  	return z.norm()
   515  }
   516  
   517  // trunc returns z = x mod 2ⁿ.
   518  func (z nat) trunc(x nat, n uint) nat {
   519  	w := (n + _W - 1) / _W
   520  	if uint(len(x)) < w {
   521  		return z.set(x)
   522  	}
   523  	z = z.make(int(w))
   524  	copy(z, x)
   525  	if n%_W != 0 {
   526  		z[len(z)-1] &= 1<<(n%_W) - 1
   527  	}
   528  	return z.norm()
   529  }
   530  
   531  func (z nat) andNot(x, y nat) nat {
   532  	m := len(x)
   533  	n := len(y)
   534  	if n > m {
   535  		n = m
   536  	}
   537  	// m >= n
   538  
   539  	z = z.make(m)
   540  	for i := 0; i < n; i++ {
   541  		z[i] = x[i] &^ y[i]
   542  	}
   543  	copy(z[n:m], x[n:m])
   544  
   545  	return z.norm()
   546  }
   547  
   548  func (z nat) or(x, y nat) nat {
   549  	m := len(x)
   550  	n := len(y)
   551  	s := x
   552  	if m < n {
   553  		n, m = m, n
   554  		s = y
   555  	}
   556  	// m >= n
   557  
   558  	z = z.make(m)
   559  	for i := 0; i < n; i++ {
   560  		z[i] = x[i] | y[i]
   561  	}
   562  	copy(z[n:m], s[n:m])
   563  
   564  	return z.norm()
   565  }
   566  
   567  func (z nat) xor(x, y nat) nat {
   568  	m := len(x)
   569  	n := len(y)
   570  	s := x
   571  	if m < n {
   572  		n, m = m, n
   573  		s = y
   574  	}
   575  	// m >= n
   576  
   577  	z = z.make(m)
   578  	for i := 0; i < n; i++ {
   579  		z[i] = x[i] ^ y[i]
   580  	}
   581  	copy(z[n:m], s[n:m])
   582  
   583  	return z.norm()
   584  }
   585  
   586  // random creates a random integer in [0..limit), using the space in z if
   587  // possible. n is the bit length of limit.
   588  func (z nat) random(rand *rand.Rand, limit nat, n int) nat {
   589  	if alias(z, limit) {
   590  		z = nil // z is an alias for limit - cannot reuse
   591  	}
   592  	z = z.make(len(limit))
   593  
   594  	bitLengthOfMSW := uint(n % _W)
   595  	if bitLengthOfMSW == 0 {
   596  		bitLengthOfMSW = _W
   597  	}
   598  	mask := Word((1 << bitLengthOfMSW) - 1)
   599  
   600  	for {
   601  		switch _W {
   602  		case 32:
   603  			for i := range z {
   604  				z[i] = Word(rand.Uint32())
   605  			}
   606  		case 64:
   607  			for i := range z {
   608  				z[i] = Word(rand.Uint32()) | Word(rand.Uint32())<<32
   609  			}
   610  		default:
   611  			panic("unknown word size")
   612  		}
   613  		z[len(limit)-1] &= mask
   614  		if z.cmp(limit) < 0 {
   615  			break
   616  		}
   617  	}
   618  
   619  	return z.norm()
   620  }
   621  
   622  // If m != 0 (i.e., len(m) != 0), expNN sets z to x**y mod m;
   623  // otherwise it sets z to x**y. The result is the value of z.
   624  // The caller may pass stk == nil to request that expNN obtain and release one itself.
   625  func (z nat) expNN(stk *stack, x, y, m nat, slow bool) nat {
   626  	if alias(z, x) || alias(z, y) {
   627  		// We cannot allow in-place modification of x or y.
   628  		z = nil
   629  	}
   630  
   631  	// x**y mod 1 == 0
   632  	if len(m) == 1 && m[0] == 1 {
   633  		return z.setWord(0)
   634  	}
   635  	// m == 0 || m > 1
   636  
   637  	// x**0 == 1
   638  	if len(y) == 0 {
   639  		return z.setWord(1)
   640  	}
   641  	// y > 0
   642  
   643  	// 0**y = 0
   644  	if len(x) == 0 {
   645  		return z.setWord(0)
   646  	}
   647  	// x > 0
   648  
   649  	// 1**y = 1
   650  	if len(x) == 1 && x[0] == 1 {
   651  		return z.setWord(1)
   652  	}
   653  	// x > 1
   654  
   655  	// x**1 == x
   656  	if len(y) == 1 && y[0] == 1 && len(m) == 0 {
   657  		return z.set(x)
   658  	}
   659  	if stk == nil {
   660  		stk = getStack()
   661  		defer stk.free()
   662  	}
   663  	if len(y) == 1 && y[0] == 1 { // len(m) > 0
   664  		return z.rem(stk, x, m)
   665  	}
   666  
   667  	// y > 1
   668  
   669  	if len(m) != 0 {
   670  		// We likely end up being as long as the modulus.
   671  		z = z.make(len(m))
   672  
   673  		// If the exponent is large, we use the Montgomery method for odd values,
   674  		// and a 4-bit, windowed exponentiation for powers of two,
   675  		// and a CRT-decomposed Montgomery method for the remaining values
   676  		// (even values times non-trivial odd values, which decompose into one
   677  		// instance of each of the first two cases).
   678  		if len(y) > 1 && !slow {
   679  			if m[0]&1 == 1 {
   680  				return z.expNNMontgomery(stk, x, y, m)
   681  			}
   682  			if logM, ok := m.isPow2(); ok {
   683  				return z.expNNWindowed(stk, x, y, logM)
   684  			}
   685  			return z.expNNMontgomeryEven(stk, x, y, m)
   686  		}
   687  	}
   688  
   689  	z = z.set(x)
   690  	v := y[len(y)-1] // v > 0 because y is normalized and y > 0
   691  	shift := nlz(v) + 1
   692  	v <<= shift
   693  	var q nat
   694  
   695  	const mask = 1 << (_W - 1)
   696  
   697  	// We walk through the bits of the exponent one by one. Each time we
   698  	// see a bit, we square, thus doubling the power. If the bit is a one,
   699  	// we also multiply by x, thus adding one to the power.
   700  
   701  	w := _W - int(shift)
   702  	// zz and r are used to avoid allocating in mul and div as
   703  	// otherwise the arguments would alias.
   704  	var zz, r nat
   705  	for j := 0; j < w; j++ {
   706  		zz = zz.sqr(stk, z)
   707  		zz, z = z, zz
   708  
   709  		if v&mask != 0 {
   710  			zz = zz.mul(stk, z, x)
   711  			zz, z = z, zz
   712  		}
   713  
   714  		if len(m) != 0 {
   715  			zz, r = zz.div(stk, r, z, m)
   716  			zz, r, q, z = q, z, zz, r
   717  		}
   718  
   719  		v <<= 1
   720  	}
   721  
   722  	for i := len(y) - 2; i >= 0; i-- {
   723  		v = y[i]
   724  
   725  		for j := 0; j < _W; j++ {
   726  			zz = zz.sqr(stk, z)
   727  			zz, z = z, zz
   728  
   729  			if v&mask != 0 {
   730  				zz = zz.mul(stk, z, x)
   731  				zz, z = z, zz
   732  			}
   733  
   734  			if len(m) != 0 {
   735  				zz, r = zz.div(stk, r, z, m)
   736  				zz, r, q, z = q, z, zz, r
   737  			}
   738  
   739  			v <<= 1
   740  		}
   741  	}
   742  
   743  	return z.norm()
   744  }
   745  
   746  // expNNMontgomeryEven calculates x**y mod m where m = m1 × m2 for m1 = 2ⁿ and m2 odd.
   747  // It uses two recursive calls to expNN for x**y mod m1 and x**y mod m2
   748  // and then uses the Chinese Remainder Theorem to combine the results.
   749  // The recursive call using m1 will use expNNWindowed,
   750  // while the recursive call using m2 will use expNNMontgomery.
   751  // For more details, see Ç. K. Koç, “Montgomery Reduction with Even Modulus”,
   752  // IEE Proceedings: Computers and Digital Techniques, 141(5) 314-316, September 1994.
   753  // http://www.people.vcu.edu/~jwang3/CMSC691/j34monex.pdf
   754  func (z nat) expNNMontgomeryEven(stk *stack, x, y, m nat) nat {
   755  	// Split m = m₁ × m₂ where m₁ = 2ⁿ
   756  	n := m.trailingZeroBits()
   757  	m1 := nat(nil).lsh(natOne, n)
   758  	m2 := nat(nil).rsh(m, n)
   759  
   760  	// We want z = x**y mod m.
   761  	// z₁ = x**y mod m1 = (x**y mod m) mod m1 = z mod m1
   762  	// z₂ = x**y mod m2 = (x**y mod m) mod m2 = z mod m2
   763  	// (We are using the math/big convention for names here,
   764  	// where the computation is z = x**y mod m, so its parts are z1 and z2.
   765  	// The paper is computing x = a**e mod n; it refers to these as x2 and z1.)
   766  	z1 := nat(nil).expNN(stk, x, y, m1, false)
   767  	z2 := nat(nil).expNN(stk, x, y, m2, false)
   768  
   769  	// Reconstruct z from z₁, z₂ using CRT, using algorithm from paper,
   770  	// which uses only a single modInverse (and an easy one at that).
   771  	//	p = (z₁ - z₂) × m₂⁻¹ (mod m₁)
   772  	//	z = z₂ + p × m₂
   773  	// The final addition is in range because:
   774  	//	z = z₂ + p × m₂
   775  	//	  ≤ z₂ + (m₁-1) × m₂
   776  	//	  < m₂ + (m₁-1) × m₂
   777  	//	  = m₁ × m₂
   778  	//	  = m.
   779  	z = z.set(z2)
   780  
   781  	// Compute (z₁ - z₂) mod m1 [m1 == 2**n] into z1.
   782  	z1 = z1.subMod2N(z1, z2, n)
   783  
   784  	// Reuse z2 for p = (z₁ - z₂) [in z1] * m2⁻¹ (mod m₁ [= 2ⁿ]).
   785  	m2inv := nat(nil).modInverse(m2, m1)
   786  	z2 = z2.mul(stk, z1, m2inv)
   787  	z2 = z2.trunc(z2, n)
   788  
   789  	// Reuse z1 for p * m2.
   790  	z = z.add(z, z1.mul(stk, z2, m2))
   791  
   792  	return z
   793  }
   794  
   795  // expNNWindowed calculates x**y mod m using a fixed, 4-bit window,
   796  // where m = 2**logM.
   797  func (z nat) expNNWindowed(stk *stack, x, y nat, logM uint) nat {
   798  	if len(y) <= 1 {
   799  		panic("big: misuse of expNNWindowed")
   800  	}
   801  	if x[0]&1 == 0 {
   802  		// len(y) > 1, so y  > logM.
   803  		// x is even, so x**y is a multiple of 2**y which is a multiple of 2**logM.
   804  		return z.setWord(0)
   805  	}
   806  	if logM == 1 {
   807  		return z.setWord(1)
   808  	}
   809  
   810  	// zz is used to avoid allocating in mul as otherwise
   811  	// the arguments would alias.
   812  	defer stk.restore(stk.save())
   813  	w := int((logM + _W - 1) / _W)
   814  	zz := stk.nat(w)
   815  
   816  	const n = 4
   817  	// powers[i] contains x^i.
   818  	var powers [1 << n]nat
   819  	for i := range powers {
   820  		powers[i] = stk.nat(w)
   821  	}
   822  	powers[0] = powers[0].set(natOne)
   823  	powers[1] = powers[1].trunc(x, logM)
   824  	for i := 2; i < 1<<n; i += 2 {
   825  		p2, p, p1 := &powers[i/2], &powers[i], &powers[i+1]
   826  		*p = p.sqr(stk, *p2)
   827  		*p = p.trunc(*p, logM)
   828  		*p1 = p1.mul(stk, *p, x)
   829  		*p1 = p1.trunc(*p1, logM)
   830  	}
   831  
   832  	// Because phi(2**logM) = 2**(logM-1), x**(2**(logM-1)) = 1,
   833  	// so we can compute x**(y mod 2**(logM-1)) instead of x**y.
   834  	// That is, we can throw away all but the bottom logM-1 bits of y.
   835  	// Instead of allocating a new y, we start reading y at the right word
   836  	// and truncate it appropriately at the start of the loop.
   837  	i := len(y) - 1
   838  	mtop := int((logM - 2) / _W) // -2 because the top word of N bits is the (N-1)/W'th word.
   839  	mmask := ^Word(0)
   840  	if mbits := (logM - 1) & (_W - 1); mbits != 0 {
   841  		mmask = (1 << mbits) - 1
   842  	}
   843  	if i > mtop {
   844  		i = mtop
   845  	}
   846  	advance := false
   847  	z = z.setWord(1)
   848  	for ; i >= 0; i-- {
   849  		yi := y[i]
   850  		if i == mtop {
   851  			yi &= mmask
   852  		}
   853  		for j := 0; j < _W; j += n {
   854  			if advance {
   855  				// Account for use of 4 bits in previous iteration.
   856  				// Unrolled loop for significant performance
   857  				// gain. Use go test -bench=".*" in crypto/rsa
   858  				// to check performance before making changes.
   859  				zz = zz.sqr(stk, z)
   860  				zz, z = z, zz
   861  				z = z.trunc(z, logM)
   862  
   863  				zz = zz.sqr(stk, z)
   864  				zz, z = z, zz
   865  				z = z.trunc(z, logM)
   866  
   867  				zz = zz.sqr(stk, z)
   868  				zz, z = z, zz
   869  				z = z.trunc(z, logM)
   870  
   871  				zz = zz.sqr(stk, z)
   872  				zz, z = z, zz
   873  				z = z.trunc(z, logM)
   874  			}
   875  
   876  			zz = zz.mul(stk, z, powers[yi>>(_W-n)])
   877  			zz, z = z, zz
   878  			z = z.trunc(z, logM)
   879  
   880  			yi <<= n
   881  			advance = true
   882  		}
   883  	}
   884  
   885  	return z.norm()
   886  }
   887  
   888  // expNNMontgomery calculates x**y mod m using a fixed, 4-bit window.
   889  // Uses Montgomery representation.
   890  func (z nat) expNNMontgomery(stk *stack, x, y, m nat) nat {
   891  	numWords := len(m)
   892  
   893  	// We want the lengths of x and m to be equal.
   894  	// It is OK if x >= m as long as len(x) == len(m).
   895  	if len(x) > numWords {
   896  		_, x = nat(nil).div(stk, nil, x, m)
   897  		// Note: now len(x) <= numWords, not guaranteed ==.
   898  	}
   899  	if len(x) < numWords {
   900  		rr := make(nat, numWords)
   901  		copy(rr, x)
   902  		x = rr
   903  	}
   904  
   905  	// Ideally the precomputations would be performed outside, and reused
   906  	// k0 = -m**-1 mod 2**_W. Algorithm from: Dumas, J.G. "On Newton–Raphson
   907  	// Iteration for Multiplicative Inverses Modulo Prime Powers".
   908  	k0 := 2 - m[0]
   909  	t := m[0] - 1
   910  	for i := 1; i < _W; i <<= 1 {
   911  		t *= t
   912  		k0 *= (t + 1)
   913  	}
   914  	k0 = -k0
   915  
   916  	// RR = 2**(2*_W*len(m)) mod m
   917  	RR := nat(nil).setWord(1)
   918  	zz := nat(nil).lsh(RR, uint(2*numWords*_W))
   919  	_, RR = nat(nil).div(stk, RR, zz, m)
   920  	if len(RR) < numWords {
   921  		zz = zz.make(numWords)
   922  		copy(zz, RR)
   923  		RR = zz
   924  	}
   925  	// one = 1, with equal length to that of m
   926  	one := make(nat, numWords)
   927  	one[0] = 1
   928  
   929  	const n = 4
   930  	// powers[i] contains x^i
   931  	var powers [1 << n]nat
   932  	powers[0] = powers[0].montgomery(one, RR, m, k0, numWords)
   933  	powers[1] = powers[1].montgomery(x, RR, m, k0, numWords)
   934  	for i := 2; i < 1<<n; i++ {
   935  		powers[i] = powers[i].montgomery(powers[i-1], powers[1], m, k0, numWords)
   936  	}
   937  
   938  	// initialize z = 1 (Montgomery 1)
   939  	z = z.make(numWords)
   940  	copy(z, powers[0])
   941  
   942  	zz = zz.make(numWords)
   943  
   944  	// same windowed exponent, but with Montgomery multiplications
   945  	for i := len(y) - 1; i >= 0; i-- {
   946  		yi := y[i]
   947  		for j := 0; j < _W; j += n {
   948  			if i != len(y)-1 || j != 0 {
   949  				zz = zz.montgomery(z, z, m, k0, numWords)
   950  				z = z.montgomery(zz, zz, m, k0, numWords)
   951  				zz = zz.montgomery(z, z, m, k0, numWords)
   952  				z = z.montgomery(zz, zz, m, k0, numWords)
   953  			}
   954  			zz = zz.montgomery(z, powers[yi>>(_W-n)], m, k0, numWords)
   955  			z, zz = zz, z
   956  			yi <<= n
   957  		}
   958  	}
   959  	// convert to regular number
   960  	zz = zz.montgomery(z, one, m, k0, numWords)
   961  
   962  	// One last reduction, just in case.
   963  	// See golang.org/issue/13907.
   964  	if zz.cmp(m) >= 0 {
   965  		// Common case is m has high bit set; in that case,
   966  		// since zz is the same length as m, there can be just
   967  		// one multiple of m to remove. Just subtract.
   968  		// We think that the subtract should be sufficient in general,
   969  		// so do that unconditionally, but double-check,
   970  		// in case our beliefs are wrong.
   971  		// The div is not expected to be reached.
   972  		zz = zz.sub(zz, m)
   973  		if zz.cmp(m) >= 0 {
   974  			_, zz = nat(nil).div(stk, nil, zz, m)
   975  		}
   976  	}
   977  
   978  	return zz.norm()
   979  }
   980  
   981  // bytes writes the value of z into buf using big-endian encoding.
   982  // The value of z is encoded in the slice buf[i:]. If the value of z
   983  // cannot be represented in buf, bytes panics. The number i of unused
   984  // bytes at the beginning of buf is returned as result.
   985  func (z nat) bytes(buf []byte) (i int) {
   986  	// This function is used in cryptographic operations. It must not leak
   987  	// anything but the Int's sign and bit size through side-channels. Any
   988  	// changes must be reviewed by a security expert.
   989  	i = len(buf)
   990  	for _, d := range z {
   991  		for j := 0; j < _S; j++ {
   992  			i--
   993  			if i >= 0 {
   994  				buf[i] = byte(d)
   995  			} else if byte(d) != 0 {
   996  				panic("math/big: buffer too small to fit value")
   997  			}
   998  			d >>= 8
   999  		}
  1000  	}
  1001  
  1002  	if i < 0 {
  1003  		i = 0
  1004  	}
  1005  	for i < len(buf) && buf[i] == 0 {
  1006  		i++
  1007  	}
  1008  
  1009  	return
  1010  }
  1011  
  1012  // bigEndianWord returns the contents of buf interpreted as a big-endian encoded Word value.
  1013  func bigEndianWord(buf []byte) Word {
  1014  	if _W == 64 {
  1015  		return Word(byteorder.BEUint64(buf))
  1016  	}
  1017  	return Word(byteorder.BEUint32(buf))
  1018  }
  1019  
  1020  // setBytes interprets buf as the bytes of a big-endian unsigned
  1021  // integer, sets z to that value, and returns z.
  1022  func (z nat) setBytes(buf []byte) nat {
  1023  	z = z.make((len(buf) + _S - 1) / _S)
  1024  
  1025  	i := len(buf)
  1026  	for k := 0; i >= _S; k++ {
  1027  		z[k] = bigEndianWord(buf[i-_S : i])
  1028  		i -= _S
  1029  	}
  1030  	if i > 0 {
  1031  		var d Word
  1032  		for s := uint(0); i > 0; s += 8 {
  1033  			d |= Word(buf[i-1]) << s
  1034  			i--
  1035  		}
  1036  		z[len(z)-1] = d
  1037  	}
  1038  
  1039  	return z.norm()
  1040  }
  1041  
  1042  // sqrt sets z = ⌊√x⌋
  1043  // The caller may pass stk == nil to request that sqrt obtain and release one itself.
  1044  func (z nat) sqrt(stk *stack, x nat) nat {
  1045  	if x.cmp(natOne) <= 0 {
  1046  		return z.set(x)
  1047  	}
  1048  	if alias(z, x) {
  1049  		z = nil
  1050  	}
  1051  
  1052  	if stk == nil {
  1053  		stk = getStack()
  1054  		defer stk.free()
  1055  	}
  1056  
  1057  	// Start with value known to be too large and repeat "z = ⌊(z + ⌊x/z⌋)/2⌋" until it stops getting smaller.
  1058  	// See Brent and Zimmermann, Modern Computer Arithmetic, Algorithm 1.13 (SqrtInt).
  1059  	// https://members.loria.fr/PZimmermann/mca/pub226.html
  1060  	// If x is one less than a perfect square, the sequence oscillates between the correct z and z+1;
  1061  	// otherwise it converges to the correct z and stays there.
  1062  	var z1, z2 nat
  1063  	z1 = z
  1064  	z1 = z1.setUint64(1)
  1065  	z1 = z1.lsh(z1, uint(x.bitLen()+1)/2) // must be ≥ √x
  1066  	for n := 0; ; n++ {
  1067  		z2, _ = z2.div(stk, nil, x, z1)
  1068  		z2 = z2.add(z2, z1)
  1069  		z2 = z2.rsh(z2, 1)
  1070  		if z2.cmp(z1) >= 0 {
  1071  			// z1 is answer.
  1072  			// Figure out whether z1 or z2 is currently aliased to z by looking at loop count.
  1073  			if n&1 == 0 {
  1074  				return z1
  1075  			}
  1076  			return z.set(z1)
  1077  		}
  1078  		z1, z2 = z2, z1
  1079  	}
  1080  }
  1081  
  1082  // subMod2N returns z = (x - y) mod 2ⁿ.
  1083  func (z nat) subMod2N(x, y nat, n uint) nat {
  1084  	if uint(x.bitLen()) > n {
  1085  		if alias(z, x) {
  1086  			// ok to overwrite x in place
  1087  			x = x.trunc(x, n)
  1088  		} else {
  1089  			x = nat(nil).trunc(x, n)
  1090  		}
  1091  	}
  1092  	if uint(y.bitLen()) > n {
  1093  		if alias(z, y) {
  1094  			// ok to overwrite y in place
  1095  			y = y.trunc(y, n)
  1096  		} else {
  1097  			y = nat(nil).trunc(y, n)
  1098  		}
  1099  	}
  1100  	if x.cmp(y) >= 0 {
  1101  		return z.sub(x, y)
  1102  	}
  1103  	// x - y < 0; x - y mod 2ⁿ = x - y + 2ⁿ = 2ⁿ - (y - x) = 1 + 2ⁿ-1 - (y - x) = 1 + ^(y - x).
  1104  	z = z.sub(y, x)
  1105  	for uint(len(z))*_W < n {
  1106  		z = append(z, 0)
  1107  	}
  1108  	for i := range z {
  1109  		z[i] = ^z[i]
  1110  	}
  1111  	z = z.trunc(z, n)
  1112  	return z.add(z, natOne)
  1113  }
  1114
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