text: // Copyright 2018 The Go Authors. All rights reserved.
  // Use of this source code is governed by a BSD-style
  // license that can be found in the LICENSE file.
  
  // This file provides the generic implementation of Sum and MAC. Other files
  // might provide optimized assembly implementations of some of this code.
  
  package poly1305
  
  import "encoding/binary"
  
  // Poly1305 [RFC 7539] is a relatively simple algorithm: the authentication tag
  // for a 64 bytes message is approximately
  //
  //     s + m[0:16] * r⁴ + m[16:32] * r³ + m[32:48] * r² + m[48:64] * r  mod  2¹³⁰ - 5
  //
  // for some secret r and s. It can be computed sequentially like
  //
  //     for len(msg) > 0:
  //         h += read(msg, 16)
  //         h *= r
  //         h %= 2¹³⁰ - 5
  //     return h + s
  //
  // All the complexity is about doing performant constant-time math on numbers
  // larger than any available numeric type.
  
  func sumGeneric(out *[TagSize]byte, msg []byte, key *[32]byte) {
  	h := newMACGeneric(key)
  	h.Write(msg)
  	h.Sum(out)
  }
  
  func newMACGeneric(key *[32]byte) (h macGeneric) {
  	initialize(key, &h.r, &h.s)
  	return
  }
  
  // macState holds numbers in saturated 64-bit little-endian limbs. That is,
  // the value of [x0, x1, x2] is x[0] + x[1] * 2⁶⁴ + x[2] * 2¹²⁸.
  type macState struct {
  	// h is the main accumulator. It is to be interpreted modulo 2¹³⁰ - 5, but
  	// can grow larger during and after rounds.
  	h [3]uint64
  	// r and s are the private key components.
  	r [2]uint64
  	s [2]uint64
  }
  
  type macGeneric struct {
  	macState
  
  	buffer [TagSize]byte
  	offset int
  }
  
  // Write splits the incoming message into TagSize chunks, and passes them to
  // update. It buffers incomplete chunks.
  func (h *macGeneric) Write(p []byte) (int, error) {
  	nn := len(p)
  	if h.offset > 0 {
  		n := copy(h.buffer[h.offset:], p)
  		if h.offset+n < TagSize {
  			h.offset += n
  			return nn, nil
  		}
  		p = p[n:]
  		h.offset = 0
  		updateGeneric(&h.macState, h.buffer[:])
  	}
  	if n := len(p) - (len(p) % TagSize); n > 0 {
  		updateGeneric(&h.macState, p[:n])
  		p = p[n:]
  	}
  	if len(p) > 0 {
  		h.offset += copy(h.buffer[h.offset:], p)
  	}
  	return nn, nil
  }
  
  // Sum flushes the last incomplete chunk from the buffer, if any, and generates
  // the MAC output. It does not modify its state, in order to allow for multiple
  // calls to Sum, even if no Write is allowed after Sum.
  func (h *macGeneric) Sum(out *[TagSize]byte) {
  	state := h.macState
  	if h.offset > 0 {
  		updateGeneric(&state, h.buffer[:h.offset])
  	}
  	finalize(out, &state.h, &state.s)
  }
  
  // [rMask0, rMask1] is the specified Poly1305 clamping mask in little-endian. It
  // clears some bits of the secret coefficient to make it possible to implement
  // multiplication more efficiently.
  const (
  	rMask0 = 0x0FFFFFFC0FFFFFFF
  	rMask1 = 0x0FFFFFFC0FFFFFFC
  )
  
  func initialize(key *[32]byte, r, s *[2]uint64) {
  	r[0] = binary.LittleEndian.Uint64(key[0:8]) & rMask0
  	r[1] = binary.LittleEndian.Uint64(key[8:16]) & rMask1
  	s[0] = binary.LittleEndian.Uint64(key[16:24])
  	s[1] = binary.LittleEndian.Uint64(key[24:32])
  }
  
  // uint128 holds a 128-bit number as two 64-bit limbs, for use with the
  // bits.Mul64 and bits.Add64 intrinsics.
  type uint128 struct {
  	lo, hi uint64
  }
  
  func mul64(a, b uint64) uint128 {
  	hi, lo := bitsMul64(a, b)
  	return uint128{lo, hi}
  }
  
  func add128(a, b uint128) uint128 {
  	lo, c := bitsAdd64(a.lo, b.lo, 0)
  	hi, c := bitsAdd64(a.hi, b.hi, c)
  	if c != 0 {
  		panic("poly1305: unexpected overflow")
  	}
  	return uint128{lo, hi}
  }
  
  func shiftRightBy2(a uint128) uint128 {
  	a.lo = a.lo>>2 | (a.hi&3)<<62
  	a.hi = a.hi >> 2
  	return a
  }
  
  // updateGeneric absorbs msg into the state.h accumulator. For each chunk m of
  // 128 bits of message, it computes
  //
  //     h₊ = (h + m) * r  mod  2¹³⁰ - 5
  //
  // If the msg length is not a multiple of TagSize, it assumes the last
  // incomplete chunk is the final one.
  func updateGeneric(state *macState, msg []byte) {
  	h0, h1, h2 := state.h[0], state.h[1], state.h[2]
  	r0, r1 := state.r[0], state.r[1]
  
  	for len(msg) > 0 {
  		var c uint64
  
  		// For the first step, h + m, we use a chain of bits.Add64 intrinsics.
  		// The resulting value of h might exceed 2¹³⁰ - 5, but will be partially
  		// reduced at the end of the multiplication below.
  		//
  		// The spec requires us to set a bit just above the message size, not to
  		// hide leading zeroes. For full chunks, that's 1 << 128, so we can just
  		// add 1 to the most significant (2¹²⁸) limb, h2.
  		if len(msg) >= TagSize {
  			h0, c = bitsAdd64(h0, binary.LittleEndian.Uint64(msg[0:8]), 0)
  			h1, c = bitsAdd64(h1, binary.LittleEndian.Uint64(msg[8:16]), c)
  			h2 += c + 1
  
  			msg = msg[TagSize:]
  		} else {
  			var buf [TagSize]byte
  			copy(buf[:], msg)
  			buf[len(msg)] = 1
  
  			h0, c = bitsAdd64(h0, binary.LittleEndian.Uint64(buf[0:8]), 0)
  			h1, c = bitsAdd64(h1, binary.LittleEndian.Uint64(buf[8:16]), c)
  			h2 += c
  
  			msg = nil
  		}
  
  		// Multiplication of big number limbs is similar to elementary school
  		// columnar multiplication. Instead of digits, there are 64-bit limbs.
  		//
  		// We are multiplying a 3 limbs number, h, by a 2 limbs number, r.
  		//
  		//                        h2    h1    h0  x
  		//                              r1    r0  =
  		//                       ----------------
  		//                      h2r0  h1r0  h0r0     <-- individual 128-bit products
  		//            +   h2r1  h1r1  h0r1
  		//               ------------------------
  		//                 m3    m2    m1    m0      <-- result in 128-bit overlapping limbs
  		//               ------------------------
  		//         m3.hi m2.hi m1.hi m0.hi           <-- carry propagation
  		//     +         m3.lo m2.lo m1.lo m0.lo
  		//        -------------------------------
  		//           t4    t3    t2    t1    t0      <-- final result in 64-bit limbs
  		//
  		// The main difference from pen-and-paper multiplication is that we do
  		// carry propagation in a separate step, as if we wrote two digit sums
  		// at first (the 128-bit limbs), and then carried the tens all at once.
  
  		h0r0 := mul64(h0, r0)
  		h1r0 := mul64(h1, r0)
  		h2r0 := mul64(h2, r0)
  		h0r1 := mul64(h0, r1)
  		h1r1 := mul64(h1, r1)
  		h2r1 := mul64(h2, r1)
  
  		// Since h2 is known to be at most 7 (5 + 1 + 1), and r0 and r1 have their
  		// top 4 bits cleared by rMask{0,1}, we know that their product is not going
  		// to overflow 64 bits, so we can ignore the high part of the products.
  		//
  		// This also means that the product doesn't have a fifth limb (t4).
  		if h2r0.hi != 0 {
  			panic("poly1305: unexpected overflow")
  		}
  		if h2r1.hi != 0 {
  			panic("poly1305: unexpected overflow")
  		}
  
  		m0 := h0r0
  		m1 := add128(h1r0, h0r1) // These two additions don't overflow thanks again
  		m2 := add128(h2r0, h1r1) // to the 4 masked bits at the top of r0 and r1.
  		m3 := h2r1
  
  		t0 := m0.lo
  		t1, c := bitsAdd64(m1.lo, m0.hi, 0)
  		t2, c := bitsAdd64(m2.lo, m1.hi, c)
  		t3, _ := bitsAdd64(m3.lo, m2.hi, c)
  
  		// Now we have the result as 4 64-bit limbs, and we need to reduce it
  		// modulo 2¹³⁰ - 5. The special shape of this Crandall prime lets us do
  		// a cheap partial reduction according to the reduction identity
  		//
  		//     c * 2¹³⁰ + n  =  c * 5 + n  mod  2¹³⁰ - 5
  		//
  		// because 2¹³⁰ = 5 mod 2¹³⁰ - 5. Partial reduction since the result is
  		// likely to be larger than 2¹³⁰ - 5, but still small enough to fit the
  		// assumptions we make about h in the rest of the code.
  		//
  		// See also https://speakerdeck.com/gtank/engineering-prime-numbers?slide=23
  
  		// We split the final result at the 2¹³⁰ mark into h and cc, the carry.
  		// Note that the carry bits are effectively shifted left by 2, in other
  		// words, cc = c * 4 for the c in the reduction identity.
  		h0, h1, h2 = t0, t1, t2&maskLow2Bits
  		cc := uint128{t2 & maskNotLow2Bits, t3}
  
  		// To add c * 5 to h, we first add cc = c * 4, and then add (cc >> 2) = c.
  
  		h0, c = bitsAdd64(h0, cc.lo, 0)
  		h1, c = bitsAdd64(h1, cc.hi, c)
  		h2 += c
  
  		cc = shiftRightBy2(cc)
  
  		h0, c = bitsAdd64(h0, cc.lo, 0)
  		h1, c = bitsAdd64(h1, cc.hi, c)
  		h2 += c
  
  		// h2 is at most 3 + 1 + 1 = 5, making the whole of h at most
  		//
  		//     5 * 2¹²⁸ + (2¹²⁸ - 1) = 6 * 2¹²⁸ - 1
  	}
  
  	state.h[0], state.h[1], state.h[2] = h0, h1, h2
  }
  
  const (
  	maskLow2Bits    uint64 = 0x0000000000000003
  	maskNotLow2Bits uint64 = ^maskLow2Bits
  )
  
  // select64 returns x if v == 1 and y if v == 0, in constant time.
  func select64(v, x, y uint64) uint64 { return ^(v-1)&x | (v-1)&y }
  
  // [p0, p1, p2] is 2¹³⁰ - 5 in little endian order.
  const (
  	p0 = 0xFFFFFFFFFFFFFFFB
  	p1 = 0xFFFFFFFFFFFFFFFF
  	p2 = 0x0000000000000003
  )
  
  // finalize completes the modular reduction of h and computes
  //
  //     out = h + s  mod  2¹²⁸
  //
  func finalize(out *[TagSize]byte, h *[3]uint64, s *[2]uint64) {
  	h0, h1, h2 := h[0], h[1], h[2]
  
  	// After the partial reduction in updateGeneric, h might be more than
  	// 2¹³⁰ - 5, but will be less than 2 * (2¹³⁰ - 5). To complete the reduction
  	// in constant time, we compute t = h - (2¹³⁰ - 5), and select h as the
  	// result if the subtraction underflows, and t otherwise.
  
  	hMinusP0, b := bitsSub64(h0, p0, 0)
  	hMinusP1, b := bitsSub64(h1, p1, b)
  	_, b = bitsSub64(h2, p2, b)
  
  	// h = h if h < p else h - p
  	h0 = select64(b, h0, hMinusP0)
  	h1 = select64(b, h1, hMinusP1)
  
  	// Finally, we compute the last Poly1305 step
  	//
  	//     tag = h + s  mod  2¹²⁸
  	//
  	// by just doing a wide addition with the 128 low bits of h and discarding
  	// the overflow.
  	h0, c := bitsAdd64(h0, s[0], 0)
  	h1, _ = bitsAdd64(h1, s[1], c)
  
  	binary.LittleEndian.PutUint64(out[0:8], h0)
  	binary.LittleEndian.PutUint64(out[8:16], h1)
  }
  