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