path: root/vendor/github.com/btcsuite/btcd/btcec/field.go

// Copyright (c) 2013-2016 The btcsuite developers
// Copyright (c) 2013-2016 Dave Collins
// Use of this source code is governed by an ISC
// license that can be found in the LICENSE file.

package btcec

// References:
//   [HAC]: Handbook of Applied Cryptography Menezes, van Oorschot, Vanstone.
//     http://cacr.uwaterloo.ca/hac/

// All elliptic curve operations for secp256k1 are done in a finite field
// characterized by a 256-bit prime.  Given this precision is larger than the
// biggest available native type, obviously some form of bignum math is needed.
// This package implements specialized fixed-precision field arithmetic rather
// than relying on an arbitrary-precision arithmetic package such as math/big
// for dealing with the field math since the size is known.  As a result, rather
// large performance gains are achieved by taking advantage of many
// optimizations not available to arbitrary-precision arithmetic and generic
// modular arithmetic algorithms.
//
// There are various ways to internally represent each finite field element.
// For example, the most obvious representation would be to use an array of 4
// uint64s (64 bits * 4 = 256 bits).  However, that representation suffers from
// a couple of issues.  First, there is no native Go type large enough to handle
// the intermediate results while adding or multiplying two 64-bit numbers, and
// second there is no space left for overflows when performing the intermediate
// arithmetic between each array element which would lead to expensive carry
// propagation.
//
// Given the above, this implementation represents the the field elements as
// 10 uint32s with each word (array entry) treated as base 2^26.  This was
// chosen for the following reasons:
// 1) Most systems at the current time are 64-bit (or at least have 64-bit
//    registers available for specialized purposes such as MMX) so the
//    intermediate results can typically be done using a native register (and
//    using uint64s to avoid the need for additional half-word arithmetic)
// 2) In order to allow addition of the internal words without having to
//    propagate the the carry, the max normalized value for each register must
//    be less than the number of bits available in the register
// 3) Since we're dealing with 32-bit values, 64-bits of overflow is a
//    reasonable choice for #2
// 4) Given the need for 256-bits of precision and the properties stated in #1,
//    #2, and #3, the representation which best accommodates this is 10 uint32s
//    with base 2^26 (26 bits * 10 = 260 bits, so the final word only needs 22
//    bits) which leaves the desired 64 bits (32 * 10 = 320, 320 - 256 = 64) for
//    overflow
//
// Since it is so important that the field arithmetic is extremely fast for
// high performance crypto, this package does not perform any validation where
// it ordinarily would.  For example, some functions only give the correct
// result is the field is normalized and there is no checking to ensure it is.
// While I typically prefer to ensure all state and input is valid for most
// packages, this code is really only used internally and every extra check
// counts.

import (
    "encoding/hex"
)

// Constants used to make the code more readable.
const (
    twoBitsMask   = 0x3
    fourBitsMask  = 0xf
    sixBitsMask   = 0x3f
    eightBitsMask = 0xff
)

// Constants related to the field representation.
const (
    // fieldWords is the number of words used to internally represent the
    // 256-bit value.
    fieldWords = 10

    // fieldBase is the exponent used to form the numeric base of each word.
    // 2^(fieldBase*i) where i is the word position.
    fieldBase = 26

    // fieldOverflowBits is the minimum number of "overflow" bits for each
    // word in the field value.
    fieldOverflowBits = 32 - fieldBase

    // fieldBaseMask is the mask for the bits in each word needed to
    // represent the numeric base of each word (except the most significant
    // word).
    fieldBaseMask = (1 << fieldBase) - 1

    // fieldMSBBits is the number of bits in the most significant word used
    // to represent the value.
    fieldMSBBits = 256 - (fieldBase * (fieldWords - 1))

    // fieldMSBMask is the mask for the bits in the most significant word
    // needed to represent the value.
    fieldMSBMask = (1 << fieldMSBBits) - 1

    // fieldPrimeWordZero is word zero of the secp256k1 prime in the
    // internal field representation.  It is used during negation.
    fieldPrimeWordZero = 0x3fffc2f

    // fieldPrimeWordOne is word one of the secp256k1 prime in the
    // internal field representation.  It is used during negation.
    fieldPrimeWordOne = 0x3ffffbf

    // primeLowBits is the lower 2*fieldBase bits of the secp256k1 prime in
    // its standard normalized form.  It is used during modular reduction.
    primeLowBits = 0xffffefffffc2f
)

// fieldVal implements optimized fixed-precision arithmetic over the
// secp256k1 finite field.  This means all arithmetic is performed modulo
// 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f.  It
// represents each 256-bit value as 10 32-bit integers in base 2^26.  This
// provides 6 bits of overflow in each word (10 bits in the most significant
// word) for a total of 64 bits of overflow (9*6 + 10 = 64).  It only implements
// the arithmetic needed for elliptic curve operations.
//
// The following depicts the internal representation:
//   -----------------------------------------------------------------
//  |        n[9]       |        n[8]       | ... |        n[0]       |
//  | 32 bits available | 32 bits available | ... | 32 bits available |
//  | 22 bits for value | 26 bits for value | ... | 26 bits for value |
//  | 10 bits overflow  |  6 bits overflow  | ... |  6 bits overflow  |
//  | Mult: 2^(26*9)    | Mult: 2^(26*8)    | ... | Mult: 2^(26*0)    |
//   -----------------------------------------------------------------
//
// For example, consider the number 2^49 + 1.  It would be represented as:
//  n[0] = 1
//  n[1] = 2^23
//  n[2..9] = 0
//
// The full 256-bit value is then calculated by looping i from 9..0 and
// doing sum(n[i] * 2^(26i)) like so:
//  n[9] * 2^(26*9) = 0    * 2^234 = 0
//  n[8] * 2^(26*8) = 0    * 2^208 = 0
//  ...
//  n[1] * 2^(26*1) = 2^23 * 2^26  = 2^49
//  n[0] * 2^(26*0) = 1    * 2^0   = 1
//  Sum: 0 + 0 + ... + 2^49 + 1 = 2^49 + 1
type fieldVal struct {
    n [10]uint32
}

// String returns the field value as a human-readable hex string.
func (f fieldVal) String() string {
    t := new(fieldVal).Set(&f).Normalize()
    return hex.EncodeToString(t.Bytes()[:])
}

// Zero sets the field value to zero.  A newly created field value is already
// set to zero.  This function can be useful to clear an existing field value
// for reuse.
func (f *fieldVal) Zero() {
    f.n[0] = 0
    f.n[1] = 0
    f.n[2] = 0
    f.n[3] = 0
    f.n[4] = 0
    f.n[5] = 0
    f.n[6] = 0
    f.n[7] = 0
    f.n[8] = 0
    f.n[9] = 0
}

// Set sets the field value equal to the passed value.
//
// The field value is returned to support chaining.  This enables syntax like:
// f := new(fieldVal).Set(f2).Add(1) so that f = f2 + 1 where f2 is not
// modified.
func (f *fieldVal) Set(val *fieldVal) *fieldVal {
    *f = *val
    return f
}

// SetInt sets the field value to the passed integer.  This is a convenience
// function since it is fairly common to perform some arithemetic with small
// native integers.
//
// The field value is returned to support chaining.  This enables syntax such
// as f := new(fieldVal).SetInt(2).Mul(f2) so that f = 2 * f2.
func (f *fieldVal) SetInt(ui uint) *fieldVal {
    f.Zero()
    f.n[0] = uint32(ui)
    return f
}

// SetBytes packs the passed 32-byte big-endian value into the internal field
// value representation.
//
// The field value is returned to support chaining.  This enables syntax like:
// f := new(fieldVal).SetBytes(byteArray).Mul(f2) so that f = ba * f2.
func (f *fieldVal) SetBytes(b *[32]byte) *fieldVal {
    // Pack the 256 total bits across the 10 uint32 words with a max of
    // 26-bits per word.  This could be done with a couple of for loops,
    // but this unrolled version is significantly faster.  Benchmarks show
    // this is about 34 times faster than the variant which uses loops.
    f.n[0] = uint32(b[31]) | uint32(b[30])<<8 | uint32(b[29])<<16 |
        (uint32(b[28])&twoBitsMask)<<24
    f.n[1] = uint32(b[28])>>2 | uint32(b[27])<<6 | uint32(b[26])<<14 |
        (uint32(b[25])&fourBitsMask)<<22
    f.n[2] = uint32(b[25])>>4 | uint32(b[24])<<4 | uint32(b[23])<<12 |
        (uint32(b[22])&sixBitsMask)<<20
    f.n[3] = uint32(b[22])>>6 | uint32(b[21])<<2 | uint32(b[20])<<10 |
        uint32(b[19])<<18
    f.n[4] = uint32(b[18]) | uint32(b[17])<<8 | uint32(b[16])<<16 |
        (uint32(b[15])&twoBitsMask)<<24
    f.n[5] = uint32(b[15])>>2 | uint32(b[14])<<6 | uint32(b[13])<<14 |
        (uint32(b[12])&fourBitsMask)<<22
    f.n[6] = uint32(b[12])>>4 | uint32(b[11])<<4 | uint32(b[10])<<12 |
        (uint32(b[9])&sixBitsMask)<<20
    f.n[7] = uint32(b[9])>>6 | uint32(b[8])<<2 | uint32(b[7])<<10 |
        uint32(b[6])<<18
    f.n[8] = uint32(b[5]) | uint32(b[4])<<8 | uint32(b[3])<<16 |
        (uint32(b[2])&twoBitsMask)<<24
    f.n[9] = uint32(b[2])>>2 | uint32(b[1])<<6 | uint32(b[0])<<14
    return f
}

// SetByteSlice packs the passed big-endian value into the internal field value
// representation.  Only the first 32-bytes are used.  As a result, it is up to
// the caller to ensure numbers of the appropriate size are used or the value
// will be truncated.
//
// The field value is returned to support chaining.  This enables syntax like:
// f := new(fieldVal).SetByteSlice(byteSlice)
func (f *fieldVal) SetByteSlice(b []byte) *fieldVal {
    var b32 [32]byte
    for i := 0; i < len(b); i++ {
        if i < 32 {
            b32[i+(32-len(b))] = b[i]
        }
    }
    return f.SetBytes(&b32)
}

// SetHex decodes the passed big-endian hex string into the internal field value
// representation.  Only the first 32-bytes are used.
//
// The field value is returned to support chaining.  This enables syntax like:
// f := new(fieldVal).SetHex("0abc").Add(1) so that f = 0x0abc + 1
func (f *fieldVal) SetHex(hexString string) *fieldVal {
    if len(hexString)%2 != 0 {
        hexString = "0" + hexString
    }
    bytes, _ := hex.DecodeString(hexString)
    return f.SetByteSlice(bytes)
}

// Normalize normalizes the internal field words into the desired range and
// performs fast modular reduction over the secp256k1 prime by making use of the
// special form of the prime.
func (f *fieldVal) Normalize() *fieldVal {
    // The field representation leaves 6 bits of overflow in each
    // word so intermediate calculations can be performed without needing
    // to propagate the carry to each higher word during the calculations.
    // In order to normalize, first we need to "compact" the full 256-bit
    // value to the right and treat the additional 64 leftmost bits as
    // the magnitude.
    m := f.n[0]
    t0 := m & fieldBaseMask
    m = (m >> fieldBase) + f.n[1]
    t1 := m & fieldBaseMask
    m = (m >> fieldBase) + f.n[2]
    t2 := m & fieldBaseMask
    m = (m >> fieldBase) + f.n[3]
    t3 := m & fieldBaseMask
    m = (m >> fieldBase) + f.n[4]
    t4 := m & fieldBaseMask
    m = (m >> fieldBase) + f.n[5]
    t5 := m & fieldBaseMask
    m = (m >> fieldBase) + f.n[6]
    t6 := m & fieldBaseMask
    m = (m >> fieldBase) + f.n[7]
    t7 := m & fieldBaseMask
    m = (m >> fieldBase) + f.n[8]
    t8 := m & fieldBaseMask
    m = (m >> fieldBase) + f.n[9]
    t9 := m & fieldMSBMask
    m = m >> fieldMSBBits

    // At this point, if the magnitude is greater than 0, the overall value
    // is greater than the max possible 256-bit value.  In particular, it is
    // "how many times larger" than the max value it is.  Since this field
    // is doing arithmetic modulo the secp256k1 prime, we need to perform
    // modular reduction over the prime.
    //
    // Per [HAC] section 14.3.4: Reduction method of moduli of special form,
    // when the modulus is of the special form m = b^t - c, highly efficient
    // reduction can be achieved.
    //
    // The secp256k1 prime is equivalent to 2^256 - 4294968273, so it fits
    // this criteria.
    //
    // 4294968273 in field representation (base 2^26) is:
    // n[0] = 977
    // n[1] = 64
    // That is to say (2^26 * 64) + 977 = 4294968273
    //
    // The algorithm presented in the referenced section typically repeats
    // until the quotient is zero.  However, due to our field representation
    // we already know at least how many times we would need to repeat as
    // it's the value currently in m.  Thus we can simply multiply the
    // magnitude by the field representation of the prime and do a single
    // iteration.  Notice that nothing will be changed when the magnitude is
    // zero, so we could skip this in that case, however always running
    // regardless allows it to run in constant time.
    r := t0 + m*977
    t0 = r & fieldBaseMask
    r = (r >> fieldBase) + t1 + m*64
    t1 = r & fieldBaseMask
    r = (r >> fieldBase) + t2
    t2 = r & fieldBaseMask
    r = (r >> fieldBase) + t3
    t3 = r & fieldBaseMask
    r = (r >> fieldBase) + t4
    t4 = r & fieldBaseMask
    r = (r >> fieldBase) + t5
    t5 = r & fieldBaseMask
    r = (r >> fieldBase) + t6
    t6 = r & fieldBaseMask
    r = (r >> fieldBase) + t7
    t7 = r & fieldBaseMask
    r = (r >> fieldBase) + t8
    t8 = r & fieldBaseMask
    r = (r >> fieldBase) + t9
    t9 = r & fieldMSBMask

    // At this point, the result will be in the range 0 <= result <=
    // prime + (2^64 - c).  Therefore, one more subtraction of the prime
    // might be needed if the current result is greater than or equal to the
    // prime.  The following does the final reduction in constant time.
    // Note that the if/else here intentionally does the bitwise OR with
    // zero even though it won't change the value to ensure constant time
    // between the branches.
    var mask int32
    lowBits := uint64(t1)<<fieldBase | uint64(t0)
    if lowBits < primeLowBits {
        mask |= -1
    } else {
        mask |= 0
    }
    if t2 < fieldBaseMask {
        mask |= -1
    } else {
        mask |= 0
    }
    if t3 < fieldBaseMask {
        mask |= -1
    } else {
        mask |= 0
    }
    if t4 < fieldBaseMask {
        mask |= -1
    } else {
        mask |= 0
    }
    if t5 < fieldBaseMask {
        mask |= -1
    } else {
        mask |= 0
    }
    if t6 < fieldBaseMask {
        mask |= -1
    } else {
        mask |= 0
    }
    if t7 < fieldBaseMask {
        mask |= -1
    } else {
        mask |= 0
    }
    if t8 < fieldBaseMask {
        mask |= -1
    } else {
        mask |= 0
    }
    if t9 < fieldMSBMask {
        mask |= -1
    } else {
        mask |= 0
    }
    lowBits -= ^uint64(mask) & primeLowBits
    t0 = uint32(lowBits & fieldBaseMask)
    t1 = uint32((lowBits >> fieldBase) & fieldBaseMask)
    t2 = t2 & uint32(mask)
    t3 = t3 & uint32(mask)
    t4 = t4 & uint32(mask)
    t5 = t5 & uint32(mask)
    t6 = t6 & uint32(mask)
    t7 = t7 & uint32(mask)
    t8 = t8 & uint32(mask)
    t9 = t9 & uint32(mask)

    // Finally, set the normalized and reduced words.
    f.n[0] = t0
    f.n[1] = t1
    f.n[2] = t2
    f.n[3] = t3
    f.n[4] = t4
    f.n[5] = t5
    f.n[6] = t6
    f.n[7] = t7
    f.n[8] = t8
    f.n[9] = t9
    return f
}

// PutBytes unpacks the field value to a 32-byte big-endian value using the
// passed byte array.  There is a similar function, Bytes, which unpacks the
// field value into a new array and returns that.  This version is provided
// since it can be useful to cut down on the number of allocations by allowing
// the caller to reuse a buffer.
//
// The field value must be normalized for this function to return the correct
// result.
func (f *fieldVal) PutBytes(b *[32]byte) {
    // Unpack the 256 total bits from the 10 uint32 words with a max of
    // 26-bits per word.  This could be done with a couple of for loops,
    // but this unrolled version is a bit faster.  Benchmarks show this is
    // about 10 times faster than the variant which uses loops.
    b[31] = byte(f.n[0] & eightBitsMask)
    b[30] = byte((f.n[0] >> 8) & eightBitsMask)
    b[29] = byte((f.n[0] >> 16) & eightBitsMask)
    b[28] = byte((f.n[0]>>24)&twoBitsMask | (f.n[1]&sixBitsMask)<<2)
    b[27] = byte((f.n[1] >> 6) & eightBitsMask)
    b[26] = byte((f.n[1] >> 14) & eightBitsMask)
    b[25] = byte((f.n[1]>>22)&fourBitsMask | (f.n[2]&fourBitsMask)<<4)
    b[24] = byte((f.n[2] >> 4) & eightBitsMask)
    b[23] = byte((f.n[2] >> 12) & eightBitsMask)
    b[22] = byte((f.n[2]>>20)&sixBitsMask | (f.n[3]&twoBitsMask)<<6)
    b[21] = byte((f.n[3] >> 2) & eightBitsMask)
    b[20] = byte((f.n[3] >> 10) & eightBitsMask)
    b[19] = byte((f.n[3] >> 18) & eightBitsMask)
    b[18] = byte(f.n[4] & eightBitsMask)
    b[17] = byte((f.n[4] >> 8) & eightBitsMask)
    b[16] = byte((f.n[4] >> 16) & eightBitsMask)
    b[15] = byte((f.n[4]>>24)&twoBitsMask | (f.n[5]&sixBitsMask)<<2)
    b[14] = byte((f.n[5] >> 6) & eightBitsMask)
    b[13] = byte((f.n[5] >> 14) & eightBitsMask)
    b[12] = byte((f.n[5]>>22)&fourBitsMask | (f.n[6]&fourBitsMask)<<4)
    b[11] = byte((f.n[6] >> 4) & eightBitsMask)
    b[10] = byte((f.n[6] >> 12) & eightBitsMask)
    b[9] = byte((f.n[6]>>20)&sixBitsMask | (f.n[7]&twoBitsMask)<<6)
    b[8] = byte((f.n[7] >> 2) & eightBitsMask)
    b[7] = byte((f.n[7] >> 10) & eightBitsMask)
    b[6] = byte((f.n[7] >> 18) & eightBitsMask)
    b[5] = byte(f.n[8] & eightBitsMask)
    b[4] = byte((f.n[8] >> 8) & eightBitsMask)
    b[3] = byte((f.n[8] >> 16) & eightBitsMask)
    b[2] = byte((f.n[8]>>24)&twoBitsMask | (f.n[9]&sixBitsMask)<<2)
    b[1] = byte((f.n[9] >> 6) & eightBitsMask)
    b[0] = byte((f.n[9] >> 14) & eightBitsMask)
}

// Bytes unpacks the field value to a 32-byte big-endian value.  See PutBytes
// for a variant that allows the a buffer to be passed which can be useful to
// to cut down on the number of allocations by allowing the caller to reuse a
// buffer.
//
// The field value must be normalized for this function to return correct
// result.
func (f *fieldVal) Bytes() *[32]byte {
    b := new([32]byte)
    f.PutBytes(b)
    return b
}

// IsZero returns whether or not the field value is equal to zero.
func (f *fieldVal) IsZero() bool {
    // The value can only be zero if no bits are set in any of the words.
    // This is a constant time implementation.
    bits := f.n[0] | f.n[1] | f.n[2] | f.n[3] | f.n[4] |
        f.n[5] | f.n[6] | f.n[7] | f.n[8] | f.n[9]

    return bits == 0
}

// IsOdd returns whether or not the field value is an odd number.
//
// The field value must be normalized for this function to return correct
// result.
func (f *fieldVal) IsOdd() bool {
    // Only odd numbers have the bottom bit set.
    return f.n[0]&1 == 1
}

// Equals returns whether or not the two field values are the same.  Both
// field values being compared must be normalized for this function to return
// the correct result.
func (f *fieldVal) Equals(val *fieldVal) bool {
    // Xor only sets bits when they are different, so the two field values
    // can only be the same if no bits are set after xoring each word.
    // This is a constant time implementation.
    bits := (f.n[0] ^ val.n[0]) | (f.n[1] ^ val.n[1]) | (f.n[2] ^ val.n[2]) |
        (f.n[3] ^ val.n[3]) | (f.n[4] ^ val.n[4]) | (f.n[5] ^ val.n[5]) |
        (f.n[6] ^ val.n[6]) | (f.n[7] ^ val.n[7]) | (f.n[8] ^ val.n[8]) |
        (f.n[9] ^ val.n[9])

    return bits == 0
}

// NegateVal negates the passed value and stores the result in f.  The caller
// must provide the magnitude of the passed value for a correct result.
//
// The field value is returned to support chaining.  This enables syntax like:
// f.NegateVal(f2).AddInt(1) so that f = -f2 + 1.
func (f *fieldVal) NegateVal(val *fieldVal, magnitude uint32) *fieldVal {
    // Negation in the field is just the prime minus the value.  However,
    // in order to allow negation against a field value without having to
    // normalize/reduce it first, multiply by the magnitude (that is how
    // "far" away it is from the normalized value) to adjust.  Also, since
    // negating a value pushes it one more order of magnitude away from the
    // normalized range, add 1 to compensate.
    //
    // For some intuition here, imagine you're performing mod 12 arithmetic
    // (picture a clock) and you are negating the number 7.  So you start at
    // 12 (which is of course 0 under mod 12) and count backwards (left on
    // the clock) 7 times to arrive at 5.  Notice this is just 12-7 = 5.
    // Now, assume you're starting with 19, which is a number that is
    // already larger than the modulus and congruent to 7 (mod 12).  When a
    // value is already in the desired range, its magnitude is 1.  Since 19
    // is an additional "step", its magnitude (mod 12) is 2.  Since any
    // multiple of the modulus is conguent to zero (mod m), the answer can
    // be shortcut by simply mulplying the magnitude by the modulus and
    // subtracting.  Keeping with the example, this would be (2*12)-19 = 5.
    f.n[0] = (magnitude+1)*fieldPrimeWordZero - val.n[0]
    f.n[1] = (magnitude+1)*fieldPrimeWordOne - val.n[1]
    f.n[2] = (magnitude+1)*fieldBaseMask - val.n[2]
    f.n[3] = (magnitude+1)*fieldBaseMask - val.n[3]
    f.n[4] = (magnitude+1)*fieldBaseMask - val.n[4]
    f.n[5] = (magnitude+1)*fieldBaseMask - val.n[5]
    f.n[6] = (magnitude+1)*fieldBaseMask - val.n[6]
    f.n[7] = (magnitude+1)*fieldBaseMask - val.n[7]
    f.n[8] = (magnitude+1)*fieldBaseMask - val.n[8]
    f.n[9] = (magnitude+1)*fieldMSBMask - val.n[9]

    return f
}

// Negate negates the field value.  The existing field value is modified.  The
// caller must provide the magnitude of the field value for a correct result.
//
// The field value is returned to support chaining.  This enables syntax like:
// f.Negate().AddInt(1) so that f = -f + 1.
func (f *fieldVal) Negate(magnitude uint32) *fieldVal {
    return f.NegateVal(f, magnitude)
}

// AddInt adds the passed integer to the existing field value and stores the
// result in f.  This is a convenience function since it is fairly common to
// perform some arithemetic with small native integers.
//
// The field value is returned to support chaining.  This enables syntax like:
// f.AddInt(1).Add(f2) so that f = f + 1 + f2.
func (f *fieldVal) AddInt(ui uint) *fieldVal {
    // Since the field representation intentionally provides overflow bits,
    // it's ok to use carryless addition as the carry bit is safely part of
    // the word and will be normalized out.
    f.n[0] += uint32(ui)

    return f
}

// Add adds the passed value to the existing field value and stores the result
// in f.
//
// The field value is returned to support chaining.  This enables syntax like:
// f.Add(f2).AddInt(1) so that f = f + f2 + 1.
func (f *fieldVal) Add(val *fieldVal) *fieldVal {
    // Since the field representation intentionally provides overflow bits,
    // it's ok to use carryless addition as the carry bit is safely part of
    // each word and will be normalized out.  This could obviously be done
    // in a loop, but the unrolled version is faster.
    f.n[0] += val.n[0]
    f.n[1] += val.n[1]
    f.n[2] += val.n[2]
    f.n[3] += val.n[3]
    f.n[4] += val.n[4]
    f.n[5] += val.n[5]
    f.n[6] += val.n[6]
    f.n[7] += val.n[7]
    f.n[8] += val.n[8]
    f.n[9] += val.n[9]

    return f
}

// Add2 adds the passed two field values together and stores the result in f.
//
// The field value is returned to support chaining.  This enables syntax like:
// f3.Add2(f, f2).AddInt(1) so that f3 = f + f2 + 1.
func (f *fieldVal) Add2(val *fieldVal, val2 *fieldVal) *fieldVal {
    // Since the field representation intentionally provides overflow bits,
    // it's ok to use carryless addition as the carry bit is safely part of
    // each word and will be normalized out.  This could obviously be done
    // in a loop, but the unrolled version is faster.
    f.n[0] = val.n[0] + val2.n[0]
    f.n[1] = val.n[1] + val2.n[1]
    f.n[2] = val.n[2] + val2.n[2]
    f.n[3] = val.n[3] + val2.n[3]
    f.n[4] = val.n[4] + val2.n[4]
    f.n[5] = val.n[5] + val2.n[5]
    f.n[6] = val.n[6] + val2.n[6]
    f.n[7] = val.n[7] + val2.n[7]
    f.n[8] = val.n[8] + val2.n[8]
    f.n[9] = val.n[9] + val2.n[9]

    return f
}

// MulInt multiplies the field value by the passed int and stores the result in
// f.  Note that this function can overflow if multiplying the value by any of
// the individual words exceeds a max uint32.  Therefore it is important that
// the caller ensures no overflows will occur before using this function.
//
// The field value is returned to support chaining.  This enables syntax like:
// f.MulInt(2).Add(f2) so that f = 2 * f + f2.
func (f *fieldVal) MulInt(val uint) *fieldVal {
    // Since each word of the field representation can hold up to
    // fieldOverflowBits extra bits which will be normalized out, it's safe
    // to multiply each word without using a larger type or carry
    // propagation so long as the values won't overflow a uint32.  This
    // could obviously be done in a loop, but the unrolled version is
    // faster.
    ui := uint32(val)
    f.n[0] *= ui
    f.n[1] *= ui
    f.n[2] *= ui
    f.n[3] *= ui
    f.n[4] *= ui
    f.n[5] *= ui
    f.n[6] *= ui
    f.n[7] *= ui
    f.n[8] *= ui
    f.n[9] *= ui

    return f
}

// Mul multiplies the passed value to the existing field value and stores the
// result in f.  Note that this function can overflow if multiplying any
// of the individual words exceeds a max uint32.  In practice, this means the
// magnitude of either value involved in the multiplication must be a max of
// 8.
//
// The field value is returned to support chaining.  This enables syntax like:
// f.Mul(f2).AddInt(1) so that f = (f * f2) + 1.
func (f *fieldVal) Mul(val *fieldVal) *fieldVal {
    return f.Mul2(f, val)
}

// Mul2 multiplies the passed two field values together and stores the result
// result in f.  Note that this function can overflow if multiplying any of
// the individual words exceeds a max uint32.  In practice, this means the
// magnitude of either value involved in the multiplication must be a max of
// 8.
//
// The field value is returned to support chaining.  This enables syntax like:
// f3.Mul2(f, f2).AddInt(1) so that f3 = (f * f2) + 1.
func (f *fieldVal) Mul2(val *fieldVal, val2 *fieldVal) *fieldVal {
    // This could be done with a couple of for loops and an array to store
    // the intermediate terms, but this unrolled version is significantly
    // faster.

    // Terms for 2^(fieldBase*0).
    m := uint64(val.n[0]) * uint64(val2.n[0])
    t0 := m & fieldBaseMask

    // Terms for 2^(fieldBase*1).
    m = (m >> fieldBase) +
        uint64(val.n[0])*uint64(val2.n[1]) +
        uint64(val.n[1])*uint64(val2.n[0])
    t1 := m & fieldBaseMask

    // Terms for 2^(fieldBase*2).
    m = (m >> fieldBase) +
        uint64(val.n[0])*uint64(val2.n[2]) +
        uint64(val.n[1])*uint64(val2.n[1]) +
        uint64(val.n[2])*uint64(val2.n[0])
    t2 := m & fieldBaseMask

    // Terms for 2^(fieldBase*3).
    m = (m >> fieldBase) +
        uint64(val.n[0])*uint64(val2.n[3]) +
        uint64(val.n[1])*uint64(val2.n[2]) +
        uint64(val.n[2])*uint64(val2.n[1]) +
        uint64(val.n[3])*uint64(val2.n[0])
    t3 := m & fieldBaseMask

    // Terms for 2^(fieldBase*4).
    m = (m >> fieldBase) +
        uint64(val.n[0])*uint64(val2.n[4]) +
        uint64(val.n[1])*uint64(val2.n[3]) +
        uint64(val.n[2])*uint64(val2.n[2]) +
        uint64(val.n[3])*uint64(val2.n[1]) +
        uint64(val.n[4])*uint64(val2.n[0])
    t4 := m & fieldBaseMask

    // Terms for 2^(fieldBase*5).
    m = (m >> fieldBase) +
        uint64(val.n[0])*uint64(val2.n[5]) +
        uint64(val.n[1])*uint64(val2.n[4]) +
        uint64(val.n[2])*uint64(val2.n[3]) +
        uint64(val.n[3])*uint64(val2.n[2]) +
        uint64(val.n[4])*uint64(val2.n[1]) +
        uint64(val.n[5])*uint64(val2.n[0])
    t5 := m & fieldBaseMask

    // Terms for 2^(fieldBase*6).
    m = (m >> fieldBase) +
        uint64(val.n[0])*uint64(val2.n[6]) +
        uint64(val.n[1])*uint64(val2.n[5]) +
        uint64(val.n[2])*uint64(val2.n[4]) +
        uint64(val.n[3])*uint64(val2.n[3]) +
        uint64(val.n[4])*uint64(val2.n[2]) +
        uint64(val.n[5])*uint64(val2.n[1]) +
        uint64(val.n[6])*uint64(val2.n[0])
    t6 := m & fieldBaseMask

    // Terms for 2^(fieldBase*7).
    m = (m >> fieldBase) +
        uint64(val.n[0])*uint64(val2.n[7]) +
        uint64(val.n[1])*uint64(val2.n[6]) +
        uint64(val.n[2])*uint64(val2.n[5]) +
        uint64(val.n[3])*uint64(val2.n[4]) +
        uint64(val.n[4])*uint64(val2.n[3]) +
        uint64(val.n[5])*uint64(val2.n[2]) +
        uint64(val.n[6])*uint64(val2.n[1]) +
        uint64(val.n[7])*uint64(val2.n[0])
    t7 := m & fieldBaseMask

    // Terms for 2^(fieldBase*8).
    m = (m >> fieldBase) +
        uint64(val.n[0])*uint64(val2.n[8]) +
        uint64(val.n[1])*uint64(val2.n[7]) +
        uint64(val.n[2])*uint64(val2.n[6]) +
        uint64(val.n[3])*uint64(val2.n[5]) +
        uint64(val.n[4])*uint64(val2.n[4]) +
        uint64(val.n[5])*uint64(val2.n[3]) +
        uint64(val.n[6])*uint64(val2.n[2]) +
        uint64(val.n[7])*uint64(val2.n[1]) +
        uint64(val.n[8])*uint64(val2.n[0])
    t8 := m & fieldBaseMask

    // Terms for 2^(fieldBase*9).
    m = (m >> fieldBase) +
        uint64(val.n[0])*uint64(val2.n[9]) +
        uint64(val.n[1])*uint64(val2.n[8]) +
        uint64(val.n[2])*uint64(val2.n[7]) +
        uint64(val.n[3])*uint64(val2.n[6]) +
        uint64(val.n[4])*uint64(val2.n[5]) +
        uint64(val.n[5])*uint64(val2.n[4]) +
        uint64(val.n[6])*uint64(val2.n[3]) +
        uint64(val.n[7])*uint64(val2.n[2]) +
        uint64(val.n[8])*uint64(val2.n[1]) +
        uint64(val.n[9])*uint64(val2.n[0])
    t9 := m & fieldBaseMask

    // Terms for 2^(fieldBase*10).
    m = (m >> fieldBase) +
        uint64(val.n[1])*uint64(val2.n[9]) +
        uint64(val.n[2])*uint64(val2.n[8]) +
        uint64(val.n[3])*uint64(val2.n[7]) +
        uint64(val.n[4])*uint64(val2.n[6]) +
        uint64(val.n[5])*uint64(val2.n[5]) +
        uint64(val.n[6])*uint64(val2.n[4]) +
        uint64(val.n[7])*uint64(val2.n[3]) +
        uint64(val.n[8])*uint64(val2.n[2]) +
        uint64(val.n[9])*uint64(val2.n[1])
    t10 := m & fieldBaseMask

    // Terms for 2^(fieldBase*11).
    m = (m >> fieldBase) +
        uint64(val.n[2])*uint64(val2.n[9]) +
        uint64(val.n[3])*uint64(val2.n[8]) +
        uint64(val.n[4])*uint64(val2.n[7]) +
        uint64(val.n[5])*uint64(val2.n[6]) +
        uint64(val.n[6])*uint64(val2.n[5]) +
        uint64(val.n[7])*uint64(val2.n[4]) +
        uint64(val.n[8])*uint64(val2.n[3]) +
        uint64(val.n[9])*uint64(val2.n[2])
    t11 := m & fieldBaseMask

    // Terms for 2^(fieldBase*12).
    m = (m >> fieldBase) +
        uint64(val.n[3])*uint64(val2.n[9]) +
        uint64(val.n[4])*uint64(val2.n[8]) +
        uint64(val.n[5])*uint64(val2.n[7]) +
        uint64(val.n[6])*uint64(val2.n[6]) +
        uint64(val.n[7])*uint64(val2.n[5]) +
        uint64(val.n[8])*uint64(val2.n[4]) +
        uint64(val.n[9])*uint64(val2.n[3])
    t12 := m & fieldBaseMask

    // Terms for 2^(fieldBase*13).
    m = (m >> fieldBase) +
        uint64(val.n[4])*uint64(val2.n[9]) +
        uint64(val.n[5])*uint64(val2.n[8]) +
        uint64(val.n[6])*uint64(val2.n[7]) +
        uint64(val.n[7])*uint64(val2.n[6]) +
        uint64(val.n[8])*uint64(val2.n[5]) +
        uint64(val.n[9])*uint64(val2.n[4])
    t13 := m & fieldBaseMask

    // Terms for 2^(fieldBase*14).
    m = (m >> fieldBase) +
        uint64(val.n[5])*uint64(val2.n[9]) +
        uint64(val.n[6])*uint64(val2.n[8]) +
        uint64(val.n[7])*uint64(val2.n[7]) +
        uint64(val.n[8])*uint64(val2.n[6]) +
        uint64(val.n[9])*uint64(val2.n[5])
    t14 := m & fieldBaseMask

    // Terms for 2^(fieldBase*15).
    m = (m >> fieldBase) +
        uint64(val.n[6])*uint64(val2.n[9]) +
        uint64(val.n[7])*uint64(val2.n[8]) +
        uint64(val.n[8])*uint64(val2.n[7]) +
        uint64(val.n[9])*uint64(val2.n[6])
    t15 := m & fieldBaseMask

    // Terms for 2^(fieldBase*16).
    m = (m >> fieldBase) +
        uint64(val.n[7])*uint64(val2.n[9]) +
        uint64(val.n[8])*uint64(val2.n[8]) +
        uint64(val.n[9])*uint64(val2.n[7])
    t16 := m & fieldBaseMask

    // Terms for 2^(fieldBase*17).
    m = (m >> fieldBase) +
        uint64(val.n[8])*uint64(val2.n[9]) +
        uint64(val.n[9])*uint64(val2.n[8])
    t17 := m & fieldBaseMask

    // Terms for 2^(fieldBase*18).
    m = (m >> fieldBase) + uint64(val.n[9])*uint64(val2.n[9])
    t18 := m & fieldBaseMask

    // What's left is for 2^(fieldBase*19).
    t19 := m >> fieldBase

    // At this point, all of the terms are grouped into their respective
    // base.
    //
    // Per [HAC] section 14.3.4: Reduction method of moduli of special form,
    // when the modulus is of the special form m = b^t - c, highly efficient
    // reduction can be achieved per the provided algorithm.
    //
    // The secp256k1 prime is equivalent to 2^256 - 4294968273, so it fits
    // this criteria.
    //
    // 4294968273 in field representation (base 2^26) is:
    // n[0] = 977
    // n[1] = 64
    // That is to say (2^26 * 64) + 977 = 4294968273
    //
    // Since each word is in base 26, the upper terms (t10 and up) start
    // at 260 bits (versus the final desired range of 256 bits), so the
    // field representation of 'c' from above needs to be adjusted for the
    // extra 4 bits by multiplying it by 2^4 = 16.  4294968273 * 16 =
    // 68719492368.  Thus, the adjusted field representation of 'c' is:
    // n[0] = 977 * 16 = 15632
    // n[1] = 64 * 16 = 1024
    // That is to say (2^26 * 1024) + 15632 = 68719492368
    //
    // To reduce the final term, t19, the entire 'c' value is needed instead
    // of only n[0] because there are no more terms left to handle n[1].
    // This means there might be some magnitude left in the upper bits that
    // is handled below.
    m = t0 + t10*15632
    t0 = m & fieldBaseMask
    m = (m >> fieldBase) + t1 + t10*1024 + t11*15632
    t1 = m & fieldBaseMask
    m = (m >> fieldBase) + t2 + t11*1024 + t12*15632
    t2 = m & fieldBaseMask
    m = (m >> fieldBase) + t3 + t12*1024 + t13*15632
    t3 = m & fieldBaseMask
    m = (m >> fieldBase) + t4 + t13*1024 + t14*15632
    t4 = m & fieldBaseMask
    m = (m >> fieldBase) + t5 + t14*1024 + t15*15632
    t5 = m & fieldBaseMask
    m = (m >> fieldBase) + t6 + t15*1024 + t16*15632
    t6 = m & fieldBaseMask
    m = (m >> fieldBase) + t7 + t16*1024 + t17*15632
    t7 = m & fieldBaseMask
    m = (m >> fieldBase) + t8 + t17*1024 + t18*15632
    t8 = m & fieldBaseMask
    m = (m >> fieldBase) + t9 + t18*1024 + t19*68719492368
    t9 = m & fieldMSBMask
    m = m >> fieldMSBBits

    // At this point, if the magnitude is greater than 0, the overall value
    // is greater than the max possible 256-bit value.  In particular, it is
    // "how many times larger" than the max value it is.
    //
    // The algorithm presented in [HAC] section 14.3.4 repeats until the
    // quotient is zero.  However, due to the above, we already know at
    // least how many times we would need to repeat as it's the value
    // currently in m.  Thus we can simply multiply the magnitude by the
    // field representation of the prime and do a single iteration.  Notice
    // that nothing will be changed when the magnitude is zero, so we could
    // skip this in that case, however always running regardless allows it
    // to run in constant time.  The final result will be in the range
    // 0 <= result <= prime + (2^64 - c), so it is guaranteed to have a
    // magnitude of 1, but it is denormalized.
    d := t0 + m*977
    f.n[0] = uint32(d & fieldBaseMask)
    d = (d >> fieldBase) + t1 + m*64
    f.n[1] = uint32(d & fieldBaseMask)
    f.n[2] = uint32((d >> fieldBase) + t2)
    f.n[3] = uint32(t3)
    f.n[4] = uint32(t4)
    f.n[5] = uint32(t5)
    f.n[6] = uint32(t6)
    f.n[7] = uint32(t7)
    f.n[8] = uint32(t8)
    f.n[9] = uint32(t9)

    return f
}

// Square squares the field value.  The existing field value is modified.  Note
// that this function can overflow if multiplying any of the individual words
// exceeds a max uint32.  In practice, this means the magnitude of the field
// must be a max of 8 to prevent overflow.
//
// The field value is returned to support chaining.  This enables syntax like:
// f.Square().Mul(f2) so that f = f^2 * f2.
func (f *fieldVal) Square() *fieldVal {
    return f.SquareVal(f)
}

// SquareVal squares the passed value and stores the result in f.  Note that
// this function can overflow if multiplying any of the individual words
// exceeds a max uint32.  In practice, this means the magnitude of the field
// being squred must be a max of 8 to prevent overflow.
//
// The field value is returned to support chaining.  This enables syntax like:
// f3.SquareVal(f).Mul(f) so that f3 = f^2 * f = f^3.
func (f *fieldVal) SquareVal(val *fieldVal) *fieldVal {
    // This could be done with a couple of for loops and an array to store
    // the intermediate terms, but this unrolled version is significantly
    // faster.

    // Terms for 2^(fieldBase*0).
    m := uint64(val.n[0]) * uint64(val.n[0])
    t0 := m & fieldBaseMask

    // Terms for 2^(fieldBase*1).
    m = (m >> fieldBase) + 2*uint64(val.n[0])*uint64(val.n[1])
    t1 := m & fieldBaseMask

    // Terms for 2^(fieldBase*2).
    m = (m >> fieldBase) +
        2*uint64(val.n[0])*uint64(val.n[2]) +
        uint64(val.n[1])*uint64(val.n[1])
    t2 := m & fieldBaseMask

    // Terms for 2^(fieldBase*3).
    m = (m >> fieldBase) +
        2*uint64(val.n[0])*uint64(val.n[3]) +
        2*uint64(val.n[1])*uint64(val.n[2])
    t3 := m & fieldBaseMask

    // Terms for 2^(fieldBase*4).
    m = (m >> fieldBase) +
        2*uint64(val.n[0])*uint64(val.n[4]) +
        2*uint64(val.n[1])*uint64(val.n[3]) +
        uint64(val.n[2])*uint64(val.n[2])
    t4 := m & fieldBaseMask

    // Terms for 2^(fieldBase*5).
    m = (m >> fieldBase) +
        2*uint64(val.n[0])*uint64(val.n[5]) +
        2*uint64(val.n[1])*uint64(val.n[4]) +
        2*uint64(val.n[2])*uint64(val.n[3])
    t5 := m & fieldBaseMask

    // Terms for 2^(fieldBase*6).
    m = (m >> fieldBase) +
        2*uint64(val.n[0])*uint64(val.n[6]) +
        2*uint64(val.n[1])*uint64(val.n[5]) +
        2*uint64(val.n[2])*uint64(val.n[4]) +
        uint64(val.n[3])*uint64(val.n[3])
    t6 := m & fieldBaseMask

    // Terms for 2^(fieldBase*7).
    m = (m >> fieldBase) +
        2*uint64(val.n[0])*uint64(val.n[7]) +
        2*uint64(val.n[1])*uint64(val.n[6]) +
        2*uint64(val.n[2])*uint64(val.n[5]) +
        2*uint64(val.n[3])*uint64(val.n[4])
    t7 := m & fieldBaseMask

    // Terms for 2^(fieldBase*8).
    m = (m >> fieldBase) +
        2*uint64(val.n[0])*uint64(val.n[8]) +
        2*uint64(val.n[1])*uint64(val.n[7]) +
        2*uint64(val.n[2])*uint64(val.n[6]) +
        2*uint64(val.n[3])*uint64(val.n[5]) +
        uint64(val.n[4])*uint64(val.n[4])
    t8 := m & fieldBaseMask

    // Terms for 2^(fieldBase*9).
    m = (m >> fieldBase) +
        2*uint64(val.n[0])*uint64(val.n[9]) +
        2*uint64(val.n[1])*uint64(val.n[8]) +
        2*uint64(val.n[2])*uint64(val.n[7]) +
        2*uint64(val.n[3])*uint64(val.n[6]) +
        2*uint64(val.n[4])*uint64(val.n[5])
    t9 := m & fieldBaseMask

    // Terms for 2^(fieldBase*10).
    m = (m >> fieldBase) +
        2*uint64(val.n[1])*uint64(val.n[9]) +
        2*uint64(val.n[2])*uint64(val.n[8]) +
        2*uint64(val.n[3])*uint64(val.n[7]) +
        2*uint64(val.n[4])*uint64(val.n[6]) +
        uint64(val.n[5])*uint64(val.n[5])
    t10 := m & fieldBaseMask

    // Terms for 2^(fieldBase*11).
    m = (m >> fieldBase) +
        2*uint64(val.n[2])*uint64(val.n[9]) +
        2*uint64(val.n[3])*uint64(val.n[8]) +
        2*uint64(val.n[4])*uint64(val.n[7]) +
        2*uint64(val.n[5])*uint64(val.n[6])
    t11 := m & fieldBaseMask

    // Terms for 2^(fieldBase*12).
    m = (m >> fieldBase) +
        2*uint64(val.n[3])*uint64(val.n[9]) +
        2*uint64(val.n[4])*uint64(val.n[8]) +
        2*uint64(val.n[5])*uint64(val.n[7]) +
        uint64(val.n[6])*uint64(val.n[6])
    t12 := m & fieldBaseMask

    // Terms for 2^(fieldBase*13).
    m = (m >> fieldBase) +
        2*uint64(val.n[4])*uint64(val.n[9]) +
        2*uint64(val.n[5])*uint64(val.n[8]) +
        2*uint64(val.n[6])*uint64(val.n[7])
    t13 := m & fieldBaseMask

    // Terms for 2^(fieldBase*14).
    m = (m >> fieldBase) +
        2*uint64(val.n[5])*uint64(val.n[9]) +
        2*uint64(val.n[6])*uint64(val.n[8]) +
        uint64(val.n[7])*uint64(val.n[7])
    t14 := m & fieldBaseMask

    // Terms for 2^(fieldBase*15).
    m = (m >> fieldBase) +
        2*uint64(val.n[6])*uint64(val.n[9]) +
        2*uint64(val.n[7])*uint64(val.n[8])
    t15 := m & fieldBaseMask

    // Terms for 2^(fieldBase*16).
    m = (m >> fieldBase) +
        2*uint64(val.n[7])*uint64(val.n[9]) +
        uint64(val.n[8])*uint64(val.n[8])
    t16 := m & fieldBaseMask

    // Terms for 2^(fieldBase*17).
    m = (m >> fieldBase) + 2*uint64(val.n[8])*uint64(val.n[9])
    t17 := m & fieldBaseMask

    // Terms for 2^(fieldBase*18).
    m = (m >> fieldBase) + uint64(val.n[9])*uint64(val.n[9])
    t18 := m & fieldBaseMask

    // What's left is for 2^(fieldBase*19).
    t19 := m >> fieldBase

    // At this point, all of the terms are grouped into their respective
    // base.
    //
    // Per [HAC] section 14.3.4: Reduction method of moduli of special form,
    // when the modulus is of the special form m = b^t - c, highly efficient
    // reduction can be achieved per the provided algorithm.
    //
    // The secp256k1 prime is equivalent to 2^256 - 4294968273, so it fits
    // this criteria.
    //
    // 4294968273 in field representation (base 2^26) is:
    // n[0] = 977
    // n[1] = 64
    // That is to say (2^26 * 64) + 977 = 4294968273
    //
    // Since each word is in base 26, the upper terms (t10 and up) start
    // at 260 bits (versus the final desired range of 256 bits), so the
    // field representation of 'c' from above needs to be adjusted for the
    // extra 4 bits by multiplying it by 2^4 = 16.  4294968273 * 16 =
    // 68719492368.  Thus, the adjusted field representation of 'c' is:
    // n[0] = 977 * 16 = 15632
    // n[1] = 64 * 16 = 1024
    // That is to say (2^26 * 1024) + 15632 = 68719492368
    //
    // To reduce the final term, t19, the entire 'c' value is needed instead
    // of only n[0] because there are no more terms left to handle n[1].
    // This means there might be some magnitude left in the upper bits that
    // is handled below.
    m = t0 + t10*15632
    t0 = m & fieldBaseMask
    m = (m >> fieldBase) + t1 + t10*1024 + t11*15632
    t1 = m & fieldBaseMask
    m = (m >> fieldBase) + t2 + t11*1024 + t12*15632
    t2 = m & fieldBaseMask
    m = (m >> fieldBase) + t3 + t12*1024 + t13*15632
    t3 = m & fieldBaseMask
    m = (m >> fieldBase) + t4 + t13*1024 + t14*15632
    t4 = m & fieldBaseMask
    m = (m >> fieldBase) + t5 + t14*1024 + t15*15632
    t5 = m & fieldBaseMask
    m = (m >> fieldBase) + t6 + t15*1024 + t16*15632
    t6 = m & fieldBaseMask
    m = (m >> fieldBase) + t7 + t16*1024 + t17*15632
    t7 = m & fieldBaseMask
    m = (m >> fieldBase) + t8 + t17*1024 + t18*15632
    t8 = m & fieldBaseMask
    m = (m >> fieldBase) + t9 + t18*1024 + t19*68719492368
    t9 = m & fieldMSBMask
    m = m >> fieldMSBBits

    // At this point, if the magnitude is greater than 0, the overall value
    // is greater than the max possible 256-bit value.  In particular, it is
    // "how many times larger" than the max value it is.
    //
    // The algorithm presented in [HAC] section 14.3.4 repeats until the
    // quotient is zero.  However, due to the above, we already know at
    // least how many times we would need to repeat as it's the value
    // currently in m.  Thus we can simply multiply the magnitude by the
    // field representation of the prime and do a single iteration.  Notice
    // that nothing will be changed when the magnitude is zero, so we could
    // skip this in that case, however always running regardless allows it
    // to run in constant time.  The final result will be in the range
    // 0 <= result <= prime + (2^64 - c), so it is guaranteed to have a
    // magnitude of 1, but it is denormalized.
    n := t0 + m*977
    f.n[0] = uint32(n & fieldBaseMask)
    n = (n >> fieldBase) + t1 + m*64
    f.n[1] = uint32(n & fieldBaseMask)
    f.n[2] = uint32((n >> fieldBase) + t2)
    f.n[3] = uint32(t3)
    f.n[4] = uint32(t4)
    f.n[5] = uint32(t5)
    f.n[6] = uint32(t6)
    f.n[7] = uint32(t7)
    f.n[8] = uint32(t8)
    f.n[9] = uint32(t9)

    return f
}

// Inverse finds the modular multiplicative inverse of the field value.  The
// existing field value is modified.
//
// The field value is returned to support chaining.  This enables syntax like:
// f.Inverse().Mul(f2) so that f = f^-1 * f2.
func (f *fieldVal) Inverse() *fieldVal {
    // Fermat's little theorem states that for a nonzero number a and prime
    // prime p, a^(p-1) = 1 (mod p).  Since the multipliciative inverse is
    // a*b = 1 (mod p), it follows that b = a*a^(p-2) = a^(p-1) = 1 (mod p).
    // Thus, a^(p-2) is the multiplicative inverse.
    //
    // In order to efficiently compute a^(p-2), p-2 needs to be split into
    // a sequence of squares and multipications that minimizes the number of
    // multiplications needed (since they are more costly than squarings).
    // Intermediate results are saved and reused as well.
    //
    // The secp256k1 prime - 2 is 2^256 - 4294968275.
    //
    // This has a cost of 258 field squarings and 33 field multiplications.
    var a2, a3, a4, a10, a11, a21, a42, a45, a63, a1019, a1023 fieldVal
    a2.SquareVal(f)
    a3.Mul2(&a2, f)
    a4.SquareVal(&a2)
    a10.SquareVal(&a4).Mul(&a2)
    a11.Mul2(&a10, f)
    a21.Mul2(&a10, &a11)
    a42.SquareVal(&a21)
    a45.Mul2(&a42, &a3)
    a63.Mul2(&a42, &a21)
    a1019.SquareVal(&a63).Square().Square().Square().Mul(&a11)
    a1023.Mul2(&a1019, &a4)
    f.Set(&a63)                                    // f = a^(2^6 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^11 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^16 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^16 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^21 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^26 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^26 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^31 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^36 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^36 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^41 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^46 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^46 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^51 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^56 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^56 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^61 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^66 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^66 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^71 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^76 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^76 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^81 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^86 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^86 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^91 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^96 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^96 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^101 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^106 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^106 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^111 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^116 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^116 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^121 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^126 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^126 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^131 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^136 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^136 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^141 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^146 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^146 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^151 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^156 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^156 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^161 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^166 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^166 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^171 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^176 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^176 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^181 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^186 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^186 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^191 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^196 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^196 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^201 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^206 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^206 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^211 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^216 - 1024)
    f.Mul(&a1023)                                  // f = a^(2^216 - 1)
    f.Square().Square().Square().Square().Square() // f = a^(2^221 - 32)
    f.Square().Square().Square().Square().Square() // f = a^(2^226 - 1024)
    f.Mul(&a1019)                                  // f = a^(2^226 - 5)
    f.Square().Square().Square().Square().Square() // f = a^(2^231 - 160)
    f.Square().Square().Square().Square().Square() // f = a^(2^236 - 5120)
    f.Mul(&a1023)                                  // f = a^(2^236 - 4097)
    f.Square().Square().Square().Square().Square() // f = a^(2^241 - 131104)
    f.Square().Square().Square().Square().Square() // f = a^(2^246 - 4195328)
    f.Mul(&a1023)                                  // f = a^(2^246 - 4194305)
    f.Square().Square().Square().Square().Square() // f = a^(2^251 - 134217760)
    f.Square().Square().Square().Square().Square() // f = a^(2^256 - 4294968320)
    return f.Mul(&a45)                             // f = a^(2^256 - 4294968275) = a^(p-2)
}