diff options
Diffstat (limited to 'cli/vendor/github.com/decred/dcrd/dcrec/secp256k1/v4/field.go')
| -rw-r--r-- | cli/vendor/github.com/decred/dcrd/dcrec/secp256k1/v4/field.go | 1680 |
1 files changed, 1680 insertions, 0 deletions
diff --git a/cli/vendor/github.com/decred/dcrd/dcrec/secp256k1/v4/field.go b/cli/vendor/github.com/decred/dcrd/dcrec/secp256k1/v4/field.go new file mode 100644 index 0000000..43e27a9 --- /dev/null +++ b/cli/vendor/github.com/decred/dcrd/dcrec/secp256k1/v4/field.go @@ -0,0 +1,1680 @@ +// Copyright (c) 2013-2014 The btcsuite developers +// Copyright (c) 2015-2021 The Decred developers +// Copyright (c) 2013-2021 Dave Collins +// Use of this source code is governed by an ISC +// license that can be found in the LICENSE file. + +package secp256k1 + +// 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 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 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 type does not perform any validation where it +// ordinarily would. See the documentation for FieldVal for more details. + +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 + + // 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 + + // These fields provide convenient access to each of the words of the + // secp256k1 prime in the internal field representation to improve code + // readability. + fieldPrimeWordZero = 0x03fffc2f + fieldPrimeWordOne = 0x03ffffbf + fieldPrimeWordTwo = 0x03ffffff + fieldPrimeWordThree = 0x03ffffff + fieldPrimeWordFour = 0x03ffffff + fieldPrimeWordFive = 0x03ffffff + fieldPrimeWordSix = 0x03ffffff + fieldPrimeWordSeven = 0x03ffffff + fieldPrimeWordEight = 0x03ffffff + fieldPrimeWordNine = 0x003fffff +) + +// FieldVal implements optimized fixed-precision arithmetic over the +// secp256k1 finite field. This means all arithmetic is performed modulo +// 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f. +// +// WARNING: Since it is so important for the field arithmetic to be extremely +// fast for high performance crypto, this type does not perform any validation +// of documented preconditions where it ordinarily would. As a result, it is +// IMPERATIVE for callers to understand some key concepts that are described +// below and ensure the methods are called with the necessary preconditions that +// each method is documented with. For example, some methods only give the +// correct result if the field value is normalized and others require the field +// values involved to have a maximum magnitude and THERE ARE NO EXPLICIT CHECKS +// TO ENSURE THOSE PRECONDITIONS ARE SATISFIED. This does, unfortunately, make +// the type more difficult to use correctly and while I typically prefer to +// ensure all state and input is valid for most code, this is a bit of an +// exception because those extra checks really add up in what ends up being +// critical hot paths. +// +// The first key concept when working with this type is normalization. In order +// to avoid the need to propagate a ton of carries, the internal representation +// provides additional overflow bits for each word of the overall 256-bit value. +// This means that there are multiple internal representations for the same +// value and, as a result, any methods that rely on comparison of the value, +// such as equality and oddness determination, require the caller to provide a +// normalized value. +// +// The second key concept when working with this type is magnitude. As +// previously mentioned, the internal representation provides additional +// overflow bits which means that the more math operations that are performed on +// the field value between normalizations, the more those overflow bits +// accumulate. The magnitude is effectively that maximum possible number of +// those overflow bits that could possibly be required as a result of a given +// operation. Since there are only a limited number of overflow bits available, +// this implies that the max possible magnitude MUST be tracked by the caller +// and the caller MUST normalize the field value if a given operation would +// cause the magnitude of the result to exceed the max allowed value. +// +// IMPORTANT: The max allowed magnitude of a field value is 64. +type FieldVal struct { + // Each 256-bit value is represented 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 + n [10]uint32 +} + +// String returns the field value as a normalized human-readable hex string. +// +// Preconditions: None +// Output Normalized: Field is not modified -- same as input value +// Output Max Magnitude: Field is not modified -- same as input value +func (f FieldVal) String() string { + // f is a copy, so it's safe to normalize it without mutating the original. + f.Normalize() + return hex.EncodeToString(f.Bytes()[:]) +} + +// Zero sets the field value to zero in constant time. A newly created field +// value is already set to zero. This function can be useful to clear an +// existing field value for reuse. +// +// Preconditions: None +// Output Normalized: Yes +// Output Max Magnitude: 1 +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 in constant time. The +// normalization and magnitude of the two fields will be identical. +// +// 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. +// +// Preconditions: None +// Output Normalized: Same as input value +// Output Max Magnitude: Same as input value +func (f *FieldVal) Set(val *FieldVal) *FieldVal { + *f = *val + return f +} + +// SetInt sets the field value to the passed integer in constant time. This is +// a convenience function since it is fairly common to perform some arithmetic +// 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. +// +// Preconditions: None +// Output Normalized: Yes +// Output Max Magnitude: 1 +func (f *FieldVal) SetInt(ui uint16) *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 in constant time. SetBytes interprets the provided +// array as a 256-bit big-endian unsigned integer, packs it into the internal +// field value representation, and returns either 1 if it is greater than or +// equal to the field prime (aka it overflowed) or 0 otherwise in constant time. +// +// Note that a bool is not used here because it is not possible in Go to convert +// from a bool to numeric value in constant time and many constant-time +// operations require a numeric value. +// +// Preconditions: None +// Output Normalized: Yes if no overflow, no otherwise +// Output Max Magnitude: 1 +func (f *FieldVal) SetBytes(b *[32]byte) uint32 { + // 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 + + // The intuition here is that the field value is greater than the prime if + // one of the higher individual words is greater than corresponding word of + // the prime and all higher words in the field value are equal to their + // corresponding word of the prime. Since this type is modulo the prime, + // being equal is also an overflow back to 0. + // + // Note that because the input is 32 bytes and it was just packed into the + // field representation, the only words that can possibly be greater are + // zero and one, because ceil(log_2(2^256 - 1 - P)) = 33 bits max and the + // internal field representation encodes 26 bits with each word. + // + // Thus, there is no need to test if the upper words of the field value + // exceeds them, hence, only equality is checked for them. + highWordsEq := constantTimeEq(f.n[9], fieldPrimeWordNine) + highWordsEq &= constantTimeEq(f.n[8], fieldPrimeWordEight) + highWordsEq &= constantTimeEq(f.n[7], fieldPrimeWordSeven) + highWordsEq &= constantTimeEq(f.n[6], fieldPrimeWordSix) + highWordsEq &= constantTimeEq(f.n[5], fieldPrimeWordFive) + highWordsEq &= constantTimeEq(f.n[4], fieldPrimeWordFour) + highWordsEq &= constantTimeEq(f.n[3], fieldPrimeWordThree) + highWordsEq &= constantTimeEq(f.n[2], fieldPrimeWordTwo) + overflow := highWordsEq & constantTimeGreater(f.n[1], fieldPrimeWordOne) + highWordsEq &= constantTimeEq(f.n[1], fieldPrimeWordOne) + overflow |= highWordsEq & constantTimeGreaterOrEq(f.n[0], fieldPrimeWordZero) + + return overflow +} + +// SetByteSlice interprets the provided slice as a 256-bit big-endian unsigned +// integer (meaning it is truncated to the first 32 bytes), packs it into the +// internal field value representation, and returns whether or not the resulting +// truncated 256-bit integer is greater than or equal to the field prime (aka it +// overflowed) in constant time. +// +// Note that since passing a slice with more than 32 bytes is truncated, it is +// possible that the truncated value is less than the field prime and hence it +// will not be reported as having overflowed in that case. It is up to the +// caller to decide whether it needs to provide numbers of the appropriate size +// or it if is acceptable to use this function with the described truncation and +// overflow behavior. +// +// Preconditions: None +// Output Normalized: Yes if no overflow, no otherwise +// Output Max Magnitude: 1 +func (f *FieldVal) SetByteSlice(b []byte) bool { + var b32 [32]byte + b = b[:constantTimeMin(uint32(len(b)), 32)] + copy(b32[:], b32[:32-len(b)]) + copy(b32[32-len(b):], b) + result := f.SetBytes(&b32) + zeroArray32(&b32) + return result != 0 +} + +// 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 in constant time. +// +// Preconditions: None +// Output Normalized: Yes +// Output Max Magnitude: 1 +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, we need to "compact" the full 256-bit value to + // the right while propagating any carries through to the high order + // word. + // + // Since this field is doing arithmetic modulo the secp256k1 prime, we + // also 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 to within one reduction how many times we would need + // to repeat as it's the uppermost bits of the high order word. Thus we + // can simply multiply the magnitude by the field representation of the + // prime and do a single iteration. After this step there might be an + // additional carry to bit 256 (bit 22 of the high order word). + t9 := f.n[9] + m := t9 >> fieldMSBBits + t9 = t9 & fieldMSBMask + t0 := f.n[0] + m*977 + t1 := (t0 >> fieldBase) + f.n[1] + (m << 6) + t0 = t0 & fieldBaseMask + t2 := (t1 >> fieldBase) + f.n[2] + t1 = t1 & fieldBaseMask + t3 := (t2 >> fieldBase) + f.n[3] + t2 = t2 & fieldBaseMask + t4 := (t3 >> fieldBase) + f.n[4] + t3 = t3 & fieldBaseMask + t5 := (t4 >> fieldBase) + f.n[5] + t4 = t4 & fieldBaseMask + t6 := (t5 >> fieldBase) + f.n[6] + t5 = t5 & fieldBaseMask + t7 := (t6 >> fieldBase) + f.n[7] + t6 = t6 & fieldBaseMask + t8 := (t7 >> fieldBase) + f.n[8] + t7 = t7 & fieldBaseMask + t9 = (t8 >> fieldBase) + t9 + t8 = t8 & fieldBaseMask + + // At this point, the magnitude is guaranteed to be one, however, the + // value could still be greater than the prime if there was either a + // carry through to bit 256 (bit 22 of the higher order word) or the + // value is greater than or equal to the field characteristic. The + // following determines if either or these conditions are true and does + // the final reduction in constant time. + // + // Also note that 'm' will be zero when neither of the aforementioned + // conditions are true and the value will not be changed when 'm' is zero. + m = constantTimeEq(t9, fieldMSBMask) + m &= constantTimeEq(t8&t7&t6&t5&t4&t3&t2, fieldBaseMask) + m &= constantTimeGreater(t1+64+((t0+977)>>fieldBase), fieldBaseMask) + m |= t9 >> fieldMSBBits + t0 = t0 + m*977 + t1 = (t0 >> fieldBase) + t1 + (m << 6) + t0 = t0 & fieldBaseMask + t2 = (t1 >> fieldBase) + t2 + t1 = t1 & fieldBaseMask + t3 = (t2 >> fieldBase) + t3 + t2 = t2 & fieldBaseMask + t4 = (t3 >> fieldBase) + t4 + t3 = t3 & fieldBaseMask + t5 = (t4 >> fieldBase) + t5 + t4 = t4 & fieldBaseMask + t6 = (t5 >> fieldBase) + t6 + t5 = t5 & fieldBaseMask + t7 = (t6 >> fieldBase) + t7 + t6 = t6 & fieldBaseMask + t8 = (t7 >> fieldBase) + t8 + t7 = t7 & fieldBaseMask + t9 = (t8 >> fieldBase) + t9 + t8 = t8 & fieldBaseMask + t9 = t9 & fieldMSBMask // Remove potential multiple of 2^256. + + // 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 +} + +// PutBytesUnchecked unpacks the field value to a 32-byte big-endian value +// directly into the passed byte slice in constant time. The target slice must +// must have at least 32 bytes available or it will panic. +// +// There is a similar function, PutBytes, which unpacks the field value into a +// 32-byte array directly. This version is provided since it can be useful +// to write directly into part of a larger buffer without needing a separate +// allocation. +// +// Preconditions: +// - The field value MUST be normalized +// - The target slice MUST have at least 32 bytes available +func (f *FieldVal) PutBytesUnchecked(b []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) +} + +// PutBytes unpacks the field value to a 32-byte big-endian value using the +// passed byte array in constant time. +// +// There is a similar function, PutBytesUnchecked, which unpacks the field value +// into a slice that must have at least 32 bytes available. This version is +// provided since it can be useful to write directly into an array that is type +// checked. +// +// Alternatively, there is also Bytes, which unpacks the field value into a new +// array and returns that which can sometimes be more ergonomic in applications +// that aren't concerned about an additional copy. +// +// Preconditions: +// - The field value MUST be normalized +func (f *FieldVal) PutBytes(b *[32]byte) { + f.PutBytesUnchecked(b[:]) +} + +// Bytes unpacks the field value to a 32-byte big-endian value in constant time. +// +// See PutBytes and PutBytesUnchecked for variants that allow an array or slice +// to be passed which can be useful to cut down on the number of allocations by +// allowing the caller to reuse a buffer or write directly into part of a larger +// buffer. +// +// Preconditions: +// - The field value MUST be normalized +func (f *FieldVal) Bytes() *[32]byte { + b := new([32]byte) + f.PutBytesUnchecked(b[:]) + return b +} + +// IsZeroBit returns 1 when the field value is equal to zero or 0 otherwise in +// constant time. +// +// Note that a bool is not used here because it is not possible in Go to convert +// from a bool to numeric value in constant time and many constant-time +// operations require a numeric value. See IsZero for the version that returns +// a bool. +// +// Preconditions: +// - The field value MUST be normalized +func (f *FieldVal) IsZeroBit() uint32 { + // 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 constantTimeEq(bits, 0) +} + +// IsZero returns whether or not the field value is equal to zero in constant +// time. +// +// Preconditions: +// - The field value MUST be normalized +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 +} + +// IsOneBit returns 1 when the field value is equal to one or 0 otherwise in +// constant time. +// +// Note that a bool is not used here because it is not possible in Go to convert +// from a bool to numeric value in constant time and many constant-time +// operations require a numeric value. See IsOne for the version that returns a +// bool. +// +// Preconditions: +// - The field value MUST be normalized +func (f *FieldVal) IsOneBit() uint32 { + // The value can only be one if the single lowest significant bit is set in + // the first word and no other bits are set in any of the other words. + // This is a constant time implementation. + bits := (f.n[0] ^ 1) | 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 constantTimeEq(bits, 0) +} + +// IsOne returns whether or not the field value is equal to one in constant +// time. +// +// Preconditions: +// - The field value MUST be normalized +func (f *FieldVal) IsOne() bool { + // The value can only be one if the single lowest significant bit is set in + // the first word and no other bits are set in any of the other words. + // This is a constant time implementation. + bits := (f.n[0] ^ 1) | 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 +} + +// IsOddBit returns 1 when the field value is an odd number or 0 otherwise in +// constant time. +// +// Note that a bool is not used here because it is not possible in Go to convert +// from a bool to numeric value in constant time and many constant-time +// operations require a numeric value. See IsOdd for the version that returns a +// bool. +// +// Preconditions: +// - The field value MUST be normalized +func (f *FieldVal) IsOddBit() uint32 { + // Only odd numbers have the bottom bit set. + return f.n[0] & 1 +} + +// IsOdd returns whether or not the field value is an odd number in constant +// time. +// +// Preconditions: +// - The field value MUST be normalized +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 in constant +// time. +// +// Preconditions: +// - Both field values being compared MUST be normalized +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 in constant +// time. 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. +// +// Preconditions: +// - The max magnitude MUST be 63 +// Output Normalized: No +// Output Max Magnitude: Input magnitude + 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 congruent to zero (mod m), the answer can + // be shortcut by simply multiplying 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 in constant time. 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. +// +// Preconditions: +// - The max magnitude MUST be 63 +// Output Normalized: No +// Output Max Magnitude: Input magnitude + 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 in constant time. This is a convenience function since it is +// fairly common to perform some arithmetic 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. +// +// Preconditions: +// - The field value MUST have a max magnitude of 63 +// Output Normalized: No +// Output Max Magnitude: Existing field magnitude + 1 +func (f *FieldVal) AddInt(ui uint16) *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 in constant time. +// +// The field value is returned to support chaining. This enables syntax like: +// f.Add(f2).AddInt(1) so that f = f + f2 + 1. +// +// Preconditions: +// - The sum of the magnitudes of the two field values MUST be a max of 64 +// Output Normalized: No +// Output Max Magnitude: Sum of the magnitude of the two individual field values +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 in +// constant time. +// +// The field value is returned to support chaining. This enables syntax like: +// f3.Add2(f, f2).AddInt(1) so that f3 = f + f2 + 1. +// +// Preconditions: +// - The sum of the magnitudes of the two field values MUST be a max of 64 +// Output Normalized: No +// Output Max Magnitude: Sum of the magnitude of the two field values +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 in constant time. 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. +// +// Preconditions: +// - The field value magnitude multiplied by given val MUST be a max of 64 +// Output Normalized: No +// Output Max Magnitude: Existing field magnitude times the provided integer val +func (f *FieldVal) MulInt(val uint8) *FieldVal { + // Since each word of the field representation can hold up to + // 32 - fieldBase 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 in constant time. 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. +// +// Preconditions: +// - Both field values MUST have a max magnitude of 8 +// Output Normalized: No +// Output Max Magnitude: 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 in constant time. 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. +// +// Preconditions: +// - Both input field values MUST have a max magnitude of 8 +// Output Normalized: No +// Output Max Magnitude: 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 +} + +// SquareRootVal either calculates the square root of the passed value when it +// exists or the square root of the negation of the value when it does not exist +// and stores the result in f in constant time. The return flag is true when +// the calculated square root is for the passed value itself and false when it +// is for its negation. +// +// 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 magnitude of the result +// will be 1. +// +// Preconditions: +// - The input field value MUST have a max magnitude of 8 +// Output Normalized: No +// Output Max Magnitude: 1 +func (f *FieldVal) SquareRootVal(val *FieldVal) bool { + // This uses the Tonelli-Shanks method for calculating the square root of + // the value when it exists. The key principles of the method follow. + // + // Fermat's little theorem states that for a nonzero number 'a' and prime + // 'p', a^(p-1) ≡ 1 (mod p). + // + // Further, Euler's criterion states that an integer 'a' has a square root + // (aka is a quadratic residue) modulo a prime if a^((p-1)/2) ≡ 1 (mod p) + // and, conversely, when it does NOT have a square root (aka 'a' is a + // non-residue) a^((p-1)/2) ≡ -1 (mod p). + // + // This can be seen by considering that Fermat's little theorem can be + // written as (a^((p-1)/2) - 1)(a^((p-1)/2) + 1) ≡ 0 (mod p). Therefore, + // one of the two factors must be 0. Then, when a ≡ x^2 (aka 'a' is a + // quadratic residue), (x^2)^((p-1)/2) ≡ x^(p-1) ≡ 1 (mod p) which implies + // the first factor must be zero. Finally, per Lagrange's theorem, the + // non-residues are the only remaining possible solutions and thus must make + // the second factor zero to satisfy Fermat's little theorem implying that + // a^((p-1)/2) ≡ -1 (mod p) for that case. + // + // The Tonelli-Shanks method uses these facts along with factoring out + // powers of two to solve a congruence that results in either the solution + // when the square root exists or the square root of the negation of the + // value when it does not. In the case of primes that are ≡ 3 (mod 4), the + // possible solutions are r = ±a^((p+1)/4) (mod p). Therefore, either r^2 ≡ + // a (mod p) is true in which case ±r are the two solutions, or r^2 ≡ -a + // (mod p) in which case 'a' is a non-residue and there are no solutions. + // + // The secp256k1 prime is ≡ 3 (mod 4), so this result applies. + // + // In other words, calculate a^((p+1)/4) and then square it and check it + // against the original value to determine if it is actually the square + // root. + // + // In order to efficiently compute a^((p+1)/4), (p+1)/4 needs to be split + // into a sequence of squares and multiplications that minimizes the number + // of multiplications needed (since they are more costly than squarings). + // + // The secp256k1 prime + 1 / 4 is 2^254 - 2^30 - 244. In binary, that is: + // + // 00111111 11111111 11111111 11111111 + // 11111111 11111111 11111111 11111111 + // 11111111 11111111 11111111 11111111 + // 11111111 11111111 11111111 11111111 + // 11111111 11111111 11111111 11111111 + // 11111111 11111111 11111111 11111111 + // 11111111 11111111 11111111 11111111 + // 10111111 11111111 11111111 00001100 + // + // Notice that can be broken up into three windows of consecutive 1s (in + // order of least to most signifcant) as: + // + // 6-bit window with two bits set (bits 4, 5, 6, 7 unset) + // 23-bit window with 22 bits set (bit 30 unset) + // 223-bit window with all 223 bits set + // + // Thus, the groups of 1 bits in each window forms the set: + // S = {2, 22, 223}. + // + // The strategy is to calculate a^(2^n - 1) for each grouping via an + // addition chain with a sliding window. + // + // The addition chain used is (credits to Peter Dettman): + // (0,0),(1,0),(2,2),(3,2),(4,1),(5,5),(6,6),(7,7),(8,8),(9,7),(10,2) + // => 2^1 2^[2] 2^3 2^6 2^9 2^11 2^[22] 2^44 2^88 2^176 2^220 2^[223] + // + // This has a cost of 254 field squarings and 13 field multiplications. + var a, a2, a3, a6, a9, a11, a22, a44, a88, a176, a220, a223 FieldVal + a.Set(val) + a2.SquareVal(&a).Mul(&a) // a2 = a^(2^2 - 1) + a3.SquareVal(&a2).Mul(&a) // a3 = a^(2^3 - 1) + a6.SquareVal(&a3).Square().Square() // a6 = a^(2^6 - 2^3) + a6.Mul(&a3) // a6 = a^(2^6 - 1) + a9.SquareVal(&a6).Square().Square() // a9 = a^(2^9 - 2^3) + a9.Mul(&a3) // a9 = a^(2^9 - 1) + a11.SquareVal(&a9).Square() // a11 = a^(2^11 - 2^2) + a11.Mul(&a2) // a11 = a^(2^11 - 1) + a22.SquareVal(&a11).Square().Square().Square().Square() // a22 = a^(2^16 - 2^5) + a22.Square().Square().Square().Square().Square() // a22 = a^(2^21 - 2^10) + a22.Square() // a22 = a^(2^22 - 2^11) + a22.Mul(&a11) // a22 = a^(2^22 - 1) + a44.SquareVal(&a22).Square().Square().Square().Square() // a44 = a^(2^27 - 2^5) + a44.Square().Square().Square().Square().Square() // a44 = a^(2^32 - 2^10) + a44.Square().Square().Square().Square().Square() // a44 = a^(2^37 - 2^15) + a44.Square().Square().Square().Square().Square() // a44 = a^(2^42 - 2^20) + a44.Square().Square() // a44 = a^(2^44 - 2^22) + a44.Mul(&a22) // a44 = a^(2^44 - 1) + a88.SquareVal(&a44).Square().Square().Square().Square() // a88 = a^(2^49 - 2^5) + a88.Square().Square().Square().Square().Square() // a88 = a^(2^54 - 2^10) + a88.Square().Square().Square().Square().Square() // a88 = a^(2^59 - 2^15) + a88.Square().Square().Square().Square().Square() // a88 = a^(2^64 - 2^20) + a88.Square().Square().Square().Square().Square() // a88 = a^(2^69 - 2^25) + a88.Square().Square().Square().Square().Square() // a88 = a^(2^74 - 2^30) + a88.Square().Square().Square().Square().Square() // a88 = a^(2^79 - 2^35) + a88.Square().Square().Square().Square().Square() // a88 = a^(2^84 - 2^40) + a88.Square().Square().Square().Square() // a88 = a^(2^88 - 2^44) + a88.Mul(&a44) // a88 = a^(2^88 - 1) + a176.SquareVal(&a88).Square().Square().Square().Square() // a176 = a^(2^93 - 2^5) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^98 - 2^10) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^103 - 2^15) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^108 - 2^20) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^113 - 2^25) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^118 - 2^30) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^123 - 2^35) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^128 - 2^40) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^133 - 2^45) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^138 - 2^50) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^143 - 2^55) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^148 - 2^60) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^153 - 2^65) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^158 - 2^70) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^163 - 2^75) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^168 - 2^80) + a176.Square().Square().Square().Square().Square() // a176 = a^(2^173 - 2^85) + a176.Square().Square().Square() // a176 = a^(2^176 - 2^88) + a176.Mul(&a88) // a176 = a^(2^176 - 1) + a220.SquareVal(&a176).Square().Square().Square().Square() // a220 = a^(2^181 - 2^5) + a220.Square().Square().Square().Square().Square() // a220 = a^(2^186 - 2^10) + a220.Square().Square().Square().Square().Square() // a220 = a^(2^191 - 2^15) + a220.Square().Square().Square().Square().Square() // a220 = a^(2^196 - 2^20) + a220.Square().Square().Square().Square().Square() // a220 = a^(2^201 - 2^25) + a220.Square().Square().Square().Square().Square() // a220 = a^(2^206 - 2^30) + a220.Square().Square().Square().Square().Square() // a220 = a^(2^211 - 2^35) + a220.Square().Square().Square().Square().Square() // a220 = a^(2^216 - 2^40) + a220.Square().Square().Square().Square() // a220 = a^(2^220 - 2^44) + a220.Mul(&a44) // a220 = a^(2^220 - 1) + a223.SquareVal(&a220).Square().Square() // a223 = a^(2^223 - 2^3) + a223.Mul(&a3) // a223 = a^(2^223 - 1) + + f.SquareVal(&a223).Square().Square().Square().Square() // f = a^(2^228 - 2^5) + f.Square().Square().Square().Square().Square() // f = a^(2^233 - 2^10) + f.Square().Square().Square().Square().Square() // f = a^(2^238 - 2^15) + f.Square().Square().Square().Square().Square() // f = a^(2^243 - 2^20) + f.Square().Square().Square() // f = a^(2^246 - 2^23) + f.Mul(&a22) // f = a^(2^246 - 2^22 - 1) + f.Square().Square().Square().Square().Square() // f = a^(2^251 - 2^27 - 2^5) + f.Square() // f = a^(2^252 - 2^28 - 2^6) + f.Mul(&a2) // f = a^(2^252 - 2^28 - 2^6 - 2^1 - 1) + f.Square().Square() // f = a^(2^254 - 2^30 - 2^8 - 2^3 - 2^2) + // // = a^(2^254 - 2^30 - 244) + // // = a^((p+1)/4) + + // Ensure the calculated result is actually the square root by squaring it + // and checking against the original value. + var sqr FieldVal + return sqr.SquareVal(f).Normalize().Equals(val.Normalize()) +} + +// Square squares the field value in constant time. 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. +// +// Preconditions: +// - The field value MUST have a max magnitude of 8 +// Output Normalized: No +// Output Max Magnitude: 1 +func (f *FieldVal) Square() *FieldVal { + return f.SquareVal(f) +} + +// SquareVal squares the passed value and stores the result in f in constant +// time. 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 squared 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. +// +// Preconditions: +// - The input field value MUST have a max magnitude of 8 +// Output Normalized: No +// Output Max Magnitude: 1 +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 in +// constant time. 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. +// +// Preconditions: +// - The field value MUST have a max magnitude of 8 +// Output Normalized: No +// Output Max Magnitude: 1 +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 multiplicative 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 multiplications 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) +} + +// IsGtOrEqPrimeMinusOrder returns whether or not the field value exceeds the +// group order divided by 2 in constant time. +// +// Preconditions: +// - The field value MUST be normalized +func (f *FieldVal) IsGtOrEqPrimeMinusOrder() bool { + // The secp256k1 prime is equivalent to 2^256 - 4294968273 and the group + // order is 2^256 - 432420386565659656852420866394968145599. Thus, + // the prime minus the group order is: + // 432420386565659656852420866390673177326 + // + // In hex that is: + // 0x00000000 00000000 00000000 00000001 45512319 50b75fc4 402da172 2fc9baee + // + // Converting that to field representation (base 2^26) is: + // + // n[0] = 0x03c9baee + // n[1] = 0x03685c8b + // n[2] = 0x01fc4402 + // n[3] = 0x006542dd + // n[4] = 0x01455123 + // + // This can be verified with the following test code: + // pMinusN := new(big.Int).Sub(curveParams.P, curveParams.N) + // var fv FieldVal + // fv.SetByteSlice(pMinusN.Bytes()) + // t.Logf("%x", fv.n) + // + // Outputs: [3c9baee 3685c8b 1fc4402 6542dd 1455123 0 0 0 0 0] + const ( + pMinusNWordZero = 0x03c9baee + pMinusNWordOne = 0x03685c8b + pMinusNWordTwo = 0x01fc4402 + pMinusNWordThree = 0x006542dd + pMinusNWordFour = 0x01455123 + pMinusNWordFive = 0x00000000 + pMinusNWordSix = 0x00000000 + pMinusNWordSeven = 0x00000000 + pMinusNWordEight = 0x00000000 + pMinusNWordNine = 0x00000000 + ) + + // The intuition here is that the value is greater than field prime minus + // the group order if one of the higher individual words is greater than the + // corresponding word and all higher words in the value are equal. + result := constantTimeGreater(f.n[9], pMinusNWordNine) + highWordsEqual := constantTimeEq(f.n[9], pMinusNWordNine) + result |= highWordsEqual & constantTimeGreater(f.n[8], pMinusNWordEight) + highWordsEqual &= constantTimeEq(f.n[8], pMinusNWordEight) + result |= highWordsEqual & constantTimeGreater(f.n[7], pMinusNWordSeven) + highWordsEqual &= constantTimeEq(f.n[7], pMinusNWordSeven) + result |= highWordsEqual & constantTimeGreater(f.n[6], pMinusNWordSix) + highWordsEqual &= constantTimeEq(f.n[6], pMinusNWordSix) + result |= highWordsEqual & constantTimeGreater(f.n[5], pMinusNWordFive) + highWordsEqual &= constantTimeEq(f.n[5], pMinusNWordFive) + result |= highWordsEqual & constantTimeGreater(f.n[4], pMinusNWordFour) + highWordsEqual &= constantTimeEq(f.n[4], pMinusNWordFour) + result |= highWordsEqual & constantTimeGreater(f.n[3], pMinusNWordThree) + highWordsEqual &= constantTimeEq(f.n[3], pMinusNWordThree) + result |= highWordsEqual & constantTimeGreater(f.n[2], pMinusNWordTwo) + highWordsEqual &= constantTimeEq(f.n[2], pMinusNWordTwo) + result |= highWordsEqual & constantTimeGreater(f.n[1], pMinusNWordOne) + highWordsEqual &= constantTimeEq(f.n[1], pMinusNWordOne) + result |= highWordsEqual & constantTimeGreaterOrEq(f.n[0], pMinusNWordZero) + + return result != 0 +} |