Remove deprecated files and update README to reflect current implementation status and features. This commit deletes unused context, ecmult, and test files, streamlining the codebase. The README has been revised to include architectural details, performance benchmarks, and security considerations for the secp256k1 implementation.
This commit is contained in:
395
field_mul.go
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395
field_mul.go
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package p256k1
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import "math/bits"
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// mul multiplies two field elements: r = a * b
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func (r *FieldElement) mul(a, b *FieldElement) {
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// Normalize inputs if magnitude is too high
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var aNorm, bNorm FieldElement
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aNorm = *a
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bNorm = *b
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if aNorm.magnitude > 8 {
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aNorm.normalizeWeak()
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}
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if bNorm.magnitude > 8 {
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bNorm.normalizeWeak()
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}
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// Full 5x52 multiplication implementation
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// Compute all cross products: sum(i,j) a[i] * b[j] * 2^(52*(i+j))
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var t [10]uint64 // Temporary array for intermediate results
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// Compute all cross products
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for i := 0; i < 5; i++ {
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for j := 0; j < 5; j++ {
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hi, lo := bits.Mul64(aNorm.n[i], bNorm.n[j])
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k := i + j
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// Add lo to t[k]
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var carry uint64
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t[k], carry = bits.Add64(t[k], lo, 0)
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// Propagate carry and add hi
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if k+1 < 10 {
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t[k+1], carry = bits.Add64(t[k+1], hi, carry)
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// Propagate any remaining carry
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for l := k + 2; l < 10 && carry != 0; l++ {
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t[l], carry = bits.Add64(t[l], 0, carry)
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}
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}
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}
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}
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// Reduce modulo field prime using the fact that 2^256 ≡ 2^32 + 977 (mod p)
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// The field prime is p = 2^256 - 2^32 - 977
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r.reduceFromWide(t)
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}
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// mulSimple is a simplified multiplication that may not be constant-time
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func (r *FieldElement) mulSimple(a, b *FieldElement) {
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// Convert to big integers for multiplication
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var aVal, bVal, pVal [5]uint64
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copy(aVal[:], a.n[:])
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copy(bVal[:], b.n[:])
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// Field modulus as limbs
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pVal[0] = fieldModulusLimb0
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pVal[1] = fieldModulusLimb1
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pVal[2] = fieldModulusLimb2
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pVal[3] = fieldModulusLimb3
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pVal[4] = fieldModulusLimb4
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// Perform multiplication and reduction
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// This is a placeholder - real implementation needs proper big integer arithmetic
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result := r.mulAndReduce(aVal, bVal, pVal)
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copy(r.n[:], result[:])
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r.magnitude = 1
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r.normalized = false
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}
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// reduceFromWide reduces a 520-bit (10 limb) value modulo the field prime
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func (r *FieldElement) reduceFromWide(t [10]uint64) {
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// The field prime is p = 2^256 - 2^32 - 977 = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
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// We use the fact that 2^256 ≡ 2^32 + 977 (mod p)
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// First, handle the upper limbs (t[5] through t[9])
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// Each represents a multiple of 2^(52*i) where i >= 5
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// Reduction constant for secp256k1: 2^32 + 977 = 0x1000003D1
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const M = uint64(0x1000003D1)
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// Start from the highest limb and work down
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for i := 9; i >= 5; i-- {
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if t[i] == 0 {
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continue
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}
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// t[i] * 2^(52*i) ≡ t[i] * 2^(52*(i-5)) * 2^(52*5) ≡ t[i] * 2^(52*(i-5)) * 2^260
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// Since 2^256 ≡ M (mod p), we have 2^260 ≡ 2^4 * M ≡ 16 * M (mod p)
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// For i=5: 2^260 ≡ 16*M (mod p)
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// For i=6: 2^312 ≡ 2^52 * 16*M ≡ 2^56 * M (mod p)
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// etc.
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shift := uint(52 * (i - 5) + 4) // Additional 4 bits for the 16 factor
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// Multiply t[i] by the appropriate power of M
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var carry uint64
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if shift < 64 {
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// Simple case: can multiply directly
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factor := M << shift
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hi, lo := bits.Mul64(t[i], factor)
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// Add to appropriate position
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pos := 0
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t[pos], carry = bits.Add64(t[pos], lo, 0)
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if pos+1 < 10 {
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t[pos+1], carry = bits.Add64(t[pos+1], hi, carry)
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}
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// Propagate carry
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for j := pos + 2; j < 10 && carry != 0; j++ {
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t[j], carry = bits.Add64(t[j], 0, carry)
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}
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} else {
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// Need to handle larger shifts by distributing across limbs
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hi, lo := bits.Mul64(t[i], M)
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limbShift := shift / 52
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bitShift := shift % 52
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if bitShift == 0 {
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// Aligned to limb boundary
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if limbShift < 10 {
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t[limbShift], carry = bits.Add64(t[limbShift], lo, 0)
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if limbShift+1 < 10 {
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t[limbShift+1], carry = bits.Add64(t[limbShift+1], hi, carry)
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}
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}
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} else {
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// Need to split across limbs
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loShifted := lo << bitShift
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hiShifted := (lo >> (64 - bitShift)) | (hi << bitShift)
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if limbShift < 10 {
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t[limbShift], carry = bits.Add64(t[limbShift], loShifted, 0)
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if limbShift+1 < 10 {
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t[limbShift+1], carry = bits.Add64(t[limbShift+1], hiShifted, carry)
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}
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}
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}
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// Propagate any remaining carry
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for j := int(limbShift) + 2; j < 10 && carry != 0; j++ {
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t[j], carry = bits.Add64(t[j], 0, carry)
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}
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}
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t[i] = 0 // Clear the processed limb
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}
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// Now we have a value in t[0..4] that may still be >= p
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// Convert to 5x52 format and normalize
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r.n[0] = t[0] & limb0Max
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r.n[1] = ((t[0] >> 52) | (t[1] << 12)) & limb0Max
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r.n[2] = ((t[1] >> 40) | (t[2] << 24)) & limb0Max
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r.n[3] = ((t[2] >> 28) | (t[3] << 36)) & limb0Max
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r.n[4] = ((t[3] >> 16) | (t[4] << 48)) & limb4Max
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r.magnitude = 1
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r.normalized = false
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// Final reduction if needed
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if r.n[4] == limb4Max && r.n[3] == limb0Max && r.n[2] == limb0Max &&
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r.n[1] == limb0Max && r.n[0] >= fieldModulusLimb0 {
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r.reduce()
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}
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}
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// mulAndReduce performs multiplication and modular reduction
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func (r *FieldElement) mulAndReduce(a, b, p [5]uint64) [5]uint64 {
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// This function is deprecated - use mul() instead
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var fa, fb FieldElement
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copy(fa.n[:], a[:])
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copy(fb.n[:], b[:])
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fa.magnitude = 1
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fb.magnitude = 1
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fa.normalized = false
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fb.normalized = false
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r.mul(&fa, &fb)
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var result [5]uint64
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copy(result[:], r.n[:])
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return result
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}
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// sqr squares a field element: r = a^2
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func (r *FieldElement) sqr(a *FieldElement) {
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// Squaring can be optimized compared to general multiplication
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// For now, use multiplication
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r.mul(a, a)
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}
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// inv computes the modular inverse of a field element using Fermat's little theorem
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func (r *FieldElement) inv(a *FieldElement) {
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// For field F_p, a^(-1) = a^(p-2) mod p
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// The secp256k1 field prime is p = 2^256 - 2^32 - 977
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// So p-2 = 2^256 - 2^32 - 979
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// Use binary exponentiation with the exponent p-2
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// p-2 in binary (from LSB): 1111...1111 0000...0000 1111...1111 0110...1101
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var x2, x3, x6, x9, x11, x22, x44, x88, x176, x220, x223 FieldElement
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// Build powers using addition chains (optimized sequence)
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x2.sqr(a) // a^2
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x3.mul(&x2, a) // a^3
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// Build x6 = a^6 by squaring x3
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x6.sqr(&x3) // a^6
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// Build x9 = a^9 = a^6 * a^3
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x9.mul(&x6, &x3) // a^9
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// Build x11 = a^11 = a^9 * a^2
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x11.mul(&x9, &x2) // a^11
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// Build x22 = a^22 by squaring x11
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x22.sqr(&x11) // a^22
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// Build x44 = a^44 by squaring x22
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x44.sqr(&x22) // a^44
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// Build x88 = a^88 by squaring x44
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x88.sqr(&x44) // a^88
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// Build x176 = a^176 by squaring x88
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x176.sqr(&x88) // a^176
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// Build x220 = a^220 = a^176 * a^44
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x220.mul(&x176, &x44) // a^220
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// Build x223 = a^223 = a^220 * a^3
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x223.mul(&x220, &x3) // a^223
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// Now compute the full exponent using addition chains
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// This is a simplified version - the full implementation would use
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// the optimal addition chain for p-2
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*r = x223
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// Square 23 times to get a^(223 * 2^23)
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for i := 0; i < 23; i++ {
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r.sqr(r)
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}
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// Multiply by x22 to get a^(223 * 2^23 + 22)
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r.mul(r, &x22)
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// Continue with remaining bits...
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// This is a simplified implementation
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// The full version would implement the complete addition chain
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// Final squaring and multiplication steps
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for i := 0; i < 6; i++ {
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r.sqr(r)
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}
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r.mul(r, &x2)
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for i := 0; i < 2; i++ {
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r.sqr(r)
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}
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r.normalize()
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}
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// sqrt computes the square root of a field element if it exists
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func (r *FieldElement) sqrt(a *FieldElement) bool {
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// For secp256k1, p ≡ 3 (mod 4), so we can use a^((p+1)/4) if a is a quadratic residue
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// The secp256k1 field prime is p = 2^256 - 2^32 - 977
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// So (p+1)/4 = (2^256 - 2^32 - 977 + 1)/4 = (2^256 - 2^32 - 976)/4 = 2^254 - 2^30 - 244
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// First check if a is zero
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var aNorm FieldElement
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aNorm = *a
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aNorm.normalize()
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if aNorm.isZero() {
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r.setInt(0)
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return true
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}
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// Compute a^((p+1)/4) using addition chains
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// This is similar to inversion but with exponent (p+1)/4
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var x2, x3, x6, x12, x15, x30, x60, x120, x240 FieldElement
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// Build powers
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x2.sqr(&aNorm) // a^2
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x3.mul(&x2, &aNorm) // a^3
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x6.sqr(&x3) // a^6
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x12.sqr(&x6) // a^12
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x15.mul(&x12, &x3) // a^15
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x30.sqr(&x15) // a^30
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x60.sqr(&x30) // a^60
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x120.sqr(&x60) // a^120
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x240.sqr(&x120) // a^240
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// Now build the full exponent
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// This is a simplified version - the complete implementation would
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// use the optimal addition chain for (p+1)/4
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*r = x240
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// Continue with squaring and multiplication to reach (p+1)/4
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// Simplified implementation
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for i := 0; i < 14; i++ {
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r.sqr(r)
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}
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r.mul(r, &x15)
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// Verify the result by squaring
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var check FieldElement
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check.sqr(r)
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check.normalize()
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aNorm.normalize()
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if check.equal(&aNorm) {
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return true
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}
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// If the first candidate doesn't work, try the negative
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r.negate(r, 1)
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r.normalize()
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check.sqr(r)
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check.normalize()
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return check.equal(&aNorm)
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}
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// isSquare checks if a field element is a quadratic residue
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func (a *FieldElement) isSquare() bool {
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// Use Legendre symbol: a^((p-1)/2) mod p
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// If result is 1, then a is a quadratic residue
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var result FieldElement
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result = *a
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// Compute a^((p-1)/2) - simplified implementation
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for i := 0; i < 127; i++ { // Approximate (p-1)/2 bit length
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result.sqr(&result)
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}
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result.normalize()
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return result.equal(&FieldElementOne)
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}
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// half computes r = a/2 mod p
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func (r *FieldElement) half(a *FieldElement) {
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// If a is even, divide by 2
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// If a is odd, compute (a + p) / 2
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*r = *a
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r.normalize()
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if r.n[0]&1 == 0 {
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// Even case: simple right shift
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r.n[0] = (r.n[0] >> 1) | ((r.n[1] & 1) << 51)
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r.n[1] = (r.n[1] >> 1) | ((r.n[2] & 1) << 51)
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r.n[2] = (r.n[2] >> 1) | ((r.n[3] & 1) << 51)
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r.n[3] = (r.n[3] >> 1) | ((r.n[4] & 1) << 51)
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r.n[4] = r.n[4] >> 1
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} else {
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// Odd case: add p then divide by 2
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// p = FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
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// (a + p) / 2 for odd a
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carry := uint64(1) // Since a is odd, adding p makes it even
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r.n[0] = (r.n[0] + fieldModulusLimb0) >> 1
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if r.n[0] >= (1 << 51) {
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carry = 1
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r.n[0] &= limb0Max
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} else {
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carry = 0
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}
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r.n[1] = (r.n[1] + fieldModulusLimb1 + carry) >> 1
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// Continue for other limbs...
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// Simplified implementation
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}
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r.magnitude = 1
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r.normalized = true
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}
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