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85 lines
2.2 KiB
Go
85 lines
2.2 KiB
Go
// Package math provides some utility functions for big integers.
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package math
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import "math/big"
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// SignedDigit obtains the signed-digit recoding of n and returns a list L of
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// digits such that n = sum( L[i]*2^(i*(w-1)) ), and each L[i] is an odd number
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// in the set {±1, ±3, ..., ±2^(w-1)-1}. The third parameter ensures that the
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// output has ceil(l/(w-1)) digits.
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//
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// Restrictions:
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// - n is odd and n > 0.
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// - 1 < w < 32.
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// - l >= bit length of n.
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//
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// References:
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// - Alg.6 in "Exponent Recoding and Regular Exponentiation Algorithms"
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// by Joye-Tunstall. http://doi.org/10.1007/978-3-642-02384-2_21
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// - Alg.6 in "Selecting Elliptic Curves for Cryptography: An Efficiency and
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// Security Analysis" by Bos et al. http://doi.org/10.1007/s13389-015-0097-y
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func SignedDigit(n *big.Int, w, l uint) []int32 {
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if n.Sign() <= 0 || n.Bit(0) == 0 {
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panic("n must be non-zero, odd, and positive")
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}
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if w <= 1 || w >= 32 {
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panic("Verify that 1 < w < 32")
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}
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if uint(n.BitLen()) > l {
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panic("n is too big to fit in l digits")
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}
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lenN := (l + (w - 1) - 1) / (w - 1) // ceil(l/(w-1))
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L := make([]int32, lenN+1)
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var k, v big.Int
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k.Set(n)
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var i uint
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for i = 0; i < lenN; i++ {
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words := k.Bits()
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value := int32(words[0] & ((1 << w) - 1))
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value -= int32(1) << (w - 1)
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L[i] = value
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v.SetInt64(int64(value))
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k.Sub(&k, &v)
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k.Rsh(&k, w-1)
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}
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L[i] = int32(k.Int64())
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return L
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}
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// OmegaNAF obtains the window-w Non-Adjacent Form of a positive number n and
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// 1 < w < 32. The returned slice L holds n = sum( L[i]*2^i ).
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//
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// Reference:
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// - Alg.9 "Efficient arithmetic on Koblitz curves" by Solinas.
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// http://doi.org/10.1023/A:1008306223194
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func OmegaNAF(n *big.Int, w uint) (L []int32) {
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if n.Sign() < 0 {
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panic("n must be positive")
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}
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if w <= 1 || w >= 32 {
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panic("Verify that 1 < w < 32")
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}
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L = make([]int32, n.BitLen()+1)
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var k, v big.Int
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k.Set(n)
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i := 0
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for ; k.Sign() > 0; i++ {
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value := int32(0)
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if k.Bit(0) == 1 {
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words := k.Bits()
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value = int32(words[0] & ((1 << w) - 1))
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if value >= (int32(1) << (w - 1)) {
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value -= int32(1) << w
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}
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v.SetInt64(int64(value))
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k.Sub(&k, &v)
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}
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L[i] = value
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k.Rsh(&k, 1)
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}
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return L[:i]
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}
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