BIG.go

/*
 * Copyright (c) 2012-2020 MIRACL UK Ltd.
 *
 * This file is part of MIRACL Core
 * (see https://github.com/miracl/core).
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *     http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

/* core BIG number class */

package SECP160R1

import (
	"math/bits"
	"strconv"

	"github.com/akakou-fork/amcl-go/miracl/core"
)

//import "fmt"

type BIG struct {
	w [NLEN]Chunk
}

type DBIG struct {
	w [2 * NLEN]Chunk
}

/***************** 64-bit specific code ****************/

/* First the 32/64-bit dependent BIG code */
/* Note that because of the lack of a 128-bit integer, 32 and 64-bit code needs to be done differently */

/* return a*b as DBIG */
func mul(a *BIG, b *BIG) *DBIG {
	c := NewDBIG()
	carry := Chunk(0)

	for i := 0; i < NLEN; i++ {
		carry = 0
		for j := 0; j < NLEN; j++ {
			carry, c.w[i+j] = muladd(a.w[i], b.w[j], carry, c.w[i+j])
		}
		c.w[NLEN+i] = carry
	}

	return c
}

/* return a^2 as DBIG */
func sqr(a *BIG) *DBIG {
	c := NewDBIG()
	carry := Chunk(0)

	for i := 0; i < NLEN; i++ {
		carry = 0
		for j := i + 1; j < NLEN; j++ {
			//if a.w[i]<0 {fmt.Printf("Negative m i in sqr\n")}
			//if a.w[j]<0 {fmt.Printf("Negative m j in sqr\n")}
			carry, c.w[i+j] = muladd(2*a.w[i], a.w[j], carry, c.w[i+j])
		}
		c.w[NLEN+i] = carry
	}

	for i := 0; i < NLEN; i++ {
		//if a.w[i]<0 {fmt.Printf("Negative m s in sqr\n")}
		top, bot := muladd(a.w[i], a.w[i], 0, c.w[2*i])

		c.w[2*i] = bot
		c.w[2*i+1] += top
	}
	c.norm()
	return c
}

func monty(md *BIG, mc Chunk, d *DBIG) *BIG {
	carry := Chunk(0)
	m := Chunk(0)
	for i := 0; i < NLEN; i++ {
		if mc == -1 {
			m = (-d.w[i]) & BMASK
		} else {
			if mc == 1 {
				m = d.w[i]
			} else {
				m = (mc * d.w[i]) & BMASK
			}
		}

		carry = 0
		for j := 0; j < NLEN; j++ {
			carry, d.w[i+j] = muladd(m, md.w[j], carry, d.w[i+j])
			//if m<0 {fmt.Printf("Negative m in monty\n")}
			//if md.w[j]<0 {fmt.Printf("Negative m in monty\n")}
		}
		d.w[NLEN+i] += carry
	}

	b := NewBIG()
	for i := 0; i < NLEN; i++ {
		b.w[i] = d.w[NLEN+i]
	}
	b.norm()
	return b
}

/* set this[i]+=x*y+c, and return high part */
func muladd(a Chunk, b Chunk, c Chunk, r Chunk) (Chunk, Chunk) {

	tp, bt := bits.Mul64(uint64(a), uint64(b)) // use math/bits intrinsic
	bot := Chunk(bt & uint64(BMASK))
	top := Chunk((tp << (64 - BASEBITS)) | (bt >> BASEBITS))
	bot += c
	bot += r
	carry := bot >> BASEBITS
	bot &= BMASK
	top += carry
	return top, bot

}

/************************************************************/

func (r *BIG) get(i int) Chunk {
	return r.w[i]
}

func (r *BIG) set(i int, x Chunk) {
	r.w[i] = x
}

func (r *BIG) xortop(x Chunk) {
	r.w[NLEN-1] ^= x
}

/* normalise BIG - force all digits < 2^BASEBITS */
func (r *BIG) norm() Chunk {
	carry := Chunk(0)
	for i := 0; i < NLEN-1; i++ {
		d := r.w[i] + carry
		r.w[i] = d & BMASK
		carry = d >> BASEBITS
	}
	r.w[NLEN-1] = (r.w[NLEN-1] + carry)
	return (r.w[NLEN-1] >> ((8 * MODBYTES) % BASEBITS))
}

/* Shift right by less than a word */
func (r *BIG) fshr(k uint) int {
	w := r.w[0] & ((Chunk(1) << k) - 1) /* shifted out part */
	for i := 0; i < NLEN-1; i++ {
		r.w[i] = (r.w[i] >> k) | ((r.w[i+1] << (BASEBITS - k)) & BMASK)
	}
	r.w[NLEN-1] = r.w[NLEN-1] >> k
	return int(w)
}

/* Shift right by less than a word */
func (r *BIG) fshl(k uint) int {
	r.w[NLEN-1] = (r.w[NLEN-1] << k) | (r.w[NLEN-2] >> (BASEBITS - k))
	for i := NLEN - 2; i > 0; i-- {
		r.w[i] = ((r.w[i] << k) & BMASK) | (r.w[i-1] >> (BASEBITS - k))
	}
	r.w[0] = (r.w[0] << k) & BMASK
	return int(r.w[NLEN-1] >> ((8 * MODBYTES) % BASEBITS)) /* return excess - only used in ff.c */
}

func NewBIG() *BIG {
	b := new(BIG)
	for i := 0; i < NLEN; i++ {
		b.w[i] = 0
	}
	return b
}

func NewBIGints(x [NLEN]Chunk) *BIG {
	b := new(BIG)
	for i := 0; i < NLEN; i++ {
		b.w[i] = x[i]
	}
	return b
}

func NewBIGint(x int) *BIG {
	b := new(BIG)
	b.w[0] = Chunk(x)
	for i := 1; i < NLEN; i++ {
		b.w[i] = 0
	}
	return b
}

func NewBIGcopy(x *BIG) *BIG {
	b := new(BIG)
	for i := 0; i < NLEN; i++ {
		b.w[i] = x.w[i]
	}
	return b
}

func NewBIGdcopy(x *DBIG) *BIG {
	b := new(BIG)
	for i := 0; i < NLEN; i++ {
		b.w[i] = x.w[i]
	}
	return b
}

/* test for zero */
func (r *BIG) iszilch() bool {
	d := Chunk(0)
	for i := 0; i < NLEN; i++ {
		d |= r.w[i]
	}
	return (1 & ((d - 1) >> BASEBITS)) != 0
}

/* set to zero */
func (r *BIG) zero() {
	for i := 0; i < NLEN; i++ {
		r.w[i] = 0
	}
}

/* Test for equal to one */
func (r *BIG) isunity() bool {
	d := Chunk(0)
	for i := 1; i < NLEN; i++ {
		d |= r.w[i]
	}
	return (1 & ((d - 1) >> BASEBITS) & (((r.w[0] ^ 1) - 1) >> BASEBITS)) != 0
}

/* set to one */
func (r *BIG) one() {
	r.w[0] = 1
	for i := 1; i < NLEN; i++ {
		r.w[i] = 0
	}
}

/* Copy from another BIG */
func (r *BIG) copy(x *BIG) {
	for i := 0; i < NLEN; i++ {
		r.w[i] = x.w[i]
	}
}

/* Copy from another DBIG */
func (r *BIG) dcopy(x *DBIG) {
	for i := 0; i < NLEN; i++ {
		r.w[i] = x.w[i]
	}
}

/* Conditional swap of two bigs depending on d using XOR - no branches */
func (r *BIG) cswap(b *BIG, d int) Chunk {
	c := Chunk(-d)
	s := Chunk(0)
	v := r.w[0] ^ b.w[1]
	va := v + v
	va >>= 1
	for i := 0; i < NLEN; i++ {
		t := c & (r.w[i] ^ b.w[i])
		t ^= v
		e := r.w[i] ^ t
		s ^= e // to force calculation of e
		r.w[i] = e ^ va
		e = b.w[i] ^ t
		s ^= e
		b.w[i] = e ^ va
	}
	return s
}

func (r *BIG) cmove(g *BIG, d int) Chunk {
	b := Chunk(-d)
	s := Chunk(0)
	v := r.w[0] ^ g.w[1]
	va := v + v
	va >>= 1
	for i := 0; i < NLEN; i++ {
		t := (r.w[i] ^ g.w[i]) & b
		t ^= v
		e := r.w[i] ^ t
		s ^= e
		r.w[i] = e ^ va
	}
	return s
}

/* general shift right */
func (r *BIG) shr(k uint) {
	n := (k % BASEBITS)
	m := int(k / BASEBITS)
	for i := 0; i < NLEN-m-1; i++ {
		r.w[i] = (r.w[m+i] >> n) | ((r.w[m+i+1] << (BASEBITS - n)) & BMASK)
	}
	r.w[NLEN-m-1] = r.w[NLEN-1] >> n
	for i := NLEN - m; i < NLEN; i++ {
		r.w[i] = 0
	}
}

/* general shift left */
func (r *BIG) shl(k uint) {
	n := k % BASEBITS
	m := int(k / BASEBITS)

	r.w[NLEN-1] = (r.w[NLEN-1-m] << n)
	if NLEN >= m+2 {
		r.w[NLEN-1] |= (r.w[NLEN-m-2] >> (BASEBITS - n))
	}
	for i := NLEN - 2; i > m; i-- {
		r.w[i] = ((r.w[i-m] << n) & BMASK) | (r.w[i-m-1] >> (BASEBITS - n))
	}
	r.w[m] = (r.w[0] << n) & BMASK
	for i := 0; i < m; i++ {
		r.w[i] = 0
	}
}

/* return number of bits */
func (r *BIG) nbits() int {
	t := NewBIGcopy(r)
	k := NLEN - 1
	t.norm()
	for k >= 0 && t.w[k] == 0 {
		k--
	}
	if k < 0 {
		return 0
	}
	bts := int(BASEBITS) * k
	c := t.w[k]
	for c != 0 {
		c /= 2
		bts++
	}
	return bts
}

func (r *BIG) Nbits() int {
	return r.nbits()
}

/* Convert to Hex String */
func (r *BIG) ToString() string {
	s := ""
	len := r.nbits()

	if len%4 == 0 {
		len /= 4
	} else {
		len /= 4
		len++

	}
	MB := int(MODBYTES * 2)
	if len < MB {
		len = MB
	}

	for i := len - 1; i >= 0; i-- {
		b := NewBIGcopy(r)

		b.shr(uint(i * 4))
		s += strconv.FormatInt(int64(b.w[0]&15), 16)
	}
	return s
}

func (r *BIG) add(x *BIG) {
	for i := 0; i < NLEN; i++ {
		r.w[i] = r.w[i] + x.w[i]
	}
}

func (r *BIG) or(x *BIG) {
	for i := 0; i < NLEN; i++ {
		r.w[i] = r.w[i] | x.w[i]
	}
}

/* return this+x */
func (r *BIG) Plus(x *BIG) *BIG {
	s := new(BIG)
	for i := 0; i < NLEN; i++ {
		s.w[i] = r.w[i] + x.w[i]
	}
	s.norm()
	return s
}

/* this+=x, where x is int */
func (r *BIG) inc(x int) {
	r.norm()
	r.w[0] += Chunk(x)
}

/* this*=c and catch overflow in DBIG */
func (r *BIG) pxmul(c int) *DBIG {
	m := NewDBIG()
	carry := Chunk(0)
	for j := 0; j < NLEN; j++ {
		carry, m.w[j] = muladd(r.w[j], Chunk(c), carry, m.w[j])
		//if c<0 {fmt.Printf("Negative c in pxmul\n")}
		//if r.w[j]<0 {fmt.Printf("Negative c in pxmul\n")}
	}
	m.w[NLEN] = carry
	return m
}

/* return this-x */
func (r *BIG) Minus(x *BIG) *BIG {
	d := new(BIG)
	for i := 0; i < NLEN; i++ {
		d.w[i] = r.w[i] - x.w[i]
	}
	return d
}

/* this-=x */
func (r *BIG) sub(x *BIG) {
	for i := 0; i < NLEN; i++ {
		r.w[i] = r.w[i] - x.w[i]
	}
}

/* reverse subtract this=x-this */
func (r *BIG) rsub(x *BIG) {
	for i := 0; i < NLEN; i++ {
		r.w[i] = x.w[i] - r.w[i]
	}
}

/* this-=x, where x is int */
func (r *BIG) dec(x int) {
	r.norm()
	r.w[0] -= Chunk(x)
}

/* this*=x, where x is small int<NEXCESS */
func (r *BIG) imul(c int) {
	for i := 0; i < NLEN; i++ {
		r.w[i] *= Chunk(c)
	}
}

/* this*=x, where x is >NEXCESS */
func (r *BIG) pmul(c int) Chunk {
	carry := Chunk(0)
	//	r.norm();
	for i := 0; i < NLEN; i++ {
		ak := r.w[i]
		r.w[i] = 0
		carry, r.w[i] = muladd(ak, Chunk(c), carry, r.w[i])
		//if c<0 {fmt.Printf("Negative c in pmul\n")}
		//if ak<0 {fmt.Printf("Negative c in pmul\n")}
	}
	return carry
}

/* convert this BIG to byte array */
func (r *BIG) tobytearray(b []byte, n int) {
	//r.norm();
	c := NewBIGcopy(r)
	c.norm()

	for i := int(MODBYTES) - 1; i >= 0; i-- {
		b[i+n] = byte(c.w[0])
		c.fshr(8)
	}
}

/* convert from byte array to BIG */
func BIG_frombytearray(b []byte, n int) *BIG {
	m := NewBIG()
	l := len(b)
	for i := 0; i < int(MODBYTES); i++ {
		m.fshl(8)
		if i < l {
			m.w[0] += Chunk(int(b[i+n] & 0xff))
		} else {
			m.w[0] += Chunk(int(0 & 0xff))
		}
	}
	return m
}

func (r *BIG) ToBytes(b []byte) {
	r.tobytearray(b, 0)
}

func FromBytes(b []byte) *BIG {
	return BIG_frombytearray(b, 0)
}

/* divide by 3 */
func (r *BIG) div3() int {
	carry := Chunk(0)
	r.norm()
	base := (Chunk(1) << BASEBITS)
	for i := NLEN - 1; i >= 0; i-- {
		ak := (carry*base + r.w[i])
		r.w[i] = ak / 3
		carry = ak % 3
	}
	return int(carry)
}

/* return a*b where result fits in a BIG */
func smul(a *BIG, b *BIG) *BIG {
	carry := Chunk(0)
	c := NewBIG()
	for i := 0; i < NLEN; i++ {
		carry = 0
		for j := 0; j < NLEN; j++ {
			if i+j < NLEN {
				carry, c.w[i+j] = muladd(a.w[i], b.w[j], carry, c.w[i+j])
			}
		}
	}
	return c
}

/* Compare a and b, return 0 if a==b, -1 if a<b, +1 if a>b. Inputs must be normalised */
func Comp(a *BIG, b *BIG) int {
	gt := Chunk(0)
	eq := Chunk(1)
	for i := NLEN - 1; i >= 0; i-- {
		gt |= ((b.w[i] - a.w[i]) >> BASEBITS) & eq
		eq &= ((b.w[i] ^ a.w[i]) - 1) >> BASEBITS
	}
	return int(gt + gt + eq - 1)
}

/* return parity */
func (r *BIG) parity() int {
	return int(r.w[0] % 2)
}

/* return n-th bit */
func (r *BIG) bit(n int) int {
	return int((r.w[n/int(BASEBITS)] & (Chunk(1) << (uint(n) % BASEBITS))) >> (uint(n) % BASEBITS))
	//	if (r.w[n/int(BASEBITS)] & (Chunk(1) << (uint(n) % BASEBITS))) > 0 {
	//		return 1
	//	}
	//	return 0
}

/* return n last bits */
func (r *BIG) lastbits(n int) int {
	msk := (1 << uint(n)) - 1
	r.norm()
	return (int(r.w[0])) & msk
}

/* set x = x mod 2^m */
func (r *BIG) mod2m(m uint) {
	wd := int(m / BASEBITS)
	bt := m % BASEBITS
	msk := (Chunk(1) << bt) - 1
	r.w[wd] &= msk
	for i := wd + 1; i < NLEN; i++ {
		r.w[i] = 0
	}
}

/* a=1/a mod 2^256. This is very fast! */
func (r *BIG) invmod2m() {
	U := NewBIG()
	b := NewBIG()
	c := NewBIG()

	U.inc(invmod256(r.lastbits(8)))

	for i := 8; i < BIGBITS; i <<= 1 {
		U.norm()
		ui := uint(i)
		b.copy(r)
		b.mod2m(ui)
		t1 := smul(U, b)
		t1.shr(ui)
		c.copy(r)
		c.shr(ui)
		c.mod2m(ui)

		t2 := smul(U, c)
		t2.mod2m(ui)
		t1.add(t2)
		t1.norm()
		b = smul(t1, U)
		t1.copy(b)
		t1.mod2m(ui)

		t2.one()
		t2.shl(ui)
		t1.rsub(t2)
		t1.norm()
		t1.shl(ui)
		U.add(t1)
	}
	U.mod2m(8 * MODBYTES)
	r.copy(U)
	r.norm()
}

func (r *BIG) ctmod(m *BIG, bd uint) {
	k := bd
	sr := NewBIG()
	c := NewBIGcopy(m)
	r.norm()

	c.shl(k)

	for {
		sr.copy(r)
		sr.sub(c)
		sr.norm()
		r.cmove(sr, int(1-((sr.w[NLEN-1]>>uint(CHUNK-1))&1)))
		if k == 0 {
			break
		}
		c.fshr(1)
		k -= 1
	}
}

/* reduce this mod m */
func (r *BIG) Mod(m *BIG) {
	k := r.nbits() - m.nbits()
	if k < 0 {
		k = 0
	}
	r.ctmod(m, uint(k))
}

func (r *BIG) ctdiv(m *BIG, bd uint) {
	k := bd
	e := NewBIGint(1)
	sr := NewBIG()
	a := NewBIGcopy(r)
	c := NewBIGcopy(m)
	r.norm()
	r.zero()

	c.shl(k)
	e.shl(k)

	for {
		sr.copy(a)
		sr.sub(c)
		sr.norm()
		d := int(1 - ((sr.w[NLEN-1] >> uint(CHUNK-1)) & 1))
		a.cmove(sr, d)
		sr.copy(r)
		sr.add(e)
		sr.norm()
		r.cmove(sr, d)
		if k == 0 {
			break
		}
		c.fshr(1)
		e.fshr(1)
		k -= 1
	}
}

/* divide this by m */
func (r *BIG) div(m *BIG) {
	k := r.nbits() - m.nbits()
	if k < 0 {
		k = 0
	}
	r.ctdiv(m, uint(k))
}

/* get 8*MODBYTES size random number */
func Random(rng *core.RAND) *BIG {
	m := NewBIG()
	var j int = 0
	var r byte = 0
	/* generate random BIG */
	for i := 0; i < 8*int(MODBYTES); i++ {
		if j == 0 {
			r = rng.GetByte()
		} else {
			r >>= 1
		}

		b := Chunk(int(r & 1))
		m.shl(1)
		m.w[0] += b
		j++
		j &= 7
	}
	return m
}

/* Create random BIG in portable way, one bit at a time */
func Randomnum(q *BIG, rng *core.RAND) *BIG {
	d := NewDBIG()
	var j int = 0
	var r byte = 0
	for i := 0; i < 2*q.nbits(); i++ {
		if j == 0 {
			r = rng.GetByte()
		} else {
			r >>= 1
		}

		b := Chunk(int(r & 1))
		d.shl(1)
		d.w[0] += b
		j++
		j &= 7
	}
	m := d.Mod(q)
	return m
}

func Randtrunc(q *BIG, trunc int, rng *core.RAND) *BIG {
	m := Randomnum(q, rng)
	if q.nbits() > trunc {
		m.mod2m(uint(trunc))
	}
	return m
}

/* return a*b mod m */
func Modmul(a1, b1, m *BIG) *BIG {
	a := NewBIGcopy(a1)
	b := NewBIGcopy(b1)
	a.Mod(m)
	b.Mod(m)
	d := mul(a, b)
	return d.ctmod(m, uint(m.nbits()))
}

/* return a^2 mod m */
func Modsqr(a1, m *BIG) *BIG {
	a := NewBIGcopy(a1)
	a.Mod(m)
	d := sqr(a)
	return d.ctmod(m, uint(m.nbits()))
}

/* return -a mod m */
func Modneg(a1, m *BIG) *BIG {
	a := NewBIGcopy(a1)
	a.Mod(m)
	a.rsub(m)
	a.norm()
	return a
}

/* return a+b mod m */
func Modadd(a1, b1, m *BIG) *BIG {
	a := NewBIGcopy(a1)
	b := NewBIGcopy(b1)
	a.Mod(m)
	b.Mod(m)
	a.add(b)
	a.norm()
	a.ctmod(m, 1)
	return a
}

/* Jacobi Symbol (this/p). Returns 0, 1 or -1 */
func (r *BIG) Jacobi(p *BIG) int {
	m := 0
	t := NewBIGint(0)
	x := NewBIGint(0)
	n := NewBIGint(0)
	zilch := NewBIGint(0)
	one := NewBIGint(1)
	if p.parity() == 0 || Comp(r, zilch) == 0 || Comp(p, one) <= 0 {
		return 0
	}
	r.norm()
	x.copy(r)
	n.copy(p)
	x.Mod(p)

	for Comp(n, one) > 0 {
		if Comp(x, zilch) == 0 {
			return 0
		}
		n8 := n.lastbits(3)
		k := 0
		for x.parity() == 0 {
			k++
			x.shr(1)
		}
		if k%2 == 1 {
			m += (n8*n8 - 1) / 8
		}
		m += (n8 - 1) * (x.lastbits(2) - 1) / 4
		t.copy(n)
		t.Mod(x)
		n.copy(x)
		x.copy(t)
		m %= 2

	}
	if m == 0 {
		return 1
	}
	return -1
}

/* this=1/this mod p. Binary method */
func (r *BIG) Invmodp(p *BIG) {
	r.Mod(p)
	if r.iszilch() {
		return
	}
	u := NewBIGcopy(r)
	v := NewBIGcopy(p)
	x1 := NewBIGint(1)
	x2 := NewBIGint(0)
	t := NewBIGint(0)
	one := NewBIGint(1)
	for Comp(u, one) != 0 && Comp(v, one) != 0 {
		for u.parity() == 0 {
			u.fshr(1)
			t.copy(x1)
			t.add(p)
			x1.cmove(t, x1.parity())
			x1.norm()
			x1.fshr(1)
		}
		for v.parity() == 0 {
			v.fshr(1)
			t.copy(x2)
			t.add(p)
			x2.cmove(t, x2.parity())
			x2.norm()
			x2.fshr(1)
		}
		if Comp(u, v) >= 0 {
			u.sub(v)
			u.norm()
			t.copy(x1)
			t.add(p)
			x1.cmove(t, (Comp(x1, x2)>>1)&1)
			x1.sub(x2)
			x1.norm()
		} else {
			v.sub(u)
			v.norm()
			t.copy(x2)
			t.add(p)
			x2.cmove(t, (Comp(x2, x1)>>1)&1)
			x2.sub(x1)
			x2.norm()
		}
	}
	r.copy(x1)
	r.cmove(x2, Comp(u, one)&1)
}

/* return this^e mod m */
func (r *BIG) Powmod(e1 *BIG, m *BIG) *BIG {
	e := NewBIGcopy(e1)
	r.norm()
	e.norm()
	a := NewBIGint(1)
	z := NewBIGcopy(e)
	s := NewBIGcopy(r)
	for true {
		bt := z.parity()
		z.fshr(1)
		if bt == 1 {
			a = Modmul(a, s, m)
		}
		if z.iszilch() {
			break
		}
		s = Modsqr(s, m)
	}
	return a
}

/* Arazi and Qi inversion mod 256 */
func invmod256(a int) int {
	var t1 int = 0
	c := (a >> 1) & 1
	t1 += c
	t1 &= 1
	t1 = 2 - t1
	t1 <<= 1
	U := t1 + 1

	// i=2
	b := a & 3
	t1 = U * b
	t1 >>= 2
	c = (a >> 2) & 3
	t2 := (U * c) & 3
	t1 += t2
	t1 *= U
	t1 &= 3
	t1 = 4 - t1
	t1 <<= 2
	U += t1

	// i=4
	b = a & 15
	t1 = U * b
	t1 >>= 4
	c = (a >> 4) & 15
	t2 = (U * c) & 15
	t1 += t2
	t1 *= U
	t1 &= 15
	t1 = 16 - t1
	t1 <<= 4
	U += t1

	return U
}

func logb2(w uint32) uint {
	v := w
	v |= (v >> 1)
	v |= (v >> 2)
	v |= (v >> 4)
	v |= (v >> 8)
	v |= (v >> 16)

	v = v - ((v >> 1) & 0x55555555)
	v = (v & 0x33333333) + ((v >> 2) & 0x33333333)
	r := uint((((v + (v >> 4)) & 0xF0F0F0F) * 0x1010101) >> 24)
	return (r)
}

// Optimized combined shift, subtract and norm
func ssn(r *BIG, a *BIG, m *BIG) int {
	n := NLEN - 1
	m.w[0] = (m.w[0] >> 1) | ((m.w[1] << (BASEBITS - 1)) & BMASK)
	r.w[0] = a.w[0] - m.w[0]
	carry := r.w[0] >> BASEBITS
	r.w[0] &= BMASK
	for i := 1; i < n; i++ {
		m.w[i] = (m.w[i] >> 1) | ((m.w[i+1] << (BASEBITS - 1)) & BMASK)
		r.w[i] = a.w[i] - m.w[i] + carry
		carry = r.w[i] >> BASEBITS
		r.w[i] &= BMASK
	}
	m.w[n] >>= 1
	r.w[n] = a.w[n] - m.w[n] + carry
	return int((r.w[n] >> uint(CHUNK-1)) & 1)
}