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dnl ******************************************************************************
dnl Copyright 2009 Paul Zimmermann and Alexander Kruppa.
dnl
dnl This file is part of the ECM Library.
dnl
dnl The ECM Library is free software; you can redistribute it and/or modify
dnl it under the terms of the GNU Lesser General Public License as published by
dnl the Free Software Foundation; either version 3 of the License, or (at your
dnl option) any later version.
dnl
dnl The ECM Library is distributed in the hope that it will be useful, but
dnl WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
dnl or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
dnl License for more details.
dnl
dnl You should have received a copy of the GNU Lesser General Public License
dnl along with the ECM Library; see the file COPYING.LIB. If not, write to
dnl the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
dnl MA 02110-1301, USA.
dnl ******************************************************************************
define(C, `
dnl')
C mp_limb_t mulredc6(mp_limb_t * z, const mp_limb_t * x, const mp_limb_t * y,
C const mp_limb_t *m, mp_limb_t inv_m);
C
C arguments:
C r3 = ptr to result z least significant limb
C r4 = ptr to input x least significant limb
C r5 = ptr to input y least significant limb
C r6 = ptr to modulus m least significant limb
C r7 = -1/m mod 2^64
C
C final carry returned in r3
include(`config.m4')
GLOBL GSYM_PREFIX`'mulredc6
GLOBL .GSYM_PREFIX`'mulredc6
.section ".opd", "aw"
.align 3
GSYM_PREFIX`'mulredc6:
.quad .GSYM_PREFIX`'mulredc6, .TOC.@tocbase, 0
.size GSYM_PREFIX`'mulredc6, 24
C Implements multiplication and REDC for two input numbers of 6 words
C The algorithm:
C (Notation: a:b:c == a * 2^128 + b * 2^64 + c)
C
C T1:T0 = x[i]*y[0] ;
C u = (T0*invm) % 2^64 ;
C cy:T1 = (m[0]*u + T1:T0) / 2^64 ; /* cy:T1 <= 2*2^64 - 4 (see note 1) */
C for (j = 1; j < len; j++)
C {
C cy:T1:T0 = x[i]*y[j] + m[j]*u + cy:T1 ;
C /* for all j result cy:T1 <= 2*2^64 - 3 (see note 2) */
C tmp[j-1] = T0;
C }
C tmp[len-1] = T1 ;
C tmp[len] = cy ; /* cy <= 1 (see note 2) */
C for (i = 1; i < len; i++)
C {
C cy:T1:T0 = x[i]*y[0] + tmp[1]:tmp[0] ;
C u = (T0*invm) % 2^64 ;
C cy:T1 = (m[0]*u + cy:T1:T0) / 2^64 ; /* cy:T1 <= 3*2^64 - 4 (see note 3) */
C for (j = 1; j < len; j++)
C {
C cy:T1:T0 = x[i]*y[j] + m[j]*u + (tmp[j+1] + cy):T1 ;
C /* for all j < (len-1), result cy:T1 <= 3*2^64 - 3
C for j = (len-1), result cy:T1 <= 2*2^64 - 1 (see note 4) */
C tmp[j-1] = T0;
C }
C tmp[len-1] = T1 ;
C tmp[len] = cy ; /* cy <= 1 for all i (see note 4) */
C }
C z[0 ... len-1] = tmp[0 ... len-1] ;
C return (tmp[len]) ;
C
C notes:
C
C 1: m[0]*u + T1:T0 <= 2*(2^64 - 1)^2 <= 2*2^128 - 4*2^64 + 2,
C so cy:T1 <= 2*2^64 - 4.
C 2: For j = 1, x[i]*y[j] + m[j]*u + cy:T1 <= 2*(2^64 - 1)^2 + 2*2^64 - 4
C <= 2*2^128 - 2*2^64 - 2 = 1:(2^64-3):(2^64-2),
C so cy:T1 <= 2*2^64 - 3. For j > 1,
C x[i]*y[j] + m[j]*u + cy:T1 <= 2*2^128 - 2*2^64 - 1 = 1:(2^64-3):(2^64-1),
C so cy:T1 <= 2*2^64 - 3 = 1:(2^64-3) holds for all j.
C 3: m[0]*u + cy:T1:T0 <= 2*(2^64 - 1)^2 + 2^128 - 1 = 3*2^128 - 4*2^64 + 1,
C so cy:T1 <= 3*2^64 - 4 = 2:(2^64-4)
C 4: For j = 1, x[i]*y[j] + m[j]*u + (tmp[j+1] + cy):T1
C <= 2*(2^64 - 1)^2 + (3*2^64 - 4) + (2^64-1)*2^64
C <= 3*2^128 - 2*2^64 - 2 = 2:(2^64-3):(2^64-2),
C so cy:T1 <= 3*2^64 - 3. For j > 1,
C x[i]*y[j] + m[j]*u + (tmp[j+1] + cy):T1 <= 2:(2^64-3):(2^64-1),
C so cy:T1 <= 3*2^64 - 3 = 2:(2^64-3) holds for all j < len - 1.
C For j = len - 1, we know from note 2 that tmp(len) <= 1 for i = 0.
C Assume this is true for index i-1, Then
C x[i]*y[len-1] + m[len-1]*u + (tmp[len] + cy):T1
C <= 2*(2^64 - 1)^2 + (3*2^64 - 3) + 2^64
C <= 2*2^128 - 1 = 1:(2^64-1):(2^64-1),
C so cy:T1 <= 1:(2^64-1) and tmp[len] <= 1 for all i by induction.
C
C Register vars: T0 = r13, T1 = r14, CY = r10, XI = r12, U = r11
C YP = r5, MP = r6, TP = r1 (stack ptr)
C
C local variables: tmp[0 ... 6] array, having 6+1 8-byte words
C The tmp array needs 6+1 entries, but tmp[6] is stored in
C r15, so only 6 entries are used in the stack.
TEXT
.align 5 C powerPC 32 byte alignment
.GSYM_PREFIX`'mulredc6:
C ########################################################################
C # i = 0 pass
C #########################################################################
C Pass for j = 0. We need to fetch x[i] from memory and compute the new u
ld r12, 0(r4) C XI = x[0]
ld r0, 0(r5) C y[0]
stdu r13, -8(r1) C save r13
mulld r8, r0, r12 C x[0]*y[0] low half
stdu r14, -8(r1) C save r14
mulhdu r9, r0, r12 C x[0]*y[0] high half
ld r0, 0(r6) C m[0]
mulld r11, r7, r8 C U = T0*invm mod 2^64
stdu r15, -8(r1) C save r15
mulld r13, r0, r11 C T0 = U*m[0] low
stdu r16, -8(r1) C save r16
li r16, 0 C set r16 to zero for carry propagation
subi r1, r1, 48 C set tmp stack space
mulhdu r14, r0, r11 C T1 = U*m[0] high
ld r0, 8(r5) C y[1]
addc r8, r8, r13 C
adde r13, r9, r14 C T0 = initial tmp(0)
addze r10, r16 C carry to CY
C CY:T1:T0 <= 2*(2^64-1)^2 <= 2^2*128 - 4*2^64 + 2, hence
C CY:T1 <= 2*2^64 - 4
C Pass for j = 1
mulld r8, r0, r12 C x[i]*y[j] low half
mulhdu r9, r0, r12 C x[i]*y[j] high half
ld r0, 8(r6) C m[j]
addc r13, r8, r13 C add low word to T0
adde r14, r9, r10 C add high word with carry + CY to T1
C T1:T0 <= 2^128 - 2*2^64 + 1 + 2*2^64 - 3 <= 2^128 - 2, no carry!
mulld r8, r0, r11 C U*m[j] low
mulhdu r9, r0, r11 C U*m[j] high
addc r8, r8, r13 C add T0 and low word
ld r0, 16(r5) C y[j+1]
adde r13, r9, r14 C add high word with carry to T1
addze r10, r16 C carry to CY
std r8, 0(r1) C store tmp[j-1]
C CY:T1:T0 <= 2^128 - 2 + 2^128 - 2*2^64 + 1 <=
C 2 * 2^128 - 2*2^64 - 1 ==> CY:T1 <= 2 * 2^64 - 3
C Pass for j = 2
mulld r8, r0, r12 C x[i]*y[j] low half
mulhdu r9, r0, r12 C x[i]*y[j] high half
ld r0, 16(r6) C m[j]
addc r13, r8, r13 C add low word to T0
adde r14, r9, r10 C add high word with carry + CY to T1
C T1:T0 <= 2^128 - 2*2^64 + 1 + 2*2^64 - 3 <= 2^128 - 2, no carry!
mulld r8, r0, r11 C U*m[j] low
mulhdu r9, r0, r11 C U*m[j] high
addc r8, r8, r13 C add T0 and low word
ld r0, 24(r5) C y[j+1]
adde r13, r9, r14 C add high word with carry to T1
addze r10, r16 C carry to CY
std r8, 8(r1) C store tmp[j-1]
C CY:T1:T0 <= 2^128 - 2 + 2^128 - 2*2^64 + 1 <=
C 2 * 2^128 - 2*2^64 - 1 ==> CY:T1 <= 2 * 2^64 - 3
C Pass for j = 3
mulld r8, r0, r12 C x[i]*y[j] low half
mulhdu r9, r0, r12 C x[i]*y[j] high half
ld r0, 24(r6) C m[j]
addc r13, r8, r13 C add low word to T0
adde r14, r9, r10 C add high word with carry + CY to T1
C T1:T0 <= 2^128 - 2*2^64 + 1 + 2*2^64 - 3 <= 2^128 - 2, no carry!
mulld r8, r0, r11 C U*m[j] low
mulhdu r9, r0, r11 C U*m[j] high
addc r8, r8, r13 C add T0 and low word
ld r0, 32(r5) C y[j+1]
adde r13, r9, r14 C add high word with carry to T1
addze r10, r16 C carry to CY
std r8, 16(r1) C store tmp[j-1]
C CY:T1:T0 <= 2^128 - 2 + 2^128 - 2*2^64 + 1 <=
C 2 * 2^128 - 2*2^64 - 1 ==> CY:T1 <= 2 * 2^64 - 3
C Pass for j = 4
mulld r8, r0, r12 C x[i]*y[j] low half
mulhdu r9, r0, r12 C x[i]*y[j] high half
ld r0, 32(r6) C m[j]
addc r13, r8, r13 C add low word to T0
adde r14, r9, r10 C add high word with carry + CY to T1
C T1:T0 <= 2^128 - 2*2^64 + 1 + 2*2^64 - 3 <= 2^128 - 2, no carry!
mulld r8, r0, r11 C U*m[j] low
mulhdu r9, r0, r11 C U*m[j] high
addc r8, r8, r13 C add T0 and low word
ld r0, 40(r5) C y[j+1]
adde r13, r9, r14 C add high word with carry to T1
addze r10, r16 C carry to CY
std r8, 24(r1) C store tmp[j-1]
C CY:T1:T0 <= 2^128 - 2 + 2^128 - 2*2^64 + 1 <=
C 2 * 2^128 - 2*2^64 - 1 ==> CY:T1 <= 2 * 2^64 - 3
C Pass for j = 5. Don't fetch new data from y[j+1].
mulld r8, r0, r12 C x[i]*y[j] low half
mulhdu r9, r0, r12 C x[i]*y[j] high half
ld r0, 40(r6) C m[j]
addc r13, r8, r13 C add low word to T0
adde r14, r9, r10 C add high word with carry + CY to T1
C T1:T0 <= 2^128 - 2*2^64 + 1 + 2*2^64 - 3 <= 2^128 - 2, no carry!
mulld r8, r0, r11 C U*m[j] low
mulhdu r9, r0, r11 C U*m[j] high
addc r8, r8, r13 C add T0 and low word
adde r13, r9, r14 C add high word with carry to T1
std r8, 32(r1) C store tmp[len-2]
addze r15, r16 C put carry in r15 (tmp[len] <= 1)
std r13, 40(r1) C store tmp[len-1]
C #########################################################################
C # i > 0 passes
C #########################################################################
li r9, 5 C outer loop count
mtctr r9
1:
C Pass for j = 0. We need to fetch x[i], tmp[i] and tmp[i+1] from memory
C and compute the new u
ldu r12, 8(r4) C x[i]
ld r0, 0(r5) C y[0]
ld r13, 0(r1) C tmp[0]
mulld r8, r0, r12 C x[i]*y[0] low half
ld r14, 8(r1) C tmp[1]
mulhdu r9, r0, r12 C x[i]*y[0] high half
addc r13, r8, r13 C T0
ld r0, 0(r6) C m[0]
mulld r11, r7, r13 C U = T0*invm mod 2^64
adde r14, r9, r14 C T1
mulld r8, r0, r11 C U*m[0] low
addze r10, r16 C CY
mulhdu r9, r0, r11 C U*m[0] high
ld r0, 8(r5) C y[1]
addc r8, r8, r13 C result = 0
adde r13, r9, r14 C T0, carry pending
C cy:T1:T0 <= 2*(2^64 - 1)^2 + 2^128 - 1 = 3*2^128 - 4*2^64 + 1,
C so cy:T1 <= 3*2^64 - 4
C Pass for j = 1
ld r14, 16(r1) C tmp[j+1]
mulld r8, r0, r12 C x[i]*y[j] low half
adde r14, r14, r10 C tmp[j+1] + CY + pending carry
addze r10, r16 C carry to CY
mulhdu r9, r0, r12 C x[i]*y[j] high half
ld r0, 8(r6) C m[j]
addc r13, r8, r13 C add low word to T0
mulld r8, r0, r11 C U*m[j] low
adde r14, r9, r14 C add high to T1
addze r10, r10 C add carry to CY
mulhdu r9, r0, r11 C U*m[j] high
addc r8, r8, r13 C add T0 and low word
ld r0, 16(r5) C y[j+1]
adde r13, r9, r14 C T1, carry pending
std r8, 0(r1) C store tmp[j-1]
C CY:T1:T0 <= 2*(2^64 - 1)^2 + (3*2^64 - 3) + (2^64-1)*2^64
C <= 3*2^128 - 2*2^64 - 1 ==> CY:T1 <= 3*2^64 - 3
C Pass for j = 2
ld r14, 24(r1) C tmp[j+1]
mulld r8, r0, r12 C x[i]*y[j] low half
adde r14, r14, r10 C tmp[j+1] + CY + pending carry
addze r10, r16 C carry to CY
mulhdu r9, r0, r12 C x[i]*y[j] high half
ld r0, 16(r6) C m[j]
addc r13, r8, r13 C add low word to T0
mulld r8, r0, r11 C U*m[j] low
adde r14, r9, r14 C add high to T1
addze r10, r10 C add carry to CY
mulhdu r9, r0, r11 C U*m[j] high
addc r8, r8, r13 C add T0 and low word
ld r0, 24(r5) C y[j+1]
adde r13, r9, r14 C T1, carry pending
std r8, 8(r1) C store tmp[j-1]
C CY:T1:T0 <= 2*(2^64 - 1)^2 + (3*2^64 - 3) + (2^64-1)*2^64
C <= 3*2^128 - 2*2^64 - 1 ==> CY:T1 <= 3*2^64 - 3
C Pass for j = 3
ld r14, 32(r1) C tmp[j+1]
mulld r8, r0, r12 C x[i]*y[j] low half
adde r14, r14, r10 C tmp[j+1] + CY + pending carry
addze r10, r16 C carry to CY
mulhdu r9, r0, r12 C x[i]*y[j] high half
ld r0, 24(r6) C m[j]
addc r13, r8, r13 C add low word to T0
mulld r8, r0, r11 C U*m[j] low
adde r14, r9, r14 C add high to T1
addze r10, r10 C add carry to CY
mulhdu r9, r0, r11 C U*m[j] high
addc r8, r8, r13 C add T0 and low word
ld r0, 32(r5) C y[j+1]
adde r13, r9, r14 C T1, carry pending
std r8, 16(r1) C store tmp[j-1]
C CY:T1:T0 <= 2*(2^64 - 1)^2 + (3*2^64 - 3) + (2^64-1)*2^64
C <= 3*2^128 - 2*2^64 - 1 ==> CY:T1 <= 3*2^64 - 3
C Pass for j = 4
ld r14, 40(r1) C tmp[j+1]
mulld r8, r0, r12 C x[i]*y[j] low half
adde r14, r14, r10 C tmp[j+1] + CY + pending carry
addze r10, r16 C carry to CY
mulhdu r9, r0, r12 C x[i]*y[j] high half
ld r0, 32(r6) C m[j]
addc r13, r8, r13 C add low word to T0
mulld r8, r0, r11 C U*m[j] low
adde r14, r9, r14 C add high to T1
addze r10, r10 C add carry to CY
mulhdu r9, r0, r11 C U*m[j] high
addc r8, r8, r13 C add T0 and low word
ld r0, 40(r5) C y[j+1]
adde r13, r9, r14 C T1, carry pending
std r8, 24(r1) C store tmp[j-1]
C CY:T1:T0 <= 2*(2^64 - 1)^2 + (3*2^64 - 3) + (2^64-1)*2^64
C <= 3*2^128 - 2*2^64 - 1 ==> CY:T1 <= 3*2^64 - 3
C Pass for j = 5. Don't fetch new data from y[j+1].
mulld r8, r0, r12 C x[i]*y[j] low half
adde r14, r15, r10 C T1 = tmp[len] + CY + pending carry
C since tmp[len] <= 1, T1 <= 3 and carry is zero
mulhdu r9, r0, r12 C x[i]*y[j] high half
ld r0, 40(r6) C m[j]
addc r13, r8, r13 C add low word to T0
mulld r8, r0, r11 C U*m[j] low
adde r14, r9, r14 C add high to T1
addze r10, r16 C CY
mulhdu r9, r0, r11 C U*m[j] high
addc r8, r8, r13 C add T0 and low word
adde r13, r9, r14 C T1, carry pending
std r8, 32(r1) C store tmp[len-2]
addze r15, r10 C store tmp[len] <= 1
std r13, 40(r1) C store tmp[len-1]
C CY:T1:T0 <= 2*(2^64 - 1)^2 + (3*2^64 - 3) + 2^64
C <= 2*2^128 - 1 ==> CY:T1 <= 2*2^64 - 1 = 1:(2^64-1)
bdnz 1b
C Copy result from tmp memory to z
ld r8, 0(r1)
ldu r9, 8(r1)
std r8, 0(r3)
stdu r9, 8(r3)
ldu r8, 8(r1)
ldu r9, 8(r1)
stdu r8, 8(r3)
stdu r9, 8(r3)
ldu r8, 8(r1)
ldu r9, 8(r1)
stdu r8, 8(r3)
stdu r9, 8(r3)
mr r3, r15 C return tmp(len)
ldu r16, 8(r1)
ldu r15, 8(r1)
ldu r14, 8(r1)
ldu r13, 8(r1)
addi r1, r1, 8
blr
.size .GSYM_PREFIX`'mulredc6, .-.GSYM_PREFIX`'mulredc6
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