Fast Carry Algorithm a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 c0.

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Fast Carry Algorithm a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 c0

a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 Step 1: calculate pi’s, gi’s c0 p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 g0 g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 pi = ai + bi gi = ai∙bi

p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 g0 g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 Step 2: Calculate global Pi’s, Gi’s c0 P0 = p0∙p1∙p2∙p3 G0 = g3+(p3∙g2)+(p3∙p2∙g1)+(p3∙p2∙p1∙g0) P1 = p4∙p5∙p6∙p7 G1 = g7+(p7∙g6)+(p7∙p6∙g5)+(p7∙p6∙p5∙g4) P2 = p8∙p9∙p10∙p11 G2 = g11+(p11∙g10)+(p11∙p10∙g9)+(p11∙p10∙p9∙g8) P3 = p12∙p13∙p14∙p15 G3 = g15+(p15∙g14)+(p15∙p14∙g13)+(p15∙p14∙p13∙g12) P0 G0 P1 G1 P2 G2 P3 G3

p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 g0 g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 Step 3: Calculate global Ci’s c0 C1 = G0+(P0∙c0) C2 = G1+(P1∙G0)+(P1∙P0∙c0) C3 = G2+(P2∙G1)+(P2∙P1∙G0)+(P2∙P1∙P0∙c0) C4 = G3+(P3∙G2)+(P3∙P2∙G1)+(P3∙P2∙P1∙G0)+(P3∙P2∙P1∙P0∙c0) P0 G0 P1 G1 P2 G2 P3 G3 C1C2C3C4

p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 g0 g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 Step 4: Calculate ri’s in groups of 4 c0 r0 = a0+b0+c0 r1 = a1+b1+g0+(p0∙c0) r2 = a2+b2+g1+(p1∙g0)+(p1∙p0∙c0) r3 = a3+b3+g2+(p2∙g1)+(p2∙p1∙g0)+(p2∙p1∙p0∙c0) P0 G0 P1 G1 P2 G2 P3 G3 C1C2C3C4 r0 r1 r2 r3

p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 g0 g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 Step 4: Calculate ri’s in groups of 4 c0 r4 = a4+b4+C1 r5 = a5+b5+g4+(p4∙C1) r6 = a6+b6+g5+(p5∙g4)+(p5∙p4∙C1) r7 = a7+b7+g6+(p6∙g5)+(p6∙p5∙g4)+(p6∙p5∙p4∙C1) P0 G0 P1 G1 P2 G2 P3 G3 C1C2C3C4 r0 r1 r2 r3 r4 r5 r6 r7

p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 g0 g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 Step 4: Calculate ri’s in groups of 4 c0 r8 = a8+b8+C2 r9 = a9+b9+g8+(p8∙C2) r10 = a10+b10+g9+(p9∙g8)+(p9∙p8∙C2) r11 = a11+b11+g10+(p10∙g9)+(p10∙p9∙g8)+(p10∙p9∙p8∙C2) P0 G0 P1 G1 P2 G2 P3 G3 C1C2C3C4 r0 r1 r2 r3r4 r5 r6 r7 r8 r9 r10 r11

p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 g0 g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 Step 4: Calculate ri’s in groups of 4 c0 r12 = a12+b12+C3 r13 = a13+b13+g12+(p12∙C3) r14 = a14+b14+g13+(p13∙g12)+(p13∙p12∙C3) r15 = a15+b15+g14+(p14∙g13)+(p14∙p13∙g12)+(p14∙p13∙p12∙C3) P0 G0 P1 G1 P2 G2 P3 G3 C1C2C3C4 r0 r1 r2 r3r4 r5 r6 r7r8 r9 r10 r11 r12 r13 r14 r15

p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 g0 g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 Example c0 a = b = = = P0 G0 P1 G1 P2 G2 P3 G3 C1C2 C3 C4

p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 g0 g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g Example 0 a = b = = = P0 G0 P1 G1 P2 G2 P3 G3 C1C2 C3 C4

p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 g0 g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g Step 1: calculate pi’s, gi’s 0 P0 G0 P1 G1 P2 G2 P3 G3 C1C2 C3 C4 pi = ai + bi gi = ai∙bi

Step 1: calculate pi’s, gi’s 0 P0 G0 P1 G1 P2 G2 P3 G3 C1C2 C3 C4 pi = ai + bi gi = ai∙bi

Step 2: Calculate global Pi’s, Gi’s 0 P0 G0 P1 G1 P2 G2 P3 G3 C1C2 C3 C4 P0 = p0∙p1∙p2∙p3 G0 = g3+(p3∙g2)+(p3∙p2∙g1)+(p3∙p2∙p1∙g0) P1 = p4∙p5∙p6∙p7 G1 = g7+(p7∙g6)+(p7∙p6∙g5)+(p7∙p6∙p5∙g4) P2 = p8∙p9∙p10∙p11 G2 = g11+(p11∙g10)+(p11∙p10∙g9)+(p11∙p10∙p9∙g8) P3 = p12∙p13∙p14∙p15 G3 = g15+(p15∙g14)+(p15∙p14∙g13)+(p15∙p14∙p13∙g12)

Step 2: Calculate global Pi’s, Gi’s C1C2 C3 C4 P0 = p0∙p1∙p2∙p3 G0 = g3+(p3∙g2)+(p3∙p2∙g1)+(p3∙p2∙p1∙g0) P1 = p4∙p5∙p6∙p7 G1 = g7+(p7∙g6)+(p7∙p6∙g5)+(p7∙p6∙p5∙g4) P2 = p8∙p9∙p10∙p11 G2 = g11+(p11∙g10)+(p11∙p10∙g9)+(p11∙p10∙p9∙g8) P3 = p12∙p13∙p14∙p15 G3 = g15+(p15∙g14)+(p15∙p14∙g13)+(p15∙p14∙p13∙g12)

Step 3: Calculate global Ci’s C1C2 C3 C4 C1 = G0+(P0∙c0) C2 = G1+(P1∙G0)+(P1∙P0∙c0) C3 = G2+(P2∙G1)+(P2∙P1∙G0)+(P2∙P1∙P0∙c0) C4 = G3+(P3∙G2)+(P3∙P2∙G1)+(P3∙P2∙P1∙G0)+(P3∙P2∙P1∙P0∙c0)

Step 3: Calculate global Ci’s C1 = G0+(P0∙c0) C2 = G1+(P1∙G0)+(P1∙P0∙c0) C3 = G2+(P2∙G1)+(P2∙P1∙G0)+(P2∙P1∙P0∙c0) C4 = G3+(P3∙G2)+(P3∙P2∙G1)+(P3∙P2∙P1∙G0)+(P3∙P2∙P1∙P0∙c0)

Step 4: Calculate ri’s in groups of r0 = a0+b0+c0 r1 = a1+b1+g0+(p0∙c0) r2 = a2+b2+g1+(p1∙g0)+(p1∙p0∙c0) r3 = a3+b3+g2+(p2∙g1)+(p2∙p1∙g0)+(p2∙p1∙p0∙c0) 0 1

Step 4: Calculate ri’s in groups of r4 = a4+b4+C1 r5 = a5+b5+g4+(p4∙C1) r6 = a6+b6+g5+(p5∙g4)+(p5∙p4∙C1) r7 = a7+b7+g6+(p6∙g5)+(p6∙p5∙g4)+(p6∙p5∙p4∙C1) 0 0

Step 4: Calculate ri’s in groups of r8 = a8+b8+C2 r9 = a9+b9+g8+(p8∙C2) r10 = a10+b10+g9+(p9∙g8)+(p9∙p8∙C2) r11 = a11+b11+g10+(p10∙g9)+(p10∙p9∙g8)+(p10∙p9∙p8∙C2)

Step 4: Calculate ri’s in groups of r12 = a12+b12+C3 r13 = a13+b13+g12+(p12∙C3) r14 = a14+b14+g13+(p13∙g12)+(p13∙p12∙C3) r15 = a15+b15+g14+(p14∙g13)+(p14∙p13∙g12)+(p14∙p13∙p12∙C3)

Final Result a = b = = = = r =