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Modulo- 𝟐𝒏 − 𝟐𝒒 − 𝟏 Parallel Prefix Addition

via Excess-Modulo Encoding of Residues

Seyed Hamed Fatemi Langroudi & Ghassem Jaberipur

Computer Science & Engineering Department

Shahid Beheshti University, Tehran, Iran

1

SBU - Tehran ENSL - Lyon

22 th IEEE symposium on Computer Arithmetic

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Prepared by :Hamed Fatemi

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via Excess-Modulo Encoding of Residue

• Contribution

• RNS in General

• Summery of Modulo Addition

• New Modulo 2𝑛 − 2𝑞 − 1 Adders

• Comparison

• Conclusion

2 ARITH 22

Outline

3



Contribution

ARITH 22

Modulo Delay(∆𝑮) Area(𝓐𝑮) EM Encoding

2𝑛 − 1 ,3- 3 + 2 log 𝑛 3𝑛 log 𝑛 + 4𝑛 0

3



Modulo-addition

Contribution

ARITH 22


2𝑛 − 1 ,3- 3 + 2 log 𝑛 3𝑛 log 𝑛 + 4𝑛 0

3



Modulo-addition

Contribution

ARITH 22

2𝑛 − 3 ,13- 4 + 2 log 𝑛 3 𝑛 − 1 log(𝑛 − 1) + 8𝑛 − 1 *0, 1, 2+


2𝑛 − 1 ,3- 3 + 2 log 𝑛 3𝑛 log 𝑛 + 4𝑛 0

3



Modulo-addition

Contribution

ARITH 22

2𝑛 − 3 ,13- 4 + 2 log 𝑛 3 𝑛 − 1 log(𝑛 − 1) + 8𝑛 − 1 *0, 1, 2+

2𝑛 − 2𝑞 − 1 ,10- 7 + 2 log 𝑛 ≤ 3𝑛 log 𝑛 + 7𝑛 − 1 + 1.5(𝑛 − 3) log(𝑛 − 3) none


2𝑛 − 1 ,3- 3 + 2 log 𝑛 3𝑛 log 𝑛 + 4𝑛 0

3



Modulo-addition

Contribution

ARITH 22

2𝑛 − 3 ,13- 4 + 2 log 𝑛 3 𝑛 − 1 log(𝑛 − 1) + 8𝑛 − 1 *0, 1, 2+


2𝑛 − 2𝑞 − 1 5 + 2 log 𝑛 ≤ 3𝑛 log 𝑛 + 7𝑛 − 1 + 1.5 𝑛 − 3 log 𝑛 − 3

+𝑛 − 3 log 𝑛 − 4 ,0,2𝑞-


2𝑛 − 1 ,3- 3 + 2 log 𝑛 3𝑛 log 𝑛 + 4𝑛 0

3



Modulo-addition

Contribution

ARITH 22

2𝑛 − 3 ,13- 4 + 2 log 𝑛 3 𝑛 − 1 log(𝑛 − 1) + 8𝑛 − 1 *0, 1, 2+


2𝑛 − 2𝑞 − 1 5 + 2 log 𝑛 ≤ 3𝑛 log 𝑛 + 7𝑛 − 1 + 1.5 𝑛 − 3 log 𝑛 − 3

+𝑛 − 3 log 𝑛 − 4 ,0,2𝑞-

𝐴 new ≤ 𝐴(,10-) 𝑛 ≤ 16

• RNS in General

4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)



ARITH 22

• RNS in General

• e.g.

4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

• RNS in General

• e.g.

• RNS Architecture

4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)

X

z1 z2 z3

Z

Binary to RNS

x1 x2 x3 y1 y2 y3

Modulo 3

operation

Modulo 4

operation

Modulo 5

operation

RNS to Binary

Y

(e.g.,{3,4,5})



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

• RNS in General

• e.g.


4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)

X

z1 z2 z3

Z

Binary to RNS

x1 x2 x3 y1 y2 y3

Modulo 3

operation

Modulo 4

operation

Modulo 5

operation

RNS to Binary

Y7 23

(e.g.,{3,4,5})



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

• RNS in General

• e.g.


4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)

X

z1 z2 z3

Z

Binary to RNS

x1 x2 x3 y1 y2 y3

Modulo 3

operation

Modulo 4

operation

Modulo 5

operation

RNS to Binary

Y7

2 3 3 1 3 2

23

(e.g.,{3,4,5})



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

• RNS in General

• e.g.


4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)

X

z1 z2 z3

Z

Binary to RNS

x1 x2 x3 y1 y2 y3

Modulo 3

operation

Modulo 4

operation

Modulo 5

operation

RNS to Binary

Y7

2 3 3 1 3 2

Modulo 3

Addition

Modulo 4

Addition

Modulo 5

Addition

23

(e.g.,{3,4,5})



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

• RNS in General

• e.g.


4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)

X

z1 z2 z3

Z

Binary to RNS

x1 x2 x3 y1 y2 y3

Modulo 3

operation

Modulo 4

operation

Modulo 5

operation

RNS to Binary

Y7

2 3 3 1 3 2

Modulo 3

Addition

Modulo 4

Addition

Modulo 5

Addition

0 2 0

30

23

(e.g.,{3,4,5})



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

• RNS in General

• e.g.


• RNS Advantage

4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

• RNS in General

• e.g.


• RNS Advantage

Addition & Multiplication

4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

• RNS in General

• e.g.


• RNS Advantage


4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

O log(𝑛

𝑘)

• RNS in General

• e.g.


• RNS Advantage


• RNS Disadvantage

4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

O log(𝑛

𝑘)

• RNS in General

• e.g.


• RNS Advantage



Comparison & Division

4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

O log(𝑛

𝑘)

• RNS in General

• e.g.


• RNS Advantage




• RNS In Application

4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

O log(𝑛

𝑘)

• RNS in General

• e.g.


• RNS Advantage





Digital Signal Processing

4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

O log(𝑛

𝑘)

• RNS in General

• e.g.


• RNS Advantage






Fault Tolerant System

4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

O log(𝑛

𝑘)

• RNS in General

• e.g.


• RNS Advantage






Fault Tolerant System

Cryptography

4

𝑅 = *𝑚𝑘−1, … ,𝑚1, 𝑚0}, 𝐷𝑅 = 𝑚𝑘𝑖=𝑘−1𝑖=0

𝑋𝜖𝑅, 𝑋 = ( 𝑋 𝑚𝑘−1, … , 𝑋 𝑚1

, 𝑋 𝑚0)



ARITH 22

𝑹 = *𝟐𝒏 − 𝟏, 𝟐𝒏, 𝟐𝒏 + 𝟏+

O log(𝑛

𝑘)

5



Modulo 2𝑛 − 1

ARITH 22

5



Modulo 2𝑛 − 1

𝑋 + 𝑌 2𝑛−1 = 𝑋 + 𝑌, if 𝑋 + 𝑌 < 2𝑛 − 1𝑋 + 𝑌 + 1 2𝑛 , if 𝑋 + 𝑌 ≥ 2𝑛 − 1

Conventional

ARITH 22

5



Modulo 2𝑛 − 1


Conventional

Adder A

Adder B

Mux

2 nX Y

2 21 n nX Y

AC

BC

X Y

1 0

1

2 1 nX Y

RCA

ARITH 22

5



Modulo 2𝑛 − 1


Conventional EM Encoding

𝑋 + 𝑌 2𝑛−1 = 𝑋 + 𝑌, if 𝑋 + 𝑌 < 2𝑛

𝑋 + 𝑌 + 1 2𝑛 , if 𝑋 + 𝑌 ≥ 2𝑛

Adder A

Adder B

Mux

2 nX Y

2 21 n nX Y

AC

BC

X Y

1 0

1

2 1 nX Y

RCA

ARITH 22

5



Modulo 2𝑛 − 1



𝑋 + 𝑌 2𝑛−1 = 𝑋 + 𝑌, if 𝑋 + 𝑌 < 2𝑛

𝑋 + 𝑌 + 1 2𝑛 , if 𝑋 + 𝑌 ≥ 2𝑛

Adder A

Adder B

Mux

2 nX Y

2 21 n nX Y

AC

BC

X Y

1 0

1

2 1 nX Y

AdderoutC

X Y

n2 1X Y

RCA RCA

ARITH 22

6



ARITH 22

Parallel Prefix Realization

6



ARITH 22


Parallel Prefix Network(PPN) pgh pgh pghpghpgh

y0y1y2y3y4y5y6y7 x0x1x2x3x4x5x6x7

h3h4h5h6h7 h2 h1

pgh pgh pghh0

G6:0 G5:0 G4:0 G3:0 G2:0 G1:0 G0:0

(g0,p0)(g1,p1)(g2,p2)(g3,p3)(g4,p4)(g5,p5)(g6,p6)(g7,p7)

G7:0

6



ARITH 22




h3h4h5h6h7 h2 h1

pgh pgh pghh0

G6:0 G5:0 G4:0 G3:0 G2:0 G1:0 G0:0


G7:0

p g h

x yxyx y

x y

6



ARITH 22




h3h4h5h6h7 h2 h1

pgh pgh pghh0

G6:0 G5:0 G4:0 G3:0 G2:0 G1:0 G0:0


G7:0

,G P

,G P

p g h

x yxyx y

x y

6



ARITH 22




h3h4h5h6h7 h2 h1

pgh pgh pghh0

G6:0 G5:0 G4:0 G3:0 G2:0 G1:0 G0:0


G7:0

,G P

,G P

p g h

x yxyx y

x y ,G P ,r rG P

( )rG P G

6



ARITH 22




h3h4h5h6h7 h2 h1

pgh pgh pghh0

G6:0 G5:0 G4:0 G3:0 G2:0 G1:0 G0:0


G7:0

,G P

,G P

p g h

x yxyx y

x y

( , )r rG P G P P

,G P ,r rG P ,G P ,r rG P

( )rG P G

6



ARITH 22


Parallel Prefix Network(PPN)

Modulo- 2𝑛 − 1 EM Encoding

Parallel prefix Adder

pgh pgh pghpghpgh


h3h4h5h6h7 h2 h1

pgh pgh pghh0

G6:0 G5:0 G4:0 G3:0 G2:0 G1:0 G0:0


G7:0

,G P

,G P

p g h

x yxyx y

x y

( , )r rG P G P P


( )rG P G

6



ARITH 22


Parallel Prefix Network(PPN)

Modulo- 2𝑛 − 1 EM Encoding

Parallel prefix Adder

pgh pgh pghpghpgh


h3h4h5h6h7 h2 h1

pgh pgh pghh0

G6:0 G5:0 G4:0 G3:0 G2:0 G1:0 G0:0


G7:0

pgh pgh pghpghpgh


c3c4c5c6

h3

c7

h4h5h6h7 h2

c2 c1

h1

pgh pgh pghh0

G7:0 (G6:0,P6:0) (G5:0,P5:0)(G4:0,P4:0) (G3:0,P3:0) (G2:0,P2:0) (G1:0,P1:0) (G0:0,P0:0)

s0s1s2s3s4s5s6s7

,G P

,G P

p g h

x yxyx y

x y

( , )r rG P G P P


( )rG P G

7



ARITH 22

7



Modulo 2𝑛 − 3

ARITH 22

7



Modulo 2𝑛 − 3

Conventional


ARITH 22

7



Modulo 2𝑛 − 3

Conventional


Adder A

Adder B

Mux

2 nX Y

2 23 n nX Y

AC

BC

X Y

1 0

3

2 3 nX Y

RCA

ARITH 22

7



Modulo 2𝑛 − 3



𝑋 + 𝑌 2𝑛−3 = 𝑋 + 𝑌, if 𝑋 + 𝑌 < 2𝑛

𝑋 + 𝑌 + 3 2𝑛 , if 𝑋 + 𝑌 ≥ 2𝑛

Adder A

Adder B

Mux

2 nX Y

2 23 n nX Y

AC

BC

X Y

1 0

3

2 3 nX Y

RCA

ARITH 22

7



Modulo 2𝑛 − 3



𝑋 + 𝑌 2𝑛−3 = 𝑋 + 𝑌, if 𝑋 + 𝑌 < 2𝑛

𝑋 + 𝑌 + 3 2𝑛 , if 𝑋 + 𝑌 ≥ 2𝑛

Adder A

Adder B

Mux

2 nX Y

2 23 n nX Y

AC

BC

X Y

1 0

3

2 3 nX Y

n-bit Adder

(n,2)-bit Adder

outC

YX

n2X Y

n2 3X Y

RCA RCA

ARITH 22

8



ARITH 22

8



ARITH 22


Modulo- 2𝑛 − 3 EM Encoding Parallel prefix Adder

8



ARITH 22



pgh pgh pghpghpgh


h3h4h5h6h7 h2 h1

pgh pgh pghh0

pgh pgh pghpghpgh

000000 w0w1w2w3w4w5w6w7

h3h4h5h6h7 h2 h1

pgh pgh pghh0

s0s1s2s3s4s5s6s7

1ac2

ac3ac4

ac5ac6

ac7ac

aaaaaaaa

bbbbbbbb

1bc2

bc3bc4

bc5bc6

bc7bc

7:0G

7:0G 7:0G

7:0G

8



ARITH 22



HA

pgh pgh pgh pghpghpg′h

HAHA HA HA HA HAHA

x0x1x2x3x4x5x6x7 y0y1y2y3y4y5y6y7

u1u2u3u4u5u6u7 v1v2v3v4v5v6v7

(g2,p2)(g3,p3)(g4,p4)(g5,p5)(g6,p6)(g 7,p7) (g1,p′1)

(G2:0,P′2:0)(G3:0,P

′3:0)(G4:0,P

′4:0)(G5:0,P

′5:0)(G6:0,P

'6:0)

G' 7:0

c3c4c5c6

h3

c7

h4h5h6h7

s2s3s4s5s6s7

(G1:0,P′1:0)

c2

p´gh

s1

c1

h2 h1

h0

s0

u0v8

HA p g h

x yxy u vuvu v1 1u v

x u1uy v1v ,G P ,r rG P

,r rG P G P PrG P G

,G P ,r rG P ,G P

,G P 0 1 1 u v u

0u

1 1u v

p´ g h p g´ h

7u 7v8v

7 7u v7 7u v 7 7 8u v v

9



ARITH 22

9



Modulo 2𝑛 − 2𝑞 − 1

ARITH 22

9




Conventional

𝑋 + 𝑌 2𝑛−2q−1 = 𝑋 + 𝑌, if 𝑋 + 𝑌 < 2𝑛 − (2𝑞 + 1)

𝑋 + 𝑌 + 2q + 1 2𝑛 , if 𝑋 + 𝑌 ≥ 2𝑛 − (2𝑞 + 1)

ARITH 22

9




Conventional

𝑋 + 𝑌 2𝑛−2q−1 = 𝑋 + 𝑌, if 𝑋 + 𝑌 < 2𝑛 − (2𝑞 + 1)

𝑋 + 𝑌 + 2q + 1 2𝑛 , if 𝑋 + 𝑌 ≥ 2𝑛 − (2𝑞 + 1)

Adder A

Adder B

Mux

2 nX Y

2 22 1n n

qX Y

AC

BC

X Y

1 0

2 1q

2 2 1n qX Y

RCA

ARITH 22

9




EM Encoding Conventional

𝑋 + 𝑌 2𝑛−2q−1 = 𝑋 + 𝑌, if 𝑋 + 𝑌 < 2𝑛 − (2𝑞 + 1)

𝑋 + 𝑌 + 2q + 1 2𝑛 , if 𝑋 + 𝑌 ≥ 2𝑛 − (2𝑞 + 1) 𝑋 + 𝑌 2𝑛−2q−1 =

𝑋 + 𝑌, if 𝑋 + 𝑌 < 2𝑛

𝑋 + 𝑌 + 2q + 1 2𝑛 , if 𝑋 + 𝑌 ≥ 2𝑛

Adder A

Adder B

Mux

2 nX Y

2 22 1n n

qX Y

AC

BC

X Y

1 0

2 1q

2 2 1n qX Y

RCA

ARITH 22

9




EM Encoding Conventional

𝑋 + 𝑌 2𝑛−2q−1 = 𝑋 + 𝑌, if 𝑋 + 𝑌 < 2𝑛 − (2𝑞 + 1)

𝑋 + 𝑌 + 2q + 1 2𝑛 , if 𝑋 + 𝑌 ≥ 2𝑛 − (2𝑞 + 1) 𝑋 + 𝑌 2𝑛−2q−1 =

𝑋 + 𝑌, if 𝑋 + 𝑌 < 2𝑛

𝑋 + 𝑌 + 2q + 1 2𝑛 , if 𝑋 + 𝑌 ≥ 2𝑛

Adder A

Adder B

Mux

2 nX Y

2 22 1n n

qX Y

AC

BC

X Y

1 0

2 1q

2 2 1n qX Y

n-bit Adder

(n,q+1)-bit Adder

outC

YX

n2X Y

n q2 2 1X Y

RCA RCA

ARITH 22

10



ARITH 22


10



ARITH 22


Modulo- 2𝑛 − 2𝑞 − 1 EM Parallel prefix Adder(𝑛 = 8, 𝑞 = 4)

pgh pgh pghpghpgh


h3h4h5h6h7 h2 h1

pgh pgh pghh0

pgh pgh pghpghpgh

000000 w0w1w2w3w4w5w6w7

h3h4h5h6h7 h2 h1

pgh pgh pghh0

s0s1s2s3s4s5s6s7

1ac2

ac3ac4

ac5ac6

ac7ac

aaaaaaaa

bbbbbbbb

1bc2

bc3bc4

bc5bc6

bc7bc

7:0G

7:0G 7:0G

7:0G

10



ARITH 22


Modulo- 2𝑛 − 2𝑞 − 1 EM Parallel prefix Adder(𝑛 = 8, 𝑞 = 4)

11



ARITH 22

𝑿 𝒙𝒏−𝟏 … 𝒙𝒒 … 𝒙𝟏 𝒙𝟎

𝒀 𝒚𝒏−𝟏 … 𝒚𝒒 … 𝒚𝟏 𝒚𝟎

11



ARITH 22

𝑿 𝒙𝒏−𝟏 … 𝒙𝒒 … 𝒙𝟏 𝒙𝟎

𝒀 𝒚𝒏−𝟏 … 𝒚𝒒 … 𝒚𝟏 𝒚𝟎

𝑬𝑨𝑪 = 𝑿 + 𝒀 𝐰𝐧−𝟏 … 𝐰𝐪 … 𝐰𝟏 𝒘𝟎 𝒘𝒏

11



ARITH 22

𝑿 𝒙𝒏−𝟏 … 𝒙𝒒 … 𝒙𝟏 𝒙𝟎

𝒀 𝒚𝒏−𝟏 … 𝒚𝒒 … 𝒚𝟏 𝒚𝟎

𝑺 𝐒𝐧−𝟏 … 𝐒𝐪 … 𝐒𝟏 𝑺𝟎

𝜹 𝑬𝑨𝑪 𝒘𝒏

𝜹 = 2q + 1 𝒘𝒏

11



ARITH 22

𝑿 𝒙𝒏−𝟏 … 𝒙𝒒 … 𝒙𝟏 𝒙𝟎

𝒀 𝒚𝒏−𝟏 … 𝒚𝒒 … 𝒚𝟏 𝒚𝟎

𝑺 𝐒𝐧−𝟏 … 𝐒𝐪 … 𝐒𝟏 𝑺𝟎

𝜹 𝑬𝑨𝑪

(𝑒. 𝑔. 𝑛 = 8 𝑞 = 4)

𝜹 = 2q + 1 𝒘𝒏

𝒘𝒏

11



ARITH 22

𝑿 𝒙𝒏−𝟏 … 𝒙𝒒 … 𝒙𝟏 𝒙𝟎

𝒀 𝒚𝒏−𝟏 … 𝒚𝒒 … 𝒚𝟏 𝒚𝟎

𝑺 𝐒𝐧−𝟏 … 𝐒𝐪 … 𝐒𝟏 𝑺𝟎

𝜹 𝑬𝑨𝑪

Input Collective value

(𝑥𝑞 , 𝑦𝑞 , 𝑤𝑛 , 𝐶𝑞) 𝐶𝑞+1

𝑋 = 10010001 4 2

𝑌 = 10011111

(𝑒. 𝑔. 𝑛 = 8 𝑞 = 4)

𝜹 = 2q + 1 𝒘𝒏

𝒘𝒏

11



ARITH 22

𝑿 𝒙𝒏−𝟏 … 𝒙𝒒 … 𝒙𝟏 𝒙𝟎

𝒀 𝒚𝒏−𝟏 … 𝒚𝒒 … 𝒚𝟏 𝒚𝟎

𝑺 𝐒𝐧−𝟏 … 𝐒𝐪 … 𝐒𝟏 𝑺𝟎

𝜹 𝑬𝑨𝑪



𝑋 = 10010001 4 2

𝑌 = 10011111

𝑋 = 10010000 3 1

𝑌 = 10011110

(𝑒. 𝑔. 𝑛 = 8 𝑞 = 4)

𝜹 = 2q + 1 𝒘𝒏

𝒘𝒏

11



ARITH 22

𝑿 𝒙𝒏−𝟏 … 𝒙𝒒 … 𝒙𝟏 𝒙𝟎

𝒀 𝒚𝒏−𝟏 … 𝒚𝒒 … 𝒚𝟏 𝒚𝟎

𝑺 𝐒𝐧−𝟏 … 𝐒𝐪 … 𝐒𝟏 𝑺𝟎

𝜹 𝑬𝑨𝑪



𝑋 = 10010001 4 2

𝑌 = 10011111

𝑋 = 10010000 3 1

𝑌 = 10011110

(𝑒. 𝑔. 𝑛 = 8 𝑞 = 4)

Problem: Variable -weight

carry on position 4

𝜹 = 2q + 1 𝒘𝒏

𝒘𝒏

12



ARITH 22

• How to solve variable-weight carry problem?

12



ARITH 22


Devise a partial carry-save preprocessing stage

12



ARITH 22



𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝑿

𝒀 𝑋 + 𝑌 2𝑛−2𝑞−1

12



ARITH 22



𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝑿′

𝒀′

𝑿

𝒀 𝑋 + 𝑌 2𝑛−2𝑞−1

ℎ𝑖 = 𝑥𝑖⨁𝑦𝑖

𝑔𝑖 = 𝑥𝑖𝑦𝑖

12



ARITH 22



𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝑿′

𝒀′

𝑿

𝒀 𝑋 + 𝑌 2𝑛−2𝑞−1



12



ARITH 22



𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝒉𝒏−𝟏 … 𝒉𝒒

𝒈𝒏−𝟏 𝒈𝒏−𝟐 … 0

𝑿′

𝒀′

𝑿

𝒀 𝑋 + 𝑌 2𝑛−2𝑞−1



12



ARITH 22



𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎


𝒈𝒏−𝟏 𝒈𝒏−𝟐 … 0

𝑿′

𝒀′

𝑿

𝒀 𝑋 + 𝑌 2𝑛−2𝑞−1



𝑒 = 𝑔𝑛−1 ∨ 𝐺𝑛−1:0′

12



ARITH 22



𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎


𝒈𝒏−𝟏 𝒈𝒏−𝟐 … 0

𝑿′

𝒀′

𝑿′ 𝒉𝒏−𝟏 … 𝒉𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒀′′ 𝒈𝒏−𝟐 … 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝜹 𝑬𝑨𝑪 𝒆 𝒆

𝑿

𝒀 𝑋 + 𝑌 2𝑛−2𝑞−1



𝑒 = 𝑔𝑛−1 ∨ 𝐺𝑛−1:0′

12



ARITH 22



𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎


𝒈𝒏−𝟏 𝒈𝒏−𝟐 … 0

𝑿′

𝒀′

𝑿′ 𝒉𝒏−𝟏 … 𝒉𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒀′′ 𝒈𝒏−𝟐 … 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎


𝒉𝐧−𝟏′ … 𝒉𝒒

′ 𝒉𝐪−𝟏′ … 𝒉𝟏

′ 𝒉𝟎′

𝒄𝒏−𝟏 … 𝒄𝒒 𝒄𝒒−𝟏 … 𝒄𝟏 𝒄𝟎

𝑿

𝒀 𝑋 + 𝑌 2𝑛−2𝑞−1



𝑒 = 𝑔𝑛−1 ∨ 𝐺𝑛−1:0′

12



ARITH 22



𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎


𝒈𝒏−𝟏 𝒈𝒏−𝟐 … 0

𝑿′

𝒀′

𝑿′ 𝒉𝒏−𝟏 … 𝒉𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒀′′ 𝒈𝒏−𝟐 … 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎



′ 𝒉𝐪−𝟏′ … 𝒉𝟏

′ 𝒉𝟎′

𝒄𝒏−𝟏 … 𝒄𝒒 𝒄𝒒−𝟏 … 𝒄𝟏 𝒄𝟎

𝑺 𝒔𝒏−𝟏 … 𝒔𝒒 𝒔𝒒−𝟏 … 𝒔𝟏 𝒔𝟎

𝑿

𝒀 𝑋 + 𝑌 2𝑛−2𝑞−1



𝑒 = 𝑔𝑛−1 ∨ 𝐺𝑛−1:0′

12



ARITH 22



𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎


𝒈𝒏−𝟏 𝒈𝒏−𝟐 … 0

𝑿′

𝒀′

𝑿′ 𝒉𝒏−𝟏 … 𝒉𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒀′′ 𝒈𝒏−𝟐 … 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎



′ 𝒉𝐪−𝟏′ … 𝒉𝟏

′ 𝒉𝟎′

𝒄𝒏−𝟏 … 𝒄𝒒 𝒄𝒒−𝟏 … 𝒄𝟏 𝒄𝟎

𝑺 𝒔𝒏−𝟏 … 𝒔𝒒 𝒔𝒒−𝟏 … 𝒔𝟏 𝒔𝟎

𝑿

𝒀

𝑐𝑖 =

𝑔𝑛−1 ∨ 𝐺𝑛−1:0′ , if 𝑖 = 0

𝐺𝑖−1:0′ ∨ 𝑃𝑖−1:0

′ (𝑔𝑛−1 ∨ 𝐺𝑛−1:0′ ), if 1 ≤ 𝑖 ≤ 𝑞

ℎ𝑞 ∨ 𝑒 𝑐𝑞 ∨ ℎ𝑞𝑒, if 𝑖 = 𝑞 + 1

𝐺𝑖−1:𝑞+1′ ∨ 𝑃𝑖−1:𝑞+1

′ 𝑐𝑞+1, if 𝑖 > 𝑞 + 1

𝑋 + 𝑌 2𝑛−2𝑞−1



𝑒 = 𝑔𝑛−1 ∨ 𝐺𝑛−1:0′

𝑋′ + 𝑌′′ 2𝑛−2𝑞−1

12



ARITH 22



𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎


𝒈𝒏−𝟏 𝒈𝒏−𝟐 … 0

𝑿′

𝒀′

𝑿′ 𝒉𝒏−𝟏 … 𝒉𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒀′′ 𝒈𝒏−𝟐 … 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎



′ 𝒉𝐪−𝟏′ … 𝒉𝟏

′ 𝒉𝟎′

𝒄𝒏−𝟏 … 𝒄𝒒 𝒄𝒒−𝟏 … 𝒄𝟏 𝒄𝟎

𝑺 𝒔𝒏−𝟏 … 𝒔𝒒 𝒔𝒒−𝟏 … 𝒔𝟏 𝒔𝟎

𝑿

𝒀

𝑐𝑖 =

𝑔𝑛−1 ∨ 𝐺𝑛−1:0′ , if 𝑖 = 0

𝐺𝑖−1:0′ ∨ 𝑃𝑖−1:0

′ (𝑔𝑛−1 ∨ 𝐺𝑛−1:0′ ), if 1 ≤ 𝑖 ≤ 𝑞


𝐺𝑖−1:𝑞+1′ ∨ 𝑃𝑖−1:𝑞+1

′ 𝑐𝑞+1, if 𝑖 > 𝑞 + 1

Problem: Carry-save

Stage is on the

critical delay path

𝑋 + 𝑌 2𝑛−2𝑞−1



𝑒 = 𝑔𝑛−1 ∨ 𝐺𝑛−1:0′

𝑋′ + 𝑌′′ 2𝑛−2𝑞−1

IWSSIP 2014 81





ARITH 22 13

IWSSIP 2014 82





ARITH 22 13

• How to solve Carry Save Stage Delay problem?

IWSSIP 2014 83





ARITH 22 13


𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝑿

𝒀

𝑿′ 𝒉𝒏−𝟏 … 𝒉𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒀′′ 𝒈𝒏−𝟐 … 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎


Finding a relative between the 𝐺, 𝑃 of 𝑋 + 𝑌 2𝑛−2𝑞−1 and 𝐺′, 𝑃′ of 𝑋′ + 𝑌′′ 2𝑛−2𝑞−1

(𝐺, 𝑃)

(𝐺′, 𝑃′)

IWSSIP 2014 84





ARITH 22 13


𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝑿

𝒀

𝑿′ 𝒉𝒏−𝟏 … 𝒉𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒀′′ 𝒈𝒏−𝟐 … 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎



(𝑮𝒊:𝒋+𝟏′ , 𝑷𝒊:𝒋+𝟏

′ 𝒉𝒋) = (𝒉𝒊𝑮𝒊−𝟏:𝒋, 𝑯𝒊:𝒋)

𝒋 ≥ 𝒒

(𝐺, 𝑃)

(𝐺′, 𝑃′)

IWSSIP 2014 85





ARITH 22 13


𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝑿

𝒀

𝑿′ 𝒉𝒏−𝟏 … 𝒉𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒀′′ 𝒈𝒏−𝟐 … 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎



𝑐𝑖 =

𝑔𝑛−1 ∨ 𝐺𝑛−1:0′ , if 𝑖 = 0

𝐺𝑖−1:0′ ∨ 𝑃𝑖−1:0

′ (𝑔𝑛−1 ∨ 𝐺𝑛−1:0′ ), if 1 ≤ 𝑖 ≤ 𝑞


𝐺𝑖−1:𝑞+1′ ∨ 𝑃𝑖−1:𝑞+1

′ 𝑐𝑞+1, if 𝑖 > 𝑞 + 1

(𝑮𝒊:𝒋+𝟏′ , 𝑷𝒊:𝒋+𝟏


𝒋 ≥ 𝒒

(𝐺, 𝑃)

(𝐺′, 𝑃′)

IWSSIP 2014 86





ARITH 22 13


𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝑿

𝒀

𝑿′ 𝒉𝒏−𝟏 … 𝒉𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒀′′ 𝒈𝒏−𝟐 … 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎



𝑐𝑖 =

𝑔𝑛−1 ∨ 𝐺𝑛−1:0′ , if 𝑖 = 0

𝐺𝑖−1:0′ ∨ 𝑃𝑖−1:0

′ (𝑔𝑛−1 ∨ 𝐺𝑛−1:0′ ), if 1 ≤ 𝑖 ≤ 𝑞


𝐺𝑖−1:𝑞+1′ ∨ 𝑃𝑖−1:𝑞+1

′ 𝑐𝑞+1, if 𝑖 > 𝑞 + 1

(𝑮𝒊:𝒋+𝟏′ , 𝑷𝒊:𝒋+𝟏


𝒋 ≥ 𝒒

(𝐺, 𝑃)

(𝐺′, 𝑃′)

IWSSIP 2014 87





ARITH 22 13


𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝑿

𝒀

𝑿′ 𝒉𝒏−𝟏 … 𝒉𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒀′′ 𝒈𝒏−𝟐 … 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎



𝑐𝑖 =

𝑔𝑛−1 ∨ 𝐺𝑛−1:0′ , if 𝑖 = 0

𝐺𝑖−1:0′ ∨ 𝑃𝑖−1:0

′ (𝑔𝑛−1 ∨ 𝐺𝑛−1:0′ ), if 1 ≤ 𝑖 ≤ 𝑞


𝐺𝑖−1:𝑞+1′ ∨ 𝑃𝑖−1:𝑞+1

′ 𝑐𝑞+1, if 𝑖 > 𝑞 + 1

𝑐𝑖 =

𝐺𝑛−1:0, if 𝑖 = 0𝐺𝑖−1:0 ∨ 𝑃𝑖−1:0𝐺𝑛−1:0, if 1 ≤ 𝑖 ≤ 𝑞

ℎ𝑞𝐺𝑞−1:0 ∨ 𝑃𝑞:0′′ 𝐺𝑛−1:0, if 𝑖 = 𝑞 + 1

ℎ𝑖−1𝐺𝑖−2:0 ∨ 𝑃𝑖−1:0′′ 𝐺𝑛−1:0, if 𝑖 > 𝑞 + 1

(𝑮𝒊:𝒋+𝟏′ , 𝑷𝒊:𝒋+𝟏


𝒋 ≥ 𝒒

𝑃𝑞:0′′ = ℎ𝑞 ∨ 𝑃𝑞−1:0 ∨ 𝐺𝑞−1:0

𝑃𝑖−1:0′′ = 𝑃′

𝑖−1:𝑞+1𝑃𝑞:0′′

(𝐺, 𝑃)

(𝐺′, 𝑃′)

IWSSIP 2014 88





ARITH 22 13


𝒙𝒏−𝟏 … 𝒙𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒚𝒏−𝟏 … 𝒚𝒒 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎

𝑿

𝒀

𝑿′ 𝒉𝒏−𝟏 … 𝒉𝒒 𝒙𝒒−𝟏 … 𝒙𝟏 𝒙𝟎

𝒀′′ 𝒈𝒏−𝟐 … 𝒚𝒒−𝟏 … 𝒚𝟏 𝒚𝟎



𝑐𝑖 =

𝑔𝑛−1 ∨ 𝐺𝑛−1:0′ , if 𝑖 = 0

𝐺𝑖−1:0′ ∨ 𝑃𝑖−1:0

′ (𝑔𝑛−1 ∨ 𝐺𝑛−1:0′ ), if 1 ≤ 𝑖 ≤ 𝑞


𝐺𝑖−1:𝑞+1′ ∨ 𝑃𝑖−1:𝑞+1

′ 𝑐𝑞+1, if 𝑖 > 𝑞 + 1

𝑐𝑖 =

𝐺𝑛−1:0, if 𝑖 = 0𝐺𝑖−1:0 ∨ 𝑃𝑖−1:0𝐺𝑛−1:0, if 1 ≤ 𝑖 ≤ 𝑞

ℎ𝑞𝐺𝑞−1:0 ∨ 𝑃𝑞:0′′ 𝐺𝑛−1:0, if 𝑖 = 𝑞 + 1

ℎ𝑖−1𝐺𝑖−2:0 ∨ 𝑃𝑖−1:0′′ 𝐺𝑛−1:0, if 𝑖 > 𝑞 + 1

(𝑮𝒊:𝒋+𝟏′ , 𝑷𝒊:𝒋+𝟏


𝒋 ≥ 𝒒

𝑃𝑞:0′′ = ℎ𝑞 ∨ 𝑃𝑞−1:0 ∨ 𝐺𝑞−1:0

𝑃𝑖−1:0′′ = 𝑃′

𝑖−1:𝑞+1𝑃𝑞:0′′

(𝐺, 𝑃)

(𝐺′, 𝑃′)

The carry-save stage is effectively off the critical delay path

14



ARITH 22

14



ARITH 22

pghp′h′ pgh pgh pgh pghp′h′

x0x1x2x3x4x5x6 y0y1y2y3y4y5y6

(g2,p2)(g3,p3)(g5,p4)(g5,p5)(g6,p6) (g1,p1)

c3c4c5c6c7

s2s3s4s5s6s7

c2

pgh

s1

c1

s0

pgh pghh′

x7y7

(g0,p0)

h6 h5

G7:0

(g7,p7)

y4x4y5x5

y6 x6

P3:0 G3:0

ℎ7′ ℎ6

′ ℎ5′ ℎ4

ℎ4

ℎ0 ℎ1 ℎ3 ℎ2

(𝑮𝟎:𝟎, 𝑷𝟎:𝟎) (𝑮𝟏:𝟎, 𝑷𝟏:𝟎) (𝑮𝟐:𝟎, 𝑷𝟐:𝟎) (𝑮𝟑:𝟎,𝑷𝟑:𝟎) (𝑮𝟑:𝟎, 𝑷𝟒:𝟎′′ ) (𝑮𝟒:𝟎, 𝑷𝟓:𝟎

′′ ) (𝑮𝟓:𝟎, 𝑷𝟔:𝟎′′ )

Eqn.8

𝑝5′ 𝑝6

′

𝑝6′

𝑝5′

𝑝5′

𝑷𝟒:𝟎′′

𝑷𝟓:𝟎′′

𝑷𝟔:𝟎′′

p g h h′

1 1i i i ix y x y i ix yi ix y

ix iy

p g h

1ix 1iy

i ix y

p g h p’ h′

i ix yi ix y

1ix 1iy

i ix y

iyix

,G P ,r rG P

,r rG P G P PrG P G

,G P ,r rG P4h 5

p6p

3:0P3:0G

4:0P

5:0P6:0

P

,G P ,r rG P

i rh G P G

ih

Eqn.8

1 1i i i ix y x y 1 1i i i ix y x y

ix iy

i ix yi ix y i ix y

,G P

,G P

Modulo (28−24 − 1)

14



ARITH 22



(g2,p2)(g3,p3)(g5,p4)(g5,p5)(g6,p6) (g1,p1)

c3c4c5c6c7

s2s3s4s5s6s7

c2

pgh

s1

c1

s0

pgh pghh′

x7y7

(g0,p0)

h6 h5

G7:0

(g7,p7)

y4x4y5x5

y6 x6

P3:0 G3:0

ℎ7′ ℎ6

′ ℎ5′ ℎ4

ℎ4

ℎ0 ℎ1 ℎ3 ℎ2

(𝑮𝟎:𝟎, 𝑷𝟎:𝟎) (𝑮𝟏:𝟎, 𝑷𝟏:𝟎) (𝑮𝟐:𝟎, 𝑷𝟐:𝟎) (𝑮𝟑:𝟎,𝑷𝟑:𝟎) (𝑮𝟑:𝟎, 𝑷𝟒:𝟎′′ ) (𝑮𝟒:𝟎, 𝑷𝟓:𝟎

′′ ) (𝑮𝟓:𝟎, 𝑷𝟔:𝟎′′ )

Eqn.8

𝑝5′ 𝑝6

′

𝑝6′

𝑝5′

𝑝5′

𝑷𝟒:𝟎′′

𝑷𝟓:𝟎′′

𝑷𝟔:𝟎′′

p g h h′

1 1i i i ix y x y i ix yi ix y

ix iy

p g h

1ix 1iy

i ix y

p g h p’ h′

i ix yi ix y

1ix 1iy

i ix y

iyix

,G P ,r rG P

,r rG P G P PrG P G

,G P ,r rG P4h 5

p6p

3:0P3:0G

4:0P

5:0P6:0

P

,G P ,r rG P

i rh G P G

ih

Eqn.8

1 1i i i ix y x y 1 1i i i ix y x y

ix iy

i ix yi ix y i ix y

,G P

,G P

pgh pgh pghpghpgh


c3c4c5c6

h3

c7

h4h5h6h7 h2

c2 c1

h1

pgh pgh pghh0


s0s1s2s3s4s5s6s7

p g h

x yxyx y

x y ,G P ,r rG P

, r rG P G P P rG P G

,G P ,r rG P ,G P

,G P

Modulo (28−24 − 1) Modulo (28−1)

14



ARITH 22



(g2,p2)(g3,p3)(g5,p4)(g5,p5)(g6,p6) (g1,p1)

c3c4c5c6c7

s2s3s4s5s6s7

c2

pgh

s1

c1

s0

pgh pghh′

x7y7

(g0,p0)

h6 h5

G7:0

(g7,p7)

y4x4y5x5

y6 x6

P3:0 G3:0

ℎ7′ ℎ6

′ ℎ5′ ℎ4

ℎ4

ℎ0 ℎ1 ℎ3 ℎ2

(𝑮𝟎:𝟎, 𝑷𝟎:𝟎) (𝑮𝟏:𝟎, 𝑷𝟏:𝟎) (𝑮𝟐:𝟎, 𝑷𝟐:𝟎) (𝑮𝟑:𝟎,𝑷𝟑:𝟎) (𝑮𝟑:𝟎, 𝑷𝟒:𝟎′′ ) (𝑮𝟒:𝟎, 𝑷𝟓:𝟎

′′ ) (𝑮𝟓:𝟎, 𝑷𝟔:𝟎′′ )

Eqn.8

𝑝5′ 𝑝6

′

𝑝6′

𝑝5′

𝑝5′

𝑷𝟒:𝟎′′

𝑷𝟓:𝟎′′

𝑷𝟔:𝟎′′

pgh pgh pghpghpgh


c3c4c5c6

h3

c7

h4h5h6h7 h2

c2 c1

h1

pgh pgh pghh0


s0s1s2s3s4s5s6s7

Modulo (28−24 − 1) Modulo (28−1)

Problem : for 𝒒 > 𝒏/𝟐 the critical delay path is one ∆𝑮 more than other case

15



15 ARITH 22

15



• How to solve extra delay critical problem for 𝑞 >𝑛

2 problem?

15 ARITH 22

15




2 problem?

Devise a mixed KS/Lander-Fischer PPN architecture ((n-1) levels use

KS architecture and the bottom level use LF )

pgh pgh pgh pghpghp h′


(g2,p2)(g3,p3)(g5,p4)(g5,p5)(g6,p6) (g1,p1)

c3c4c5c6c7

s2s3s4s5s6s7

c2

pgh

s1

c1

s0

pgh pghh′

x7y7

(g0,p0)

h6

G7:0

(g7,p7)

y5 x5y6 x6

p4

g4

ℎ0 ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6′ ℎ7

′

(𝑮𝟎:𝟎, 𝑷𝟎:𝟎) (𝑮𝟏:𝟎, 𝑷𝟏:𝟎) (𝑮𝟐:𝟎, 𝑷𝟐:𝟎) (𝑮𝟑:𝟎, 𝑷𝟑:𝟎) (𝑮𝟒:𝟎, 𝑷𝟒:𝟎) (𝑮𝟒:𝟎, 𝑷𝟓:𝟎′′ ) (𝑮𝟓:𝟎, 𝑷𝟔:𝟎

′′ )

𝑮𝟑:𝟎 ∨ 𝑷𝟑:𝟎

𝑮𝟏:𝟎 ∨ 𝑷𝟏:𝟎

𝑮𝟏:𝟎 ∨ 𝑷𝟏:𝟎

𝒑𝟎

Eqn.9

ℎ5 𝑝6

′

𝑷𝟓:𝟎′′

𝑷𝟔:𝟎′′

𝐺3:0⋁𝑃3:0

𝑝6′

p g h h′

1 1i i i ix y x y

i ix yi ix y

ix iy

p g h

1ix 1iy

i ix y

p g h p’ h′

i ix yi ix y

1ix 1iy

i ix y

iyix

,G P ,r rG P

,r rG P G P PrG P G


i rh G P G

ih

1 1i i i ix y x y

1 1i i i ix y x y

ix iy

i ix yi ix y i ix y

,G P

,G P

5h 6p

4P

5:0P

6:0P

Eqn.9

,G P ,r rG P

( )r rG P G P rG P G

r rG P 4g

3:0 3:0G P

15 ARITH 22

15




2 problem?





(g2,p2)(g3,p3)(g5,p4)(g5,p5)(g6,p6) (g1,p1)

c3c4c5c6c7

s2s3s4s5s6s7

c2

pgh

s1

c1

s0

pgh pghh′

x7y7

(g0,p0)

h6

G7:0

(g7,p7)

y5 x5y6 x6

p4

g4

ℎ0 ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6′ ℎ7

′

(𝑮𝟎:𝟎, 𝑷𝟎:𝟎) (𝑮𝟏:𝟎, 𝑷𝟏:𝟎) (𝑮𝟐:𝟎, 𝑷𝟐:𝟎) (𝑮𝟑:𝟎, 𝑷𝟑:𝟎) (𝑮𝟒:𝟎, 𝑷𝟒:𝟎) (𝑮𝟒:𝟎, 𝑷𝟓:𝟎′′ ) (𝑮𝟓:𝟎, 𝑷𝟔:𝟎

′′ )

𝑮𝟑:𝟎 ∨ 𝑷𝟑:𝟎

𝑮𝟏:𝟎 ∨ 𝑷𝟏:𝟎

𝑮𝟏:𝟎 ∨ 𝑷𝟏:𝟎

𝒑𝟎

Eqn.9

ℎ5 𝑝6

′

𝑷𝟓:𝟎′′

𝑷𝟔:𝟎′′

𝐺3:0⋁𝑃3:0

𝑝6′

p g h h′

1 1i i i ix y x y

i ix yi ix y

ix iy

p g h

1ix 1iy

i ix y

p g h p’ h′

i ix yi ix y

1ix 1iy

i ix y

iyix

,G P ,r rG P

,r rG P G P PrG P G


i rh G P G

ih

1 1i i i ix y x y

1 1i i i ix y x y

ix iy

i ix yi ix y i ix y

,G P

,G P

5h 6p

4P

5:0P

6:0P

Eqn.9

,G P ,r rG P


r rG P 4g

3:0 3:0G P

KS

LF

15 ARITH 22

15




2 problem?





(g2,p2)(g3,p3)(g5,p4)(g5,p5)(g6,p6) (g1,p1)

c3c4c5c6c7

s2s3s4s5s6s7

c2

pgh

s1

c1

s0

pgh pghh′

x7y7

(g0,p0)

h6

G7:0

(g7,p7)

y5 x5y6 x6

p4

g4

ℎ0 ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6′ ℎ7

′

(𝑮𝟎:𝟎, 𝑷𝟎:𝟎) (𝑮𝟏:𝟎, 𝑷𝟏:𝟎) (𝑮𝟐:𝟎, 𝑷𝟐:𝟎) (𝑮𝟑:𝟎, 𝑷𝟑:𝟎) (𝑮𝟒:𝟎, 𝑷𝟒:𝟎) (𝑮𝟒:𝟎, 𝑷𝟓:𝟎′′ ) (𝑮𝟓:𝟎, 𝑷𝟔:𝟎

′′ )

𝑮𝟑:𝟎 ∨ 𝑷𝟑:𝟎

𝑮𝟏:𝟎 ∨ 𝑷𝟏:𝟎

𝑮𝟏:𝟎 ∨ 𝑷𝟏:𝟎

𝒑𝟎

Eqn.9

ℎ5 𝑝6

′

𝑷𝟓:𝟎′′

𝑷𝟔:𝟎′′

𝐺3:0⋁𝑃3:0

𝑝6′

p g h h′

1 1i i i ix y x y

i ix yi ix y

ix iy

p g h

1ix 1iy

i ix y

p g h p’ h′

i ix yi ix y

1ix 1iy

i ix y

iyix

,G P ,r rG P

,r rG P G P PrG P G


i rh G P G

ih

1 1i i i ix y x y

1 1i i i ix y x y

ix iy

i ix yi ix y i ix y

,G P

,G P

5h 6p

4P

5:0P

6:0P

Eqn.9

,G P ,r rG P


r rG P 4g

3:0 3:0G P

Using twin nodes (Computing 𝐺𝑖:0 and 𝐺𝑖:0 ∨ 𝑃𝑖:0 as a same time)

15 ARITH 22

16



ARITH 22

16



Design Delay (∆𝑮) Area (𝓐𝑮)

[4] 4 log 𝑛 + 8 6𝑛 log 𝑛 + 5𝑛 + 1

[5] 4 log 𝑛 + 12 3𝑛 log 𝑛 + 10𝑛 + 1

[6] 2 log (𝑛 − 1) + 7 3 𝑛 − 1 log 𝑛 +

3 𝑛 − 1 log 𝑛 − 1 + 5𝑛 + 1

[7] 2 log 𝑛 − 1 +

2 log (𝑛 − 2) + 8

3 𝑛 − 2 log 𝑛 − 2 −

log (𝑛 − 1) + 7𝑛 + 27

[8] 2 log (𝑛 − 1) + 7 3 𝑛 log (𝑛 − 1) + 13𝑛 − 5

[9] 2 log 𝑛 + 7 3𝑛 − 1 log𝑛 + 11.5𝑛 + 1

[10] 2 log 𝑛 + 7 3𝑛 log 𝑛 + 6𝑛 − 3𝑞 + 2 + 𝒜𝑐

[11] (𝒒 = 𝒏 − 𝟐) 2 log 𝑛 + 5 3𝑛 log𝑛 + 1.5𝑛 log 𝑛 − 1

+ 7𝑛 + 2 log 𝑛−1 −1

[13]-RPP (𝒒 = 𝟏) 2 log (𝑛 − 1) + 6 3 𝑛 − 1 log(𝑛 − 1) + 8𝑛 − 1

New (𝒒 ≤ 𝒏/𝟐) 2 log 𝑛 + 5 3 𝑛 − 1 log 𝑛 +

7𝑛 − 3𝑞 − 1 + 𝒜𝑃′′

New (𝒒 > 𝒏/𝟐) 2 log 𝑛 + 5 3(𝑛 − 1) log 𝑛 + 5.5𝑛 −

3𝑞 + 2 log 𝑛 + 𝒜𝑃′′

[3]-RPP 2 log 𝑛 + 5 3𝑛 log 𝑛 + 4𝑛

∆𝑮: Delay of Simple Gate

𝓐𝑮: Area of Simple Gate

Analytical gate Level Evaluation

ARITH 22

17



ARITH 22

17



ARITH 22

Synthesis: Synopsys Design Compiler Technology file: 130 nm

17



ARITH 22


𝑞 1 2 3 4 5 6

Least delay (𝒏𝒔) 0.71 0.71 0.72 0.71 0.70 0.70

Area (𝝁𝒎𝟐) 1405 1268 1224 1129 1110 910

Power (𝝁𝒘) 449 408 383 353 326 264

Performance measures of the New design (𝑛 = 8, 1 ≤ 𝑞 ≤ 6)

17



ARITH 22

Design Least delay

(𝒏𝒔) Ratio Area

(𝝁𝒎𝟐) Ratio AT

Power

𝝁𝒘 Ratio PDP

[6] 0.77 1.08 1828 1.30 1.41 577 1.28 1.39

[8] 0.84 1.18 1501 1.07 1.26 499 1.11 1.31

[9] 0.83 1.17 1524 1.08 1.27 500 1.11 1.30

[10] 0.84 1.18 1504 1.07 1.27 500 1.11 1.32

[13] 0.80 1.12 1400 1.00 1.12 446 0.99 1.12

New 0.71 1.00 1405 1.00 1.00 449 1.00 1.00


(n=8,q=1)

17



ARITH 22

Design Least delay

(𝒏𝒔) Ratio Area


Power

𝝁𝒘 Ratio PDP

[6] 0.77 1.08 1828 1.30 1.41 577 1.28 1.39

[8] 0.84 1.18 1501 1.07 1.26 499 1.11 1.31

[9] 0.83 1.17 1524 1.08 1.27 500 1.11 1.30

[10] 0.84 1.18 1504 1.07 1.27 500 1.11 1.32

[13] 0.80 1.12 1400 1.00 1.12 446 0.99 1.12

New 0.71 1.00 1405 1.00 1.00 449 1.00 1.00


(n=8,q=6)

(n=8,q=1)

Design Least delay

(𝒏𝒔) Ratio

Area


Power

𝝁𝒘 Ratio PDP

[6] 0.77 1.10 1825 2.00 2.21 550 2.08 2.29

[8] 0.84 1.20 1404 1.54 1.85 434 1.64 1.97

[9] 0.88 1.26 1338 1.47 1.85 454 1.72 2.16

[10] 0.75 1.07 1480 1.63 1.74 452 1.71 1.83

[11] 0.73 1.04 1394 1.53 1.60 442 1.67 1.75

New 0.70 1.00 910 1.00 1.00 264 1.00 1.00

17



ARITH 22


Desig

n

Least delay

(𝒏𝒔) Ratio

Area


Power

𝝁𝒘 Ratio PDP

[6] 0.88 1.03 4589 1.41 1.46 1383 1.40 1.44

[8] 0.90 1.06 3725 1.15 1.21 1190 1.20 1.27

[9] 0.92 1.08 3828 1.18 1.28 1295 1.31 1.41

[10] 0.97 1.14 3394 1.04 1.19 1075 1.08 1.24

[13] 0.89 1.05 3414 1.05 1.10 1018 1.03 1.08

New 0.85 1.00 3248 1.00 1.00 991 1.00 1.00

(n=16,q=1)

17



ARITH 22


Desig

n

Least delay

(𝒏𝒔) Ratio

Area


Power

𝝁𝒘 Ratio PDP

[6] 0.88 1.03 4589 1.41 1.46 1383 1.40 1.44

[8] 0.90 1.06 3725 1.15 1.21 1190 1.20 1.27

[9] 0.92 1.08 3828 1.18 1.28 1295 1.31 1.41

[10] 0.97 1.14 3394 1.04 1.19 1075 1.08 1.24

[13] 0.89 1.05 3414 1.05 1.10 1018 1.03 1.08

New 0.85 1.00 3248 1.00 1.00 991 1.00 1.00

Desig

n

Least delay

(𝒏𝒔) Ratio

Area


Power

𝝁𝒘 Ratio PDP

[6] 0.90 1.06 4538 2.18 2.31 1360 2.34 2.48

[8] 0.91 1.07 3988 1.91 2.05 1262 2.17 2.32

[9] 0.90 1.06 3855 1.85 1.96 1341 2.31 2.44

[10] 0.89 1.05 3394 1.63 1.71 940 1.62 1.69

[11] 0.87 1.02 3829 1.84 1.88 1226 2.11 2.16

New 0.85 1.00 2083 1.00 1.00 581 1.00 1.00

(n=16,q=1)

(n=16,q=14)

18



ARITH 22

18



ARITH 22

Conclusion

• Design and implement a new Modulo- 𝟐𝒏 − 𝟐𝒒 − 𝟏 parallel prefix adder

based on only one interim sum which has the same delay complexity as

the modulo- 𝟐𝒏 − 𝟏 adder .

18



ARITH 22

Conclusion




• Future work

18



ARITH 22

Conclusion




• Future work

Design and implement Modulo-(𝟐𝒏 − 𝟐𝒒 − 𝟏) multiplier

Questions

19



ARITH 22

sbu - tehran ensl - lyonarith22.gforge.inria.fr/slides/11-langroudi.pdfmodulo- − − parallel...

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