Exponentiation and

1
Lecture 9
Exponentiation and Division
2
Modular Multiplication
Special Cases
k bits
a
a
a x
p pH 2k pL
x
x
pH
pL
p
a x mod 2k pL
a x mod 2k-1 pL pH carry
a x mod 2k1 pL - pH - borrow
3
Modular Multiplication
Special Case (1)
a x mod 2k-1 (pH 2k pL) mod (2k-1)
(pH (2k mod 2k-1) pL) mod
(2k-1) pH pL mod
(2k-1)
pH pL if pH pL lt 2k
- 1
pH pL - (2k-1) if pH pL ? 2k - 1
pL pH carry

carry carry from addition pL pH
4
Fig. 12.15 Design of a 4 4 modulo-15 multiplier.
5
Fig. 12.16 One way to design of a 4 4 modulo-13
multiplier.
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Exponentiation Y XE mod N
Right-to-left binary exponentiation
Left-to-right binary exponentiation
E (eL-1, eL-2, , e1, e0)2
Y 1 S X for i0 to L-1 if (ei
1) Y Y ? S mod N S S2 mod N
Y 1 for iL-1 downto 0 Y Y2 mod N
if (ei 1) Y Y ? X mod N
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Exponentiation Example Y 712 mod 11
Right-to-left binary exponentiation
Left-to-right binary exponentiation
12 (1 1 0 0)2
i 3 2 1 0 ei
1 1 0 0 Y 1 7 2 4
5
i 0 1 2 3 ei
0 0 1
1 Sbefore 7 5 3
9 Yafter 1 1 1 3
5 Safter 7 5 3 9 4
Sbefore - S before round i is computed Safter -
S after round i is computed
8
Right-to-Left Binary Exponentiation in Hardware
X
1
enable
S
Y
E
SQR
MUL
output
9
Left-to-Right Binary Exponentiation in Hardware
1
Y
X
Control Logic
E
MUL
output
10
Notation
z Dividend z2k-1z2k-2 . . . z2 z1 z0 d
Divisor dk-1dk-2 . . . d1 d0 q
Quotient qk-1qk-2 . . . q1 q0 s
Remainder sk-1sk-2 . . . s1 s0 (s
z - dq)
11
Basic Equations of Division
z d q s
s lt d
sign(s) sign(z)
z gt 0 0 ? s lt d
z lt 0 - d lt s ? 0
12
Unsigned Integer Division Overflow
Condition for no overflow
z q d s lt (2k-1) d d d 2k
z zH 2k zL lt d 2k
zH lt d
13
Sequential Integer Division Basic Equations
s(0) z s(j) 2 s(j-1) - qk-j (2k d) s(k)
2k s
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Fig. 13.2 Examples of sequential division with
integer and fractional operands.
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Sequential Integer Division Justification
s(1) 2 z - qk-1 (2k d) s(2) 2(2 z - qk-1 (2k
d)) - qk-2 (2k d) s(3) 2(2(2 z - qk-1 (2k d)) -
qk-2 (2k d)) - qk-3 (2k d) . . . . . .
s(k) 2(. . . 2(2(2 z - qk-1 (2k d)) - qk-2
(2k d)) - qk-3 (2k d) . . . - q0 (2k d)
2k z - (2k d) (qk-1 2k-1 qk-2 2k-2
qk-3 2k-3 q020) 2k z - (2k d) q
2k (z - d q) 2k s
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Unsigned Fractional Division
zfrac Dividend .z-1z-2 . . .
Z-(2k-1)z-2k dfrac Divisor .d-1d-2 . .
. d-(k-1) d-k qfrac Quotient .q-1q-2 .
. . q-(k-1) q-k sfrac Remainder
.0000s-(k1) . . . s-(2k-1) s-2k
k bits
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Integer vs. Fractional Division
For Integers
z q d s
? 2-2k
z 2-2k (q 2-k) (d 2-k) s (2-2k)
For Fractions
zfrac qfrac dfrac sfrac
where
zfrac z 2-2k dfrac d 2-k
qfrac q 2-k sfrac s 2-2k
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Unsigned Fractional Division Overflow
Condition for no overflow
zfrac lt dfrac
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Sequential Fractional Division Basic Equations
sfrac(0) zfrac s(j) 2 s(j-1) - q-j
dfrac s(k)frac 2k sfrac
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Sequential Fractional Division Justification
s(1) 2 zfrac - q-1 dfrac s(2) 2(2 zfrac - q-1
dfrac) - q-2 dfrac s(3) 2(2(2 zfrac - q-1
dfrac) - q-2 dfrac) - q-3 dfrac . . . . . .
s(k) 2(. . . 2(2(2 zfrac - q-1 dfrac) - q-2
dfrac) - q-3 dfrac . . . - q-k dfrac
2k zfrac - dfrac (q-1 2k-1 q-2 2k-2
q-3 2k-3 q-k20) 2k zfrac - dfrac
2k (q-1 2-1 q-2 2-2 q-3 2-3 q-k2-k)
2k zfrac - (2k dfrac) qfrac 2k (zfrac -
dfrac qfrac) 2k sfrac
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Restoring Unsigned Integer Division
s(0) z for j 1 to k if 2 s(j-1) - 2k
d gt 0 qk-j 1 s(j) 2 s(j-1)
- qk-j (2k d) else qk-j 0
s(j) 2 s(j-1)
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Fig. 13.5 Shift/subtract sequential restoring
divider.
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Fig. 13.6 Example of restoring unsigned division.
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Restoring Signed Integer Division
z
d
z
d
sign(z)
sign(d)
Unsigned division
sign(s) sign(z)
sign(z) sign(d)

q
s
sign(q)
-
sign(z) ? sign(d)
q
s
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Non-Restoring Unsigned Integer Division
s(0) z qk 1 for j 1 to k if qk-(j-1)
1 s(j) 2 s(j-1) - 2k d else
s(j) 2 s(j-1) 2k d if s(j) gt
0 qk-j 1 else qk-j 0
if s(k) lt 0 s(k) s(k) 2k d
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Non-Restoring Unsigned Integer Division
Justification
Restoring division
Non-Restoring division
s(j) 2 s(j-1) s(j1) 2 s(j) - 2k d
4 s(j-1) - 2k d
s(j) 2 s(j-1) - 2k d s(j1) 2 s(j) 2k d
2 (2 s(j-1) - 2k d) 2k d
4 s(j-1) - 2k d
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Fig. 13.7 Example of nonrestoring unsigned
division.
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Fig. 13.8 Partial remainder variations for
restoring andnonrestoring division.
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Fig. 13.10 Shift-subtract sequential nonrestoring
divider.
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Non-Restoring Signed Integer Division
s(0) z for j 1 to k if sign(s(j-1))
sign(d) qk-j 1 s(j) 2
s(j-1) - 2k d 2 s(j-1) - qk-j (2k d) else
qk-j -1 s(j) 2 s(j-1) 2k d
2 s(j-1) - qk-j (2k d)
Correction_step q BSD_2s_comp_conversion(q)
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Fig. 13.9 Example of nonrestoring signed division.
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Non-Restoring Signed Integer Division
Correction step
z q d s
z (q-1) d (sd) z q d s
z (q1) d (s-d) z q d s
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BSD ? 2s Complement Conversion
q (qk-1 qk-2 . . . q1 q0)BSD (pk-1 pk-2
. . . p1 p0 1)2s complement
where
Example
qBSD
1 -1 1 1
qi
pi
p
-1
0
1 0 1 1
1
1
q2scomp
0 0 1 1 1 0 1 1 1
no overflow if pk-2 pk-1 (qk-1 ? qk-2)
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Fig. 14.8 Block diagram of a radix-2 divider with
partialremainder in stored-carry form.
35
Using Carry-Save Adders with the Dividers
sum u u1u0.u-1u-2u-3u-4.u-k carry v
v1v0.v-1v-2v-3v-4.v-k
t u1u0.u-1u-2 v1v0.v-1v-2
u v - t 00.00u-3u-4.u-k
00.00v-3v-4.v-k
lt
0 ?
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Using Carry-Save Adders with the Dividers
-
0
t
-
t lt
-
? t lt 0
t ? 0
-
? uv lt
uv ? 0
uv lt 0
q-j -1
q-j 1
q-j 0
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Classification of Dividers
Array Dividers
Dividers by Convergence
Sequential
Radix-2
High-radix

• Restoring
• Non-restoring
• regular
• SRT
• using carry save adders
• SRT using carry save adders

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Fig. 15.7 Restoring array divider composed of
controlledsubtractor cells.
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Fig. 15.8 Nonrestoring array divider built of
controlledadd/subtract cells.
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Fig. 15.9 Sequential radix-2 multiply/divide unit.