FPGA based Hardware Accleration for Elliptic Curve Cryptography ...

3.4. FINITE FIELD ARITHMETIC 23
Thus, we can calculate the number of gates of an+ -bit CKM with the following recurrences:
e)'U 3 K+ˆ
6+*
I + 6 c
3 K+ˆ†¦«+ ´ c e('U 8 K+ˆ
6+*
I + 6 c
c:)?þe('U 8 K+ˆ†¦ + ´ c
+¢)Wþe)'pU
€ 3 K+ˆ
6+* c + 6 c
þ v
€ 3 K+ˆ†¦«+ ´ c e)'U $ K+ˆ
6+*
I + 6 c
+ _ ¨)?þe)'pU-$PK+ˆ†¦ + ´ c
v
With the master method [19] it can easily be shown that all of these recurrences belong to the complexity
õ7K+¨öø÷Cù 8 class . The ofv
€ 3 number gates exactly+¨öø÷Cù 8 is . By substituting a 3-input XOR by
e('U 3
two
and a 4-input XOR threee)'pU 3 by gates, an upper bound on requirede('U 3 the count is be given
k+Nöø÷Cù 8 by . Some gate counts for multipliers of various operand bit widths are summarized in Tab. 3.1 and
illustrated in Fig. 3.4.
v
€ 3
e('U 3
e('U 8 e('U-$
Table 3.1:
Bit Width 1 2 4 8 16 32 64
1 3 9 27 81 243 729
0 2 10 38 130 422 1330
0 0 2 12 50 180 602
0 1 4 13 40 121 364
,! 1 6 25 90 301 966 3025
3500
3000
2500
2000
gate count
1500
1000
500
0
4
8
16
bit width
32
64
AND2
gate type
XOR4
XOR3
SUM
XOR2
Figure 3.4: Combinational Karatsuba Multiplier gate count
3.4.5 MSK Pattern Generation
As detailed in Sec. 2.3.3 there are three different MSK patterns which are build based on the output of the
CKM. A simplified illustration of the corresponding architecture is shown in part (e) of Fig. 3.2. In practice,
the pattern creation is implemented by various multiplexers. Moreover, in case of the& œ" 8
scheme, the
patterns will exceed the bit width( of the affiliated"!ƒe
8 #Y and therefore have to be reduced before they