FPGA based Hardware Accleration for Elliptic Curve Cryptography ...

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3.4. FINITE FIELD ARITHMETIC 24
can be added to the intermediate result. This leads to approximately k+ additional e)'U 3 gates for a
! #" 8
design. In general, an optimized multiplexer structure leads to the following resource requirement:
3 K+ˆ 6 + -!-e 3$#YZK+ˆ 6 k+ v
€ 3 K+ˆ 6 k+
e('U
3.4.6 Interleaved Polynomial Reduction
A first naive design approach may perform the polynomial reduction after the calculation of the complete
multiplication. According to Eqn. 2.9, such a architecture would requirek( ) >#e('U 3 gates. Furthermore,
this would lead to datapaths and multiplexers of sizek( _ c . To keep the datapaths on a maximum size of
( a method of interleaved reduction has been developed.
When utilizing the MSK multiplication scheme based on + a -bit CKM the maximum degree of each
intermediate is( )Q+ result with+/. ( , . only( )Q+ Therefore, bit values have to be reduced in each
iteration. Regarding this fact Eqn. 2.9 reads as
} 6 =­ Yª ô Ì
Ò ±K5 ± Ü =­ Y
±ÈÇÉ® Ò ±z5 ± ) P­ Yª ô
±ÈÇÉ®
±–Ç Ò ±CK5 ±­. ),5 㪠±­. Í7ÏÐ
@ (3.1)
which results in a total of onlyk+]e('U 3 gates for the polynomial reduction. The particular terms (1...3)
of Eqn. 3.1 are structured according to Fig. 3.5 in order to calculate the reduction. Within the MSK scheme,
this kind of interleaved reduction of degree ( )²+ polynomials is performed each time the intermediate
result is shifted left by+ bit.
n−1 0
(1)
n−1+m
(3)
n−1+m
(2)
n
n
6 =­ Y
±ÈÇÉ®#Ò ±5 ±
±–ÇÉ®Ò ±ª 5 㪠±
±ÈÇÉ®Ò ±ª 5 ±
) ô ­ Y
) ô ­ Y
åYºæ
å3 æ
å8 æ
¢ £h¤ ¥
¢ £h¤ ¥
¢ £h¤ ¥
Result Register (n bit)
Figure 3.5: Structure of the polynomial reduction of(*),+ bit
The gatecount calculation for part (f) of Fig. 3.2 is as follows: As already mentioned, the reduction of
degree()*+ polynomials takesk+!e)'pU 3 gates. Additionally,+
¯(:K(œ@ ñ + _ Fe)'U 3 gates are required
to perform the accumulation of the actual MSK pattern to the current result value. Finallyþk(D"!ƒe 3$#Y are
needed to choose if the result of an addition, square or multiply operation is stored in the result register.
Summing up, this results in the following resource requirement:
npn%K(b 6 ( e('U 3 K+ˆ 6 + ¯(:K(œ@ ñ + _ `)?k+ -!-e 3$#YBK(b 6dñ (