FPGA based Hardware Accleration for Elliptic Curve Cryptography ...

v E k
Input:
v ­ Y
Ø moxµBÙ 3 ( _ c
Output:
2.2. FINITE FIELD ARITHMETIC 11
Algorithm 4 Finite Field Inversion
Ø rx~yÚm v
whileØ ³ I
do
st
ṕ´ Ø
// right shift byØ
bits spmo(
st Ø r9xKy
for¯
fromc toKs ṕ´ cS do
qm
_ œqBr;Pst=zq // perform Û3 square operations
end for
qpmonpn
_ wr9xKyKst Ø rxKyX@gq
ifs is odd then
yœm‚npn
npn _ #qkr;PstPKyg yœm
Ø r9xKyœm npn _ wr9xKyKyX@ v
else
st
Ø r9xKyœmoy
end if
st
m Ø _ c Ø
end while
Ø rx~yÚmonpn _ #qkr9;=stPKst Ø r9xKyg
returnst
Ø rx~y
st
v 3CÝ ­ Y ͉Ï4Ð v ­ Y cƒÜ v 3CÝ ­ 3 ͉Ï4Ð (2.7)
Ü
Inversion can therefore be simply computed by repeated FF-Square and FF-Mult operations like it is shown
in Alg. 4. The algorithm in particular benefits from the fact in that squaring is much cheaper than
multiplication. The total number multiplications¢K(b of required for one FF inversion is given by
6]ÞKßÏà 3 K( _ cSºá)?¬bK( _ cS _ c#\
½K(b
2.2.8 Polynomial Reduction
As mentioned above, the basic arithmetic operations take place in-hÁ5ÃÂ . In case of multiplication and
squaring the resulting polynomial has to be reduced. According to Eqn. 2.5 the maximum degree of the
multiplication result} 6 v |
withv
@ |¤E isk( _ . The subsequent polynomial reduction of}
modulo is based on the equivalence
Ü P­ Y Ì
±–Çɮ⠱~5 ± ͉Ï4Ð \ (2.8)
5