FPGA based Hardware Accleration for Elliptic Curve Cryptography ...

Ymonpn _ #qkr9;=stPKe'Y[ l
l
monpn _ #qkr9;=stPuiY[ 3
3 monpn _ #qkr9;=stPl 3 l
l
monpn _ 2rxKyhl 3 @A> Y
3 monpn _ #qkr9;=stPzf Y l
l
monpn _ vƒ{={ l 3 @ l Y l Ymonpn _ 2rxKyhz;9@Xi
3
l Ymonpn _ vƒ{={ l YS@ l 3 8
l Ymonpn _ 2rxKyhl YZ@Ce 8
2.1. ELLIPTIC CURVE ARITHMETIC 7
Algorithm 2 EC-Double
Input:
6 KeNYS@gfDYh@XijY[ E -k
Output:Q)<
6 ¦ E ¦.
8 monpn _ 2rx~yl YZ@ l 3
e
i
m‚npn _ œqBr;Pst=l YX 8
e 8 m‚npn _ v-{P{ Ke 8 @ l Y
f 8 monpn _ 2rxKyhl Y @Xi 8
8 monpn _ vƒ{={ zf 8 @ l Y[
returnKe
f
@gf 8 @Xi 8 8
cations, 8 additions and 4 square operations. The computation of (4 EC-Double ) requires multiplications,
4 additions and 5 squares. All these operations have to be done in the underlying finite field.
2.1.3 EC point multiplication (žbŸ¡ )
Since the points on an elliptic curve/
form an additive group, there is no inner group operation like the
multiplication. Even so repeated point additions such as
§Z 6 U 6
Ë /
and
E'©ª
, are usually considered as the operation called EC point multiplication.
Based on this operation, a discrete logarithm problem for elliptic curves can be formulated. A problem,
that is considered to be a secure cryptographic function. A secure cryptographic function in this terms
with@gU
means, the ofU calculation of and out can be performed quite efficient while it is hardly possible
compute
to
only andU if known. are is called the discrete ofU logarithm to base the .
The level of security, ECC provides directly follows from the bitwidth the numbers in the underlying
finite field. Currently bitwidths ranging from 113 bit (for low security applications) up to about 409 bit (for
very high security applications) are utilized.
The hierarchy of arithmetics for an EC point multiplication is depicted in Fig. 2.2. The level top algorithm
is performed by repeated EC-Add and EC-Double operations. The EC operations in turn are composed
of basic operations in the underlying field. The proposed finite field arithmetic is capable to compute the
FF-Add and FF-Square operations within one clock cycle. The operation FF-Mult is more costly. The
number of clock cycles for its computation depends on the number of segments used in the FF multiplier
(see Sec. 2.3.3 for details).
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times
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