FPGA based Hardware Accleration for Elliptic Curve Cryptography ...
FPGA based Hardware Accleration for Elliptic Curve Cryptography ...
FPGA based Hardware Accleration for Elliptic Curve Cryptography ...
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2.2.4 Addition in-·‰¸º¹ »<br />
2.2.7 Inversion in·7¸À¹ »<br />
Ò<br />
Ì<br />
2.2. FINITE FIELD ARITHMETIC 10<br />
the polynomial divisionv<br />
†k : It is a polynomial of degree less than( . The computation of the canonical<br />
representative is called polynomial reduction.<br />
This leads to the following definitions of the basic arithmetic operations that are similar to the operations<br />
defined in-hÁ5. except that an additional reduction is necessary whenever the degree of the resulting<br />
polynomial is³<br />
( .<br />
Given polynomialsv<br />
@ |ÊE withv 6 Ä = Y ±ÈÇÉ® ; ±5 ± and|Ë6 Ä = Y ±ÈÇÉ® > ±5 ±<br />
two , the addition operation<br />
is defined as<br />
¾ |!6 = Y v z; ± ¾W> ±º5 ±Î͉Ï4Ð \ (2.4)<br />
±ÈÇÉ®<br />
From Eqn. 2.4 thatv<br />
¾ v 6¢I<br />
follows allv E <strong>for</strong> . The additive inverse is there<strong>for</strong>e the identity<br />
function, i.e., addition and subtraction are identical Sincev<br />
¾ |<br />
operations. will be of a maximum<br />
of( _ c <strong>for</strong>v<br />
@ |½E -¦<br />
degree<br />
, no reduction step has to be per<strong>for</strong>med in the case of addition.<br />
2.2.5 in·7¸À¹ »<br />
Multiplication<br />
The multiplication of polynomialsv<br />
@ |¤E - two is given by<br />
denoting<br />
|Ñ6 3 = 3 Ì v ±5 ±ÓÍ7Ï4Ð (2.5)<br />
±–ÇÉ®%Ò<br />
6 ¦ Ô<br />
±ÈÇÉ® ;=±¿W> ¦ ± <strong>for</strong><br />
I7Õ Õ k( _ 4@<br />
¦<br />
with P as the corresponding prime polynomial ;.± 6ÖI<br />
and >X± 6×I<br />
,<br />
¯%³ ( <strong>for</strong> . Ä 3 = 3<br />
Since<br />
maximum ofk( _ degree the of( _ c reduction bits has to be per<strong>for</strong>med.<br />
±ÈÇÉ® Ò ±z5 ±<br />
has a<br />
2.2.6 in·7¸À¹ »<br />
Squaring<br />
Squaring is a special case of multiplication. By inserting Eqn. 2.4 into Eqn. 2.5 it can be simplified to<br />
3 6 = Y Ì v ;=±~5 3 ± ͉Ï4Ð (2.6)<br />
±ÈÇÉ®<br />
Like in the case of multiplication, a of( _ c maximum bits have to be reduced while per<strong>for</strong>ming a square<br />
operation.<br />
As stated in Sec. 2.1 the inversion is a complex operation that is computed only once a in<br />
62I<br />
Operation.<br />
, Fermat’s Little Theorem can be<br />
To compute the multiplicative inverse <strong>for</strong> elementv E ,v G<br />
an<br />
applied:
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