FPGA based Hardware Accleration for Elliptic Curve Cryptography ...

6 v | }
v Yg5 ¦î 3 ¾ v ®SaP| Yg5 î 3 ¾ | ®S 6
v Y | Yg5 ¾ v Y | ®[5 î 3 ¾ v ® | Yg5 ¦î 3 ¾ v ® | ®
6
6 v Y | Y
ú
ü v Y | Y
ü v ® | ®
Y 0ó6 v Y | Y l
3 0ó6 v Ya¾ v ®Sb=| Ya¾ | ®S l
v Y | ®Ú¾ v ® | YD¾ v Y | Yb¾ v ® | ® 6
l
8
8
8
2.3. SEQUENTIAL MULTIPLICATION SCHEMES 15
Fig. 2.4a illustrates the schoolbook multiplication for 6°ñ
. The gray boxes represent the results of
the degree+ respective polynomial multiplications, which are denoted next to the boxes. The horizontal
position of a box indicates its ò 5 0ó6 59ô offset . The ordering of the partial products by
¯
decreasing
allows for the accumulation of the final result in a shift register and the application of an interleaved
reduction scheme as detailed in Sec. 3.4.6.
5 ±
with ò
2.3.2 Polynomial Karatsuba Multiplication
In 1963 A. Karatsuba and Y. Ofman developed an algorithm of complexity õ7K(Úöø÷Cù 8 that computes the
product of two( -bit integers [17].
Like the Schoolbook Multiplication, this algorithm divides the operands into two equal parts. Adopting
the arithmetical operations tohÁ5. leads to
¢ £h¤ ¥
¢ £¤ ¥
¢ £h¤ ¥
¢ £h¤ ¥
¢ £h¤ ¥
úû 5 ¾ÆÁ–v Ya¾ v ®Sh| Ya¾ | ®S
úBý ÂÈ5 î 3 ¾ v ® | ®
úû
úBý
6 l Y 5 ¾²Ál 3 ü l Y ü l
8 ÂÈ5 ¦î 3 ¾ l
8
(2.11)
6 l Y 5 ¾²Ál Y ¾ l 3 ¾ l
8
given by
8 ÂÈ5 ¦î 3 ¾ l
withl
YZ@ l 3 andl
v ® | ®k\ 0ó6
Thus, the final product can be computed by 3 multiplications and 2 additions degree(b†¦ of polynomials
and 4 additions degree( of polynomials as illustrated in Fig. 2.5. Again, since the addition can be computed
combinationally in the same cycle as a partial multiplication, the complete multiplication takes 3 cycles
only. By splitting the into
6 ±
factors segments any¯ EN©
(for ), the product can be withþ=öø÷Cù ¦
computed
multiplications by a recursive application of this scheme.
Due to the need to store intermediate results and to maintain a stack, recursive algorithms are not appropriate
for hardware implementations. The recursion has thus to be unrolled. Fig. 2.4b shows the resulting
degree+
multiplication scheme for an unrolled recursion of the Karatsuba with
6Šñ
multiplication . Each pattern
in Fig. 2.4b, which is additionally surrounded by a gray box, can be composed from one partial multiplication.
The labels at the right side of the boxes determine the indices of the segments, whose sums have been
multiplied. E.g., the label "13" denotes the termv
Ya¾ v
8 .
6 v Y | ®Ú¾ v ® | YD¾ l Ya¾ l
8 b=| Yb¾ |