Prime Field ECDSA Signature Processing for Reconfigurable Embedded Systems

Algorithm 5

Simultaneous multiple point multiplication.

Input:  Point 𝑃,𝑄∈𝐸; Integers 𝑘=∑^𝑙−1_𝑖=0𝑘_𝑖2^𝑖 and

    𝑟=∑^𝑙−1_𝑖=0𝑟_𝑖2^𝑖 with 𝑘_𝑖,𝑟_𝑖∈{0,1} and 𝑘_𝑙−1∨𝑟_𝑙−1=1.

Output:  Point 𝑋=𝑘𝑃+𝑟𝑄∈𝐸.

(1)  Precomputation: 𝑃+𝑄

(2)  𝑋=𝑂 (point at infinity)

(3)  for  𝑖=𝑙−2 downto 0  do

(4)   𝑋=2𝑋

(5)   𝑋=𝑋+(𝑘_𝑖𝑃+𝑟_𝑖𝐺)

(6)  end  for

(7)  return 𝑋