Page 215 - Hardware Implementation of Finite-Field Arithmetic

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Operations over GF (2 )—Polynomial Bases 195

register_w: process(reset, inic, clk)
begin
if reset = ‘1’ then w <= (others => ‘0’);
elsif clk’event and clk = ‘1’ then
if inic = ‘1’ then
w(0) <= A(0);
square: for i in 1 to (M-1)/2 loop
w(2*i-1) <= ‘0’; w(2*i) <= A(i);
end loop;
if M mod 2 = 0 then w(M-1) <= ‘0’; end if;
elsif ce_c = ‘1’ then w <= new_w; end if;
end if;
end process register_w;
z <= w;
counter: process(reset, clk)
begin
if reset = ‘1’ then count <= 0;
elsif clk’ event and clk = ‘1’ then
if inic = ‘1’ then count <= 0;
elsif shift_r = ‘1’ then count <= count+1; end if;
end if;
end process counter;
control_unit: process(clk, reset, current_state)
begin
case current_state is
when 0 to 1 => inic <= ‘0’; shift_r <=’0’; done <=’1’;
ce_c <=’0’;
when 2 => inic <= ‘1’; shift_r <= ‘0’; done <= ‘0’;
ce_c <= ‘0’;
when 3 => inic <= ‘0’; shift_r <= ‘1’; done <= ‘0’;
ce_c <= ‘1’;
end case;
if reset = ‘1’ then current_state <= 0;
elsif clk’event and clk = ‘1’ then
case current_state is
when 0 => if start = ‘0’ then current_state <= 1;
end if;
when 1 => if start = ‘1’ then current_state <= 2;
end if;
when 2 => current_state <= 3;
when 3 => if count = (M+1)/2-1 then current_state
<= 0; end if;
end case;
end if;
end process control_unit;

7.3 Exponentiation
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Let a be a n arbitrary element of a finite field GF(2 ), and e an arbitrary
positive integer. Then, field exponentiation is defined as the problem
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e
of finding an element b ∈ GF(2 ) so that the equation b = a holds. In
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general, an arbitrary integer power of an element a ∈ GF(2 ) can be