Find the value of 445 in Z11 a using the fact that 45 3 mid

Find the value of 4^45 in Z_11 (a) using the fact that 45 = 3 middot 3 middot 5 (b) using repeated squaring

Solution

a) using the fact that 45=3.3.5

4⁴⁵ = 4^3.3.5 = (((4³)^3)^5) = (64³)^5 = 266144⁵ = 1.237940039*10²⁷

b) using repeated squaring 4⁴⁵ ---->

4²= 16

4⁴ = 16*16 = 256

4⁸ = 256*256 = 65536

4¹⁶ = 65536*65536 = 4294967296

4³² = 4294967296*4294967296 = 1.844674407*10¹⁹

45 = 32+8+4+1

so 4⁴⁵ = 4³² * 4⁸ * 4⁴ * 4¹ = 1.844674407*10¹⁹ * 65536 * 256 *4 = 1.237940039*10²⁷