1. Sidebar – Computing Large Numbers mod n In public key cryptography, we have to compute large numbers, like: 643 mod 13 Calculating such large numbers on a hand calculator (or on the PC calculator often results in getting a floating point approximation. This isn’t much use since we need exact arithmetic. These kinds of problems can be decomposed in to simpler forms by direct manipulation. One method is repeated squaring. Big Numbers

2. Large Numbers mod n For 643 mod 13, observe = (243 mod 13) x (343 mod 13). Also consider: 21 mod 13 = 2 mod 13 = 2 22 mod 13 = 4 mod 13 = 4 24 mod 13 = 42 mod 13 = 16 mod 13 = 3 28 mod 13 = (24)2 mod 13 = 32 mod 13 = 9 216 mod 13 = (28)2 mod 13 = 92 mod 13 = 81 mod 13 = 3 232 mod 13 = (216)2 mod 13 = 32 mod 13 = 9 So… 243 mod 13 = (232 x 28 x 22 x 21) mod 13 = (9 x 9 x 4 x 2) mod 13 = mod 13 = 11 Big Numbers

3. Large Numbers mod n For 343 mod 13: 31 mod 13 = 3 mod 13 = 3 32 mod 13 = 9 mod 13 = 9 34 mod 13 = 92 mod 13 = 81 mod 13 = 3 38 mod 13 = (34)2 mod 13 = 32 mod 13 = 9 316 mod 13 = (38)2 mod 13 = 92 mod 13 = 81 mod 13 = 3 332 mod 13 = (316)2 mod 13 = 32 mod 13 = 9 So… 343 mod 13 = (332 x 38 x 32 x 31) mod 13 = (9 x 9 x 9 x 3) mod 13 = 2187 mod 13 = 3 And 243 x 343 mod 13 = (11 x 3) mod 13 = 33 mod 13 = 7 Big Numbers