1. Modular Arithmetic Warmup

2. Computing powers What is 3 2 (mod 7)? 3 2 = 9 = 2 (mod 7) What is 3 25 (mod 7)? 3 25 = (3 12 ) 2 ×3 3 12 = (3 6 ) 2 3 6 = (3 3 ) 2 3 3 = 3 2 ×3 = 2×3 = 6 (mod 7) 3 6 = 6 2 = 1 (mod 7) 3 12 = 1 2 = 1 (mod 7) 3 25 = 1 2 × 3 = 3 (mod 7)

3. Discrete Logarithms So 25 is a base-7 discrete logarithm of 3 since 3 25 = 3 (mod 7) What is log 123 1332267249740427269160992250729781217879996026 812642202428082373938226374627511507048987816590193 29899261348951831735003? Easy using Wolfram alpha: –log b a = log a/log b But what is a discrete base 123 log of 1 (mod 7)? 57 is an answer since 123 57 = 190323892820061038451570321532825888268570860973234 600346868910562603767803930215292712545227170471284 65906993118819286×7+1 But how would you ever know? And if the base and the modulus get bigger ….