Discrrete mathematics for computer science crypto warmup

Modular Arithmetic Warmup Computing powers • • • • • • • • • • What is 32 (mod 7)? 32 = = (mod 7) What is 325 (mod 7)? 325 = (312)2×3 312 = (36)2 36 = (33)2 33 = 32×3 = 2×3 = (mod 7) 36 = 62 = (mod 7) 312 = 12 = (mod 7) 325 = 12 × = (mod 7) Discrete Logarithms • So 25 is a base-7 discrete logarithm of since 325 = (mod 7) • What is log123133226724974042726916099225072978121787999602 6812642202428082373938226374627511507048987816590 19329899261348951831735003? • Easy using Wolfram alpha: – logba = log a/log b • But what is a discrete base 123 log of (mod 7)? • 57 is an answer since 12357 = 1903238928200610384515703215328258882685708609732 3460034686891056260376780393021529271254522717047 128465906993118819286ì7+1 But how would you ever know? And if the base and the modulus get bigger …