Discrrete mathematics for computer science crypto warmup

Modular Arithmetic Warmup.

Modular Arithmetic

Warmup

Computing powers

• What is 32 (mod 7)?

• 32 = 9 = 2 (mod 7)

• What is 325 (mod 7)?

• 325 = (312)2×3

• 312 = (36)2

• 36 = (33)2

• 33 = 32×3 = 2×3 = 6 (mod 7)

• 36 = 62 = 1 (mod 7)

• 312 = 12 = 1 (mod 7)

• 325 = 12 × 3 = 3 (mod 7)

Discrete Logarithms

• So 25 is a base-7 discrete logarithm of 3 since 325 = 3 (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 1 (mod 7)?

• 57 is an answer since 12357 =

1903238928200610384515703215328258882685708609732 3460034686891056260376780393021529271254522717047 128465906993118819286×7+1

• But how would you ever know?

• And if the base and the modulus get bigger …