Discrrete mathematics for computer science crypto warmup

Modular Arithmetic
Warmup


Computing powers
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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 ….