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Modular Arithmetic Warmup

Modular Arithmetic Warmup. 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).

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Modular Arithmetic Warmup

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  1. Modular Arithmetic Warmup

  2. 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)

  3. Discrete Logarithms • So 25 is a base-7 discrete logarithm of 3 since 325 = 3 (mod 7) • What is log123133226724974042726916099225072978121787999602681264220242808237393822637462751150704898781659019329899261348951831735003? • 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 = 19032389282006103845157032153282588826857086097323460034686891056260376780393021529271254522717047128465906993118819286×7+1 • But how would you ever know? • And if the base and the modulus get bigger ….

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