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Modular Arithmetic Warmup

This article delves into modular arithmetic, focusing on computing powers and discrete logarithms. We explore how to calculate (32 mod 7) and (325 mod 7), illustrating the results step-by-step. We'll also discuss the concept of discrete logarithms and demonstrate how to find a base-7 logarithm for a given number using examples. By understanding these concepts, you'll gain insights into the applications of modular arithmetic, which are fundamental in areas such as cryptography and computer science.

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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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