1 / 3

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).

gaura
Download Presentation

Modular Arithmetic Warmup

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 ….

More Related