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Understanding Divisibility and Modular Arithmetic in Computing

This lecture from CSE 311 discusses foundational concepts in computability, focusing on divisibility and modular arithmetic. It addresses the Division Theorem, congruence relations, and arithmetic operations within a finite domain. Key ideas include the importance of integers in computing, examples of modular arithmetic operations, and practical applications in cryptography. The lecture also touches on vital terms in set theory that relate to computing, such as cardinality and Venn diagrams, providing a comprehensive overview for students interested in the mathematics underpinning computer science.

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Understanding Divisibility and Modular Arithmetic in Computing

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  1. CSE 311 Foundations of Computing I Lecture 10 Divisibility and Modular Arithmetic Autumn 2011 CSE 311

  2. Announcements • Reading assignments • Today: • 4.1-4.2 7th Edition • 3.4, 3.6 up to p. 227 6th Edition • 2.4, 2.5 up to p. 177 5th Edition • Homework 4 • Coming soon . . . CSE 311

  3. Highlights from last lecture • Set theory and ties to logic • Lots of terminology • Complement, Universe of Discourse, Cartesian Product, Cardinality, Power Set, Empty Set, N, Z, Z+, Q, R, Subset, Proper Subset, Venn Diagram, Set Difference, Symmetric Difference, De Morgan’s Laws, Distributive Laws • Bit vector representation of characteristic functions • Bitwise operations for Set operations CSE 311

  4. Unix/Linux file permissions • ls–l drwxr-xr-x ... Documents/ -rw-r--r-- ... file1 • Permissions maintained as bit vectors • Letter means bit is 1 -means bit is 0. • How is chmodog+rimplemented? CSE 311

  5. A simple identity • If x and y are bits: (x y)  y = ? • What if x and y are bit-vectors? CSE 311

  6. Private Key Cryptography • Alice wants to be able to communicate message secretly to Bob so that eavesdropper Eve who hears their conversation, cannot tell what Alice’s message is • Alice and Bob can get together and privately share a secret key K ahead of time. CSE 311

  7. One-time pad • Alice and Bob privately share random n-bit vector K • Eve does not know K • Later, Alice has n-bit message m to send to Bob • Alice computes C = m  K • Alice sends C to Bob • Bob computes m = C  K which is (m  K)  K • Eve cannot figure out m from C unless she can guess K CSE 311

  8. Russell’s Paradox S = { x | xx } /

  9. Number Theory (and applications to computing) • Branch of Mathematics with direct relevance to computing • Many significant applications • Cryptography • Hashing • Security • Important tool set

  10. Modular Arithmetic • Arithmetic over a finite domain • In computing, almost all computations are over a finite domain

  11. What are the values computed? [-128, 127] public void Test1() { byte x = 250; byte y = 20; byte z = (byte) (x + y); Console.WriteLine(z); } public void Test2() { sbyte x = 120; sbyte y = 20; sbyte z = (sbyte) (x + y); Console.WriteLine(z); } 14 -116

  12. CSE 311

  13. Arithmetic mod 7 • a +7 b = (a + b) mod 7 • a 7 b = (a  b) mod 7

  14. Divisibility Integers a, b, with a ≠ 0, we say that a divides b is there is an integer k such that b = ak. The notation a | b denotes a divides b. CSE 311

  15. Division Theorem Let a be an integer and d a positive integer. Then there are unique integers q and r, with 0 ≤ r < d, such that a = dq + r. q = adivdr = amodd Note: r ≥ 0 even if a < 0. Not quite the same as a%d CSE 311

  16. Modular Arithmetic Let a and b be integers, and m be a positive integer. We say a is congruent to b modulo m if m divides a – b. We use the notation a ≡ b (mod m) to indicate that a is congruent to b modulo m. CSE 311

  17. Modular arithmetic Let a and b be integers, and let m be a positive integer. Then a ≡ b (mod m) if and only if a mod m = b mod m. CSE 311

  18. Modular arithmetic • Let m be a positive integer. If a ≡ b (mod m) and c ≡ d (mod m), then • a + c ≡ b + d (mod m) and • ac ≡ bd (mod m) CSE 311

  19. Example Let n be an integer, prove that n2 ≡ 0 (mod 4) or n2 ≡ 1 (mod 4) CSE 311

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