MODINT: Compact modular arithmetic Java class library for cellular phones, and its application to Secure E-Voting

Jul 16, 2013

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MODINT: Compact modular arithmetic Java class library for cellular phones, and its application to Secure E-Voting

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Modular Arithmetic

Modular Arithmetic. Several important cryptosystems make use of modular arithmetic. This is when the answer to a calculation is always in the range 0 – m where m is the modulus .

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Modular Arithmetic

Modular Arithmetic. Created by previous Honors Pre-Calculus students. Division Algorithm. If a and b are integers where b > 0, There exist integers, q and r, with the property that: a = bq + r where 0 ≤ r < b. For example, if a = 17 and b = 5,

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Modular (Remainder) Arithmetic

n = qk + r (for some k; r < k) eg 37 = (2)(17) + 3 Divisibility notation: 17 | 37 - 3. n mod k = r 37 mod 17 = 3. Modular (Remainder) Arithmetic. Sets of Remainders. x -2 (5 – 2) -1 (5 – 1) * 0 1 2 3 4 5 6 7 8. x mod 5 3 4 0 1 2 3 4 0 1 2 3.

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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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6. Modular Arithmetic

6. Modular Arithmetic. After this module. Modular mathematical system Modular Arithmetic and Multiplication Caley’s table. Congruence of Numbers. If a and b are integers and m is a positive integer, then “ a is congruent to b modulo m” if m divides a-b

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Modular Arithmetic

Modular Arithmetic. Shirley Moore CS4390/5390 Fall 2013 http://svmoore.pbworks.com/ September 3, 2013 . Agenda. Complete pre-course survey if you haven’t already Discuss homework from last class (15 min) Homework 1 to turn in (due Sept. 12) (5 min)

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Secure Network Coding and its Application

Secure Network Coding and its Application. Yongdae Kim University of Minnesota. Organization. Introduction to Network Coding Practical Network Coding Secure Network Coding Structured File Sharing Conclusion. Network Coding: What is it?. B. Traditional Coding/Routing

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Modular Arithmetic

Modular Arithmetic. Shirley Moore CS4390/5390 Fall 2013 http://svmoore.pbworks.com/ September 5, 2013 . Agenda. Intro to Matlab by Rogelio Long (cont.) (15 min) Discuss homework from last class (15 min ) Modular arithmetic ( 20 min ) Fermat’s Little Theorem (20 min)

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Cellular Phones

Cellular Phones. Holly Mote.

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CONGRUENCES AND MODULAR ARITHMETIC

CONGRUENCES AND MODULAR ARITHMETIC. Congruence and Modular Arithmetic Definition: a is congruent to b mod n means that n∣a -b, (a-b) is divisible by n. Notation: a ≡ b (mod n) , a, b, n ∈ I, n ≠ b Ex . 42 ≡ 30 (mod 3)

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Modular Arithmetic

Modular Arithmetic. p-1. 1. ´ p. n|m means that m is a an integer multiple of n. We say that “n divides m”. 7|21 7 divides 21. 2|4 2 divides 4 7|21 7 divides 21. (a mod n) means the remainder when a is divided by n. If ad + r = n, 0 · r < n Then r = (a mod n) and d = (a div n).

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Modular Arithmetic

Modular Arithmetic. Lecture 9: Oct 5. Modular Arithmetic. Def : a  b (mod n) iff n|( a - b) iff a mod n = b mod n. e.g. 12  2 (mod 10) 107  207 (mod 10) 7  1 (mod 2) 1  -1 (mod 2) 13  -1 (mod 7) -15  0 (mod 5). Modular Addition.

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Modular Arithmetic I

Modular Arithmetic I. Lecture 9: Oct 4. This Lecture. Modular arithmetic is an arithmetic about remainders. It is very useful in coding theory and cryptography. In this lecture we will focus on additions and multiplications, while in the next lecture we will talk about “divisions”.

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Modular Arithmetic

Modular Arithmetic. “ Clock” Arithmetic. For integers x and n, x mod n is the remainder of x  n. 0. Examples 7 mod 6 = 1 33 mod 5 = 3 33 mod 6 = 3 51 mod 17 = 0 17 mod 6 = 5. 1. 5. arithmetic mod 6. 2. 4. 3. 11 mod 7 = 4 , since 11 = 7X1 +4.

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Modular Arithmetic II

Modular Arithmetic II. Lecture 10: Oct 6. This Lecture. Last time we talked about modular additions and modular multiplications. This time we will talk about modular “divisions”, the inverse of multiplications. This is an important operation that makes modular arithmetic interesting.

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Modular Arithmetic

Modular Arithmetic. Lecture 8. Modular Arithmetic. Def : a  b (mod n) iff n|( a - b) iff a mod n = b mod n. Modular Addition. Lemma : If a  c (mod n), and b  d (mod n) then a+b  c+d (mod n). Modular Multiplication.

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Modular Arithmetic

Modular Arithmetic. Peter Lam Discrete Math CS. Introduction to Modular Arithmetic. Sometimes Referred to Clock Arithmetic Remainder is Used as Part of Value i.e Clocks 24 Hours in a Day However, Time is Divided to Two Twelve Hour Periods 22 Hours is 12 + 10 or Ten O'clock.

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Library 2.0: Its Application in Library

Library 2.0: Its Application in Library. Presented by Apollo K. Abungu at the Web2.0/Library 2.0 Workshop. 15 th -17 th June 2011, IRDP-DODOMA. Contents. Introduction Web/Library 2.0 Application in Information centres Conclusion Hands on. Introduction Web 2.0.

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Modular Arithmetic

Modular Arithmetic. Lecture 13: Oct 24. (based on slides in MIT 6.042). Last Time. Euclid’s algorithm to compute gcd(a,b). gcd(a,b) = sa + tb. Use this to solve Die Hard’s water jug problem. How to find s and t so that gcd(a,b) = sa + tb??. Method: apply Euclidean algorithm, finding

437 views • 24 slides

Modular Arithmetic

Modular Arithmetic. Dec 28. This Lecture. Basic rule of modular addition and modular multiplication. The Quotient-Remainder Theorem. For b > 0 and any a , there are unique numbers q ::= quotient( a , b ), r ::= remainder( a , b ), such that a = qb + r and 0  r < b.

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Cellular Phones Accessories-Custom cellular

The custom cellular offer various types of Cellular Phones Accessories to know about us just visit at http://www.customcellular.com/html/cellular_accessories.html

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