B FOR 204 Lecture Notes - Lecture 7: Plaintext, Secure Communication, Public-Key Cryptography

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2 Jun 2018

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Math Stuff

Co-Prime

• Two numbers x and y are said to be co-primes -

• IF and only IF the only positive number that divides them both is 1.

• All prime numbers by definition are coprime in pairs.

o E.g. 2, 3, 5, 7, 11, 13, 17

• But non-prime numbers can also be coprime

o E.g. 14 and 15

o 14 = 7 * 2 * 1

o 15 = 3 * 5 * 1

o No multiples in common except 1

Modulo

• The modulo operation finds the remainder after division of one number by

another

• Also called modulus

• If dividend is x and divisor is n, remainder is y, then modulo operation is

represented as:

y = x modulo n = y

y = x mod n

• e.g. 15 mod 7 = 1, because 15-7*2 =1

Modulo – 10 minutes exercise

31 (mod 19) =

32 (mod 19) =

33 (mod 19) =

34 (mod 19) =

35 (mod 19) =

36 (mod 19) =

37 (mod 19) =

Modulo Exercise Answers

31 (mod 19) = 3

32 (mod 19) = 9 (mod 19) = 9

33 (mod 19) = 27 (mod 19) = 8

34 (mod 19) = 32 * 32 (mod 19) = 9 * 9 (mod 19) = 5

35 (mod 19) = 32 * 32 * 3 (mod 19) = 9 * 9 * 3 (mod 19) = 15

36 (mod 19) = 32 * 32 * 32 (mod 19) = 9 * 9 * 9 (mod 19) = 7

Discrete Logarithm – Primitive Root

• The primitive roots are defined only for prime numbers.

• A primitive root r of a prime number p is a number whose powers generate all the

numbers from 1 to p-1.

• It is written mathematically as

r mod p, r2 mod p, r3 mod p, ……, rp-1 mod p

are distinct and consist of integers from 1 through p in some permutation.

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Document Summary

Co-prime: two numbers x and y are said to be co-primes , all prime numbers by definition are coprime in pairs. If and only if the only positive number that divides them both is 1: e. g. 2, 3, 5, 7, 11, 13, 17: but non-prime numbers can also be coprime, e. g. 14 and 15: 14 = 7 * 2 * 1, 15 = 3 * 5 * 1, no multiples in common except 1. Modulo another: also called modulus, the modulo operation finds the remainder after division of one number by. If dividend is x and divisor is n, remainder is y, then modulo operation is represented as: y = x modulo n = y y = x mod n: e. g. 15 mod 7 = 1, because 15-7*2 =1. 36 (mod 19) = 32 * 32 * 32 (mod 19) = 9 * 9 * 9 (mod 19) = 7.

Related Questions

1.Create a list that contains 10 successive integers starting with the value in the variable start. For example, if start is 7, the resulting list should be [7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Store the list in a variable named my_list. Assume that the value of start has already been set.

2.Create a list that is made up of the first half of the elements of a list called original_list. Store the new list in a variable called half_list. Assume that the original list has already been initialized and contains an even number of elements.

3. Write a predicate function called same_ends that accepts a list of integers and an integer (n) as arguments. Return True if the first n numbers in the list are the same as the last n numbers in the list, and False otherwise. For example, if the list was [1, 2, 3, 4, 99, 1, 2, 3, 4], the function would return True if n is 4 and False if n is 3. Assume that n is between 1 and the length of the list, inclusive.

4.Write a function called highest_values that accepts as arguments a list of numeric values and a single integer (n). The function should return a list made up of the largest n values from the argument list in order from lowest to highest. For example, if the argument list is [46, 82, 25, 61, 70, 33, 54] and n is 3, the return value should be [61, 70, 82]. Assume that the argument list contains at least n elements.

5.Assume that a list of integers stored in a variable named original contains exactly two elements that are zero. Write code that creates a separate list that contains the elements from original that are between the two zeros. Do not include the zeros in the new list. Store the list in a variable named between_zeros.

6. Write a function called price_range that accepts a list of floating-point numbers representing prices as an argument and returns the the difference between the highest price and the lowest price. Do NOT use the built-in functions max or min in your solution. Assume that all prices are positive values and that none of them exceed 10,000.00.

7. Write a function called count_vowels that accepts a string argument that represents a word and returns the number of vowels that are in the word. The vowels are A, E, I, O, and U (ignore the 'sometimes Y' rule). Count both uppercase and lowercase vowels.

Assume that the variable table holds a two-dimensional list of integers. Write code that computes the sum of all elements in the table, assigning the result to the variable table_sum.

emeraldhorse214

B FOR 204 Lecture Notes - Lecture 7: Plaintext, Secure Communication, Public-Key Cryptography

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