How do you do division in modular arithmetic

How do you do division in modular arithmetic mod#
How do you do division in modular arithmetic password#

Output of the du command is passed to cut command using pipe redirection. This is where cut command is used to extract the first column (i.e.

The output has both the file size and the file name. The du command gives you the size of the file along. I do hope you are also familiar with the cut and du commands. You must be familiar with arguments in bash scripts by now. LetвЂ™s create a bash script named addition.sh that will simply add two file sizes (in bytes) and display the output. Performing addition and subtraction in bash scripts Integer division (without decimal numbers) To refresh your memory, here are the arithmetic operators in bash: Operator In this chapter of the bash beginner series, you will learn to use bash operators to carry out various arithmetic calculations.

How do you do division in modular arithmetic password#

With this I will wrap up this tutorial on modular arithmetic, in next blog we will dig deeper into the block ciphers which I promised to continue in my last blog.While writing your bash scripts, you will often find yourself wanting to figure out the result of an arithmetic calculation to determine a remaining disk space, file sizes, password expiry dates, number of hosts, network bandwidth, etc. (in the example above gcd (5, 9) = 1, so that the inverse of 5 exists modulo 9).

How do you do division in modular arithmetic mod#

NOTE:В The inverse of a number вЂњa mod nвЂќ only exists if and only if: Now, do all the numbers in a set of integers Z * p have inverses? And the answer is NO. The inverse of 7 mod 9 is 4 since 7 x 4 в‰Ў 28 в‰Ў 1 mod 9, hence: The inverse a -1 of a number a is defined such that: if we want to divide b by we will do the following: To perform a modular division, we compute inversion for which we multiply by its inverse rather than actually divide. If gcd (x, y) = 1, the x and y are relatively prime.ДЇor all x, y в€€ Z, there exist a, b в€€ Z such that:ДЄ and b can be found efficiently using the Extended Euclid Algorithm (EEA). But for now understand and remember this definition. GCD plays a very important role when we are trying to compute inverse or when we try to perform modular division in a group (we will address the topic of group and field theory separately.).

12 в‰Ў -6 mod 9 в†’ -6 is a valid remainder since 9 divides (-6-12)Дefinition : For x, y в€€ Z, d = gcd(x, y), where d is the largest number that divides both x and y.

12 в‰Ў 21 mod 9 в†’ 21 is a valid remainder since 9 divides (21-12).

12 в‰Ў 3 mod 9 в†’ 3 is a valid remainder since 9 divides (3-12).

It is somewhat surprising that for every given modulus m and number a, there are (infinitely) many valid remainders. Hence, in modular arithmetic for the case 12В в‰Ў 3 mod 9,В in modulus 9 group of integers, the integers 12 and 3 are congruent, means they can be used in place of each other.

Let a =12 and m = 9, so if 12 is divided by 9 the remainder is 3, hence we write this as : 12В в‰Ў 3 mod 9.

LetвЂ™s see a few examples to make this clear: The congruence relation may be rewritten as So, for a positive integer n, two numbers aВ and bВ are said to be congruent modulo n, if their difference aВ в€’ bВ is anВ integer multipleВ of nВ (that is, if there is an integer kВ such that aВ в€’ bВ = kn). We read this as вЂњa is congruent to b mod nвЂќ. Now, lets see how it works in mathematics. You never leave the integer set of n numbers in a modulo n, rather you wrap around. This is the rule behind modular arithmetic. Interestingly, even though the numbers are incremented every hour we never leave the set of integers:ДЎ, 2, 3, вЂ¦ 11, 12, 1, 2, 3, вЂ¦ 11, 12, 1, 2, 3, вЂ¦ So, if you have your breakfast in the morning at 8:00 a.m and work kept you very busy so that you could have your lunch only 6 hours later, it will be 2 by that time. We have two sets of 12-hour periods each. Now, this 24 hrs is divided into two time sets using modular arithmetic. We all know we have 24 hours on earth for each day. Do you ever give it a second thought that as to how we have same time at different events at the day. At lunch, your mother calls up and asks you to pick her at 6:00 pm from your auntвЂ™s house. Suppose in the morning you get up at 6:00 am, get ready and leave for office. And a familiar example is the use of clock. LetвЂ™s understand the concept of modular arithmetic with a daily life example before talking about its mathematical syntax etc. Modular Arithmetic is a system of arithmetic for integers where numbers вЂњwrap aroundвЂќ upon reaching a certain value вЂ“ the modulus. So, letвЂ™s begin without any further delay. It is a branch of mathematics that lies at the heart of cryptography and moving any further without itвЂ™s knowledge would be like entering a battle without sufficient arms with you. ItвЂ™s knowledge is extremely important for asymmetric cryptography (RSA, Elliptic Curve etc.). But then I realized that I need to introduce one more topic before I move any further and that is Modular Arithmetic. In my last post, I ended with a note to start with DES in my next post.