It only takes a minute to sign up. EDIT: I've been confusing this the whole time. Thanks for pointing that. I can't find any information on this and I really want to know.
We discussed in class number theory and after making some proofs on division algorithms we made some modular arithmetic. Is this right? Why can't I find this anywhere? I'm totally lost. There is a Wikipedia entry on modular arithmetic which is a better introduction to that subject, but lacks the specialization to crypto and discussion on bitstrings.
The following discussion is exclusively about Java except for restrictions of the language in some Java Cards, which may not support type long or even intand thus somewhat off-topic here, but well. This makes code clearer, simpler, faster, and for arrays conserves memory. That technique is used because, before Java 7, there was no unsigned type in Java.
This does not hold for the full range of input operands for two independent reasons:. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Ask Question.
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Next Product. Related products. Read more. Get a Quote!This operator gets a remainder. It provides a way to execute code once every several iterations of a loop. It uses the percentage sign character in the lexical syntax. Modulo, notes. Modulo has some unique properties. As with all low-level operations, it has a specific cost. We analyze modulo division in the C language. An example. Modulo division is expressed with the percentage sign.
It is implemented with "rem" in the intermediate language. Rem takes the top 2 values on the evaluation stack. Then: Rem performs the computation that returns the remainder of the division.
It pushes that value onto the evaluation stack. Here: This example demonstrates the math behind modulo. The expressions here are turned into constants during the C compilation step. Important: We see that and divide into parts of 90 with a remainder of Note: If the first argument to the predefined modulo operator is 81 and the second operand is 80, the expression evaluates to a value of 1.
You can apply modulo in a loop to achieve an interval or step effect. If you use a modulo operation on the loop index variable, you can execute code at an interval. Note: This example shows how to write to the screen every ten iterations in the for-loop. Important: The 3 numbers in the condition in the if-statement can have any values, but we cannot divide by 0. Performance, table. The time required for modulo division depends on hardware and other factors.
Some analysis from Microsoft is helpful. This page provides a table listing times required. Estimated costs of instructions: Add: 1 ns Subtract: 1 ns Multiply: 2. Error, divide by zero. If you use modulo by 0, you will get a compile error or a runtime exception.
The denominator must never be zero. Modulo has several common uses in programs. You can use modulo division in loops to only execute code every several iterations. This can improve real code. Note: We do not often need to compute numeric remainders for user consumption. The regular division operator may be more useful here.Method 1 Naive A Naive method is to try all numbers from 1 to m.
Below is implementation of this method. Since we know that a and m are relatively prime, we can put value of gcd as 1. Below is the implementation of above idea. We have discussed three methods to find multiplicative inverse modulo m. Applications: Computation of the modular multiplicative inverse is an essential step in RSA public-key encryption method.
This article is contributed by Ankur. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. Attention reader! Writing code in comment? Please use ide. Python 3 program to find modular. A naive method to find modulor. This code is contributed by Nikita Tiwari. WriteLine modInverse a, m. Iterative Python 3 program to find. Returns modulo inverse of a with. Algorithm Assumption: a and m are.
Update x and y. Make x positive.In computingthe modulo operation finds the remainder or signed remainder after division of one number by another called the modulus of the operation.
Given two positive numbers, a and na modulo n abbreviated as a mod n is the remainder of the Euclidean division of a by nwhere a is the dividend and n is the divisor.
Modular multiplicative inverse
For example, the expression "5 mod 2" would evaluate to 1 because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0 because the division of 9 by 3 has a quotient of 3 and leaves a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3. Doing the division with a calculator will not show the result referred to here by this operation; the quotient will be expressed as a decimal fraction. Although typically performed with a and n both being integers, many computing systems allow other types of numeric operands.
See modular arithmetic for an older and related convention applied in number theory. When either a or n is negative, the naive definition breaks down, and programming languages differ in how these values are defined. In mathematicsthe result of the modulo operation is an equivalence classand any member of the class may be chosen as representative; however, the usual representative is the least positive residuethe smallest non-negative integer that belongs to that class, i.
However, other conventions are possible. Computers and calculators have various ways of storing and representing numbers; thus their definition of the modulo operation depends on the programming language or the underlying hardware. In nearly all computing systems, the quotient q and the remainder r of a divided by n satisfy.
However, this still leaves a sign ambiguity if the remainder is nonzero: two possible choices for the remainder occur, one negative and the other positive, and two possible choices for the quotient occur. Usually, in number theory, the positive remainder is always chosen, but programming languages choose depending on the language and the signs of a or n.
See the table for details. Boute argues that Euclidean division is superior to the other ones in terms of regularity and useful mathematical properties, although floored division, promoted by Knuth, is also a good definition. Despite its widespread use, truncated division is shown to be inferior to the other definitions. As neither floor division nor Euclidean division offer this symmetry, Boute's judgement is at least incomplete.
When the result of a modulo operation has the sign of the dividend, it can lead to surprising mistakes. For example, to test if an integer is odd, one might be inclined to test if the remainder by 2 is equal to One correct alternative is to test that the remainder is not 0 because remainder 0 is the same regardless of the signs :.
Some calculators have a mod function button, and many programming languages have a similar function, expressed as mod anfor example. Modulo operations might be implemented such that a division with a remainder is calculated each time.The modulus operator also informally known as the remainder operator is an operator that returns the remainder after doing an integer division.
Modulus only works with integer operands. The modulus operator can also work with negative operands. Note that the parameters and return value of function pow are of type double. Due to rounding errors in floating point numbers, the results of pow may not be precise even if you pass it integers or whole numbers.
In the vast majority of cases, integer exponentiation will overflow the integral type. Quiz time. Show Solution. See lesson 4. And in this case, since we only use variable isEven once, we might as well eliminate the variable:.
My problem with this function is that it's not intuitive for other users. The function name is "isEven", but returns 0 false if the number is even, and 1 true if the number is odd. A better name for the function would be isOddbut since the quiz asks for a function by the name isEvenmy solution would be as follows:.
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This code does what the name suggests, i. It can still be a bit of a hassle to understand intuitively from an outsider's perspective by just looking at the code, which is I assume why Alex and Nascardriver chose their solution instead:. I wasn't caring that much about how to make code readable and easy to understand. Name required. Email will not be published required. Save my name, email, and website in this browser for the next time I comment. Enter your search terms Submit search form.
By Alex on August 17th, last modified by nascardriver on June 11th, The modulus operator The modulus operator also informally known as the remainder operator is an operator that returns the remainder after doing an integer division. Warning In the vast majority of cases, integer exponentiation will overflow the integral type. Question 1 What does the following expression evaluate to?
Question 2 Write a program that asks the user to input an integer, and tells the user whether the number is even or odd. Write a function called isEven that returns true if an integer passed to it is even, and false otherwise. Use the modulus operator to test whether the integer parameter is even.
Your program should match the following output: Enter an integer: 5 5 is odd Show Solution.Enter two numbers, with the first number a being the dividend while the second smaller number n is the divisor. People use modular arithmetic all the time, especially with anything involving remainders, time and calendar schedules. Modular arithmetic, sometimes called clock arithmetic, is a calculation that involves a number that resets itself to zero each time a whole number greater than 1, which is the mod, is reached.
An example of this is the hour digital clock, which resets itself to 0 at midnight. The standard format for mod is: a mod n Where a is the value that is divided by n. Instead of its decimal form 0. One might presume the mod function generates the same values as positive numbers when one number is negative.
This is actually not the case. For instance, if you have mod 60the remainder is But if you have mod 60the remainder is Why does this happen? But withwe subtract a number with a greater absolute valueso the mod function generates a positive value. The resulting remainder is also smaller compared to when both numbers are positive. According to Britannicathe concept of modular arithmetic has been used by ancient civilizations such as the Indians and Chinese.
Moreover, modular arithmetic was used to solve astronomical and seasonal calculations which were problems associated with natural and man-made cycles. In Western mathematics, German mathematician and physicist Carl Friedrich Gauss did the first systematic study of modular arithmetic.
Gauss is regarded as one of the most influential figures in modern mathematics. In the number theory, scholars analyze the properties of natural numbers, which are whole numbers like -1, -2, 0, 1, 2, and so on. Their objective is to discover unexpected mathematical patterns and interactions between natural numbers.
The residues are added by finding the arithmetic sum of the numbers, and the mod is subtracted from the sum as many times as possible.
This diminishes the sum to a number M, which is between 0 and N — 1.Is the modulus operator (%) useful?
For instance, if we add the sum of 2, 4, 3 and 7, the sum is congruent to 6 mod This means 16 divided by 10 leaves a remainder of 6. This means 13 divided by 12 leaves a remainder of 1. Since we have 24 hours in a day, it makes sense to refer to time in a hour fashion.
This is the principle behind the military time systembeginning at midnight with hours, and ending the hour at 11PM with hours. The military uses this to coordinate with bases and other personnel located in different time zones.
Moreover, all pilots commercial or otherwise use the hour clock to avoid confusion while traveling between time zones. How is this connected with modulo?
This is why the hour standard time uses modulo. Unless specified otherwise, a 4am meeting is absurd, unless you work at night and have online meetings with clients from other time zones. Mod is useful for organizing large information. Books are tracked using modular arithmetic to calculate checksums for international standard book numbers ISBN.
Ina digit ISBN number system which was previously 10 was introduced to help manufacturers identify a large volume of books.