File Name: modular arithmetic problems and solutions .zip
- Modular Arithmetic
- Modular Arithmetic (May 2012)
- High School Mathematics Extensions/Primes/Modular Arithmetic
- Modular Arithmetic Problems And Solutions Pdf
Many complex cryptographic algorithms are actually based on fairly simple modular arithmetic. In modular arithmetic, the numbers we are dealing with are just integers and the operations used are addition, subtraction, multiplication and division. The only difference between modular arithmetic and the arithmetic you learned in your primary school is that in modular arithmetic all operations are performed regarding a positive integer, i.
Before going into modular arithmetic, let's review some basic concepts. This establishes a natural congruence relation on the integers. For example:. The remainders 3 and 2 are not the same. You need to be careful with negative numbers.
They are usually not congruent to their positive counter parts, as you can see in the above examples. Congruence is an equivalence relation , if a and b are congruent modulo n , then they have no difference in modular arithmetic under modulo n. Because of this, in modular n arithmetic we usually use only n numbers 0, 1, 2, All the other numbers can be found congruent to one of the n numbers.
So how to perform arithmetic operations with moduli? For addition , subtraction and multiplication , it is quite simple: calculate as in ordinary arithmetic and reduce the result to the smallest positive reminder by dividing the modulus. But for division, it is not so simple because division is not defined for every number. That means that it is not always possible to perform division in modular arithmetic. First of all, as in ordinary arithmetic, division by zero is not defined so 0 cannot be the divisor.
The tricky bit is that the multiples of the modulus are congruent to 0. For example, 6, -6, 12, , Secondly, going back to the very basics: what does "division" mean in ordinary arithmetic?
So division is defined through multiplication. But you run into problems extending this to modular arithmetic. From the above table, we can find that 2 and only 2 satisfies this equation. This time division is not uniquely defined, because there are two numbers that can multiply by 2 to give 4. In such cases, division is not allowed.
Then when modular division is defined? When the multiplicative inverse or just inverse of the divisor exists. An integer can have either one or no inverse. The inverse of a can be another integer or a itself. In the above table, we can see that 1 has an inverse, which is itself and 5 also has an inverse which is also itself. But 2, 3 and 4 do not have inverses. Whether an integer has the inverse or not depends on the integer itself and also the modulus.
Compare the follwing table to table You can see that when the modulus is 6, 2 has no inverse. But when the modulus is 5, the inverse of 2 is 3. The rule is that the inverse of an integer a exists iff a and the modulus n are coprime. That is, the only positive integer which divides both a and n is 1. In particular, when n is prime , then every integer except 0 and the multiples of n is coprime to n , so every number except 0 has a corresponding inverse under modulo n.
You can verify this rule with table 1 and 2. Sometimes it is easy to determine whether two integers are coprime. But most of the time it is not easy. For example, are and 63 coprime? You may not be able to answer immediately. Fortunately, we can use the Euclidean algorithm to find out. The Euclidean algorithm describes how to find what is called the greatest common divisor gcd of two positive integers. Of course, if the gcd of two integers is 1, they are coprime.
Let me show you by an example. We start with two positive integers and The first step of the Euclidean algorithm is to divide the bigger integer by the smaller one , so we have:. Then divide the divisor in last step by the remainder :. Continue to divide the previous divisors by the remainders, until the remainder is 0 :. The divisor in the last step is the gcd of the two input integers.
To see why the algorithm works, we follow the division steps backwards. From the last step, we know that 21 divides Because 21 divides both 42 and 21, it must also divide Since 21 divides both 63 and , it is indeed a common divisor of those two integers. Now we need to prove that it is the greatest. The proof is based on a theorem which says:. If this is true, then 21 must be the gcd why? Figuring this out is left to you as an exercise.
Now let's start:. So the Euclidean algorithm indeed outputs the gcd. If the gcd is 1, we can conclude a and b are coprime. Knowing that an integer a and a modulus n are coprime is not enough. How can we find the multiplicative inverse of a? This can be done by running an extended version of Euclidean algorithm which tracks x when computing the gcd. When the remainder becomes 0, continue the calculation of x for one more round.
The final x is the inverse. Here is an example that shows how to find the inverse of 15 when the modulus is Modular arithmetic calculator addition, multiplication and exponentiation only. GCD, multiplicative Inverse calculator on the bottom of the page. Modular Arithmetic. Modular arithmetic calculator addition, multiplication and exponentiation only GCD, multiplicative Inverse calculator on the bottom of the page.
Modular Arithmetic (May 2012)
Residue arithmetic. Modular arithmetic is almost the same as the usual arithmetic of whole numbers. The main difference is that operations involve remainders after division by a specified number the modulus rather than the integers themselves. Modular arithmetic is a key ingredient of many public key cryptosystems. An important property of these structures is that they appear to be randomly permuted by operations such as exponentiation, but the permutation is often easily reversed by another exponentiation. For suitably chosen cases, these operations enable encryption and decryption or signature generation and verification.
High School Mathematics Extensions/Primes/Modular Arithmetic
Exercise 2. What is the time hours after 7 a. What is the time 15 hours before 11 p.
Modular arithmetic is a system in which all numbers up to some positive integer, n say, are used. So if you were to start counting you would go 0, 1, 2, 3, Once 2 n has been reached the number is reset to 0 again, and so on.
Modular arithmetic has been a major concern of mathematicians for at least years, and is still a very active topic of current research.
Modular Arithmetic Problems And Solutions Pdf
Many complex cryptographic algorithms are actually based on fairly simple modular arithmetic. In modular arithmetic, the numbers we are dealing with are just integers and the operations used are addition, subtraction, multiplication and division. The only difference between modular arithmetic and the arithmetic you learned in your primary school is that in modular arithmetic all operations are performed regarding a positive integer, i. Before going into modular arithmetic, let's review some basic concepts. This establishes a natural congruence relation on the integers. For example:. The remainders 3 and 2 are not the same.
Since N - 1 is always a coprime with N, then according to Problem 5, the last row must be a permutation of the first one. It's a very specific permutation. Let m be coprime to N. Let a and b be two different remainders of division by N. By Problem 5, all rows and, analogously, all columns are permutations of the first row.
This set is called the standard residue system mod m, and answers to modular arithmetic problems will usually be simplified to a number in this range. Example.
Emphasis on problem solving, reasoning, and proving. The following theorem says that two numbers being congruent modulo m. Putnam Exam Problems and Related Topics. Previously MA Furthermore when you convert between military time and standard time, you're performing modular arithmetic. Pick an appropriate modulus for each. Modular arithmetic operations and sequences of numbers You can perform elementary number-theoretic operations to find the LCM, GCD, modulus, quotient, and remainder.
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Com-puting and working with remainders is called modular arithmetic. On primality testing.
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