# How To Use Inverse Modular Arithmetic? (Best solution)

Modular inverses

1. The inverse of a number A is 1/A since A * 1/A = 1 (e.g. the inverse of 5 is 1/5)
2. All real numbers other than 0 have an inverse.
3. Multiplying a number by the inverse of A is equivalent to dividing by A (e.g. 10/5 is the same as 10* 1/5)

## What is modular inverse used for?

Modular multiplicative inverses are used to obtain a solution of a system of linear congruences that is guaranteed by the Chinese Remainder Theorem.

## What is modular inverse of a number?

A modular inverse of an integer ( modulo ) is the integer such that. A modular inverse can be computed in the Wolfram Language using PowerMod[b, -1, m]. Every nonzero integer has an inverse (modulo ) for a prime and not a multiple of.. For example, the modular inverses of 1, 2, 3, and 4 (mod 5) are 1, 3, 2, and 4.

## How do you find modular exponentiation?

Modular exponentiation can be performed with a negative exponent e by finding the modular multiplicative inverse d of b modulo m using the extended Euclidean algorithm. That is: c = be mod m = de mod m, where e < 0 and b ⋅ d ≡ 1 (mod m). Modular exponentiation is efficient to compute, even for very large integers.

## What is multiplicative inverse in modular arithmetic?

The modular inverse of a number refers to the modular multiplicative inverse. For any integer a such that (a, p) = 1 there exists another integer b such that ab≡ 1 (mod p). The integer b is called the multiplicative inverse of a which is denoted as b = a1.

## Can you divide in modular arithmetic?

Can we always do modular division? The answer is “NO”. In modular arithmetic, not only 4/0 is not allowed, but 4/12 under modulo 6 is also not allowed. The reason is, 12 is congruent to 0 when modulus is 6.

## How do you do inverse mod on a calculator?

For every number x from this set, calculate a * x mod m, i.e., the remainder from the division of a * x by m. The modular multiplicative inverse of a modulo m is the value of x for which this remainder is equal to 1.

## What is the value of mod 7?

15 MOD 10 is equal to 5 (because 15 – (15 DIV 10) equals 5 — the remainder of the division is 5) 20 MOD 7 = 6. 21 MOD 7 = 0.

## What is the inverse of 7 mod 11?

7x≡1≡12≡23≡34≡45≡56(mod11). Then from 7x≡56(mod11), we can cancel 7, obtaining x≡8(mod11). Hence, −3 is the inverse of 7(mod11).

## What is the inverse of 19 mod 141?

Therefore, the modular inverse of 19 mod 141 is 52.

## How do you reduce modular arithmetic?

In modular arithmetic, when we say “reduced modulo,” we mean whatever result we obtain, we divide it by n, and report only the smallest possible nonnegative residue. The next theorem is fundamental to modular arithmetic. Let n≥2 be a fixed integer. If a≡b (mod n) and c≡d (mod n), then a+c≡b+d(modn),ac≡bd(modn).

## Modular multiplicative inverse

Given two numbers ‘a’ and’m,’ compute the modular multiplicative inverse of ‘a’ under the modulo of’m’ by multiplying them together. The modular multiplicative inverse is a positive integer ‘x’ that has the property that. an x = 1 an x = 1 an x = 1 an x = 1 an x = 1 an x = 1 (mod m) The value of x should be in the range of integer modulo m, i.e., inside the integer modulo m range. (It should be noted that x cannot be zero since a*0 mod m would never be 1). The multiplicative inverse of “a modulo m” occurs if and only if a and m are relatively prime (i.e., if gcd(a, m) = 1).

Examples: a = 3, m = 11 are the input values.

(under 11).

Input values: a = 10, m = 17, 12 as a final result Because (10*12) mod 17 = 1, 12 is the modulo inverse of the number 10.

Method No.

For each integer x, determine whether (a*x) percent m equals 1.

## C++

Includeiostreamusingnamespacestd;intmod Inverse(inta,intm)intmain()

## Java

Import java.io.*;classGFGpublicstaticvoidmain(String args);classGFGpublicstaticvoidmain(String args)

## Python3

DefmodInverse(a, m):forxinrange(1, m): DefmodInverse(a, m):forxinrange(1, m): In the case of ( (((a percent m)*(x percent m) percent m) percent m=1): returnxreturn-1a=3m=11print(modInverse(a, m)) returnxreturn-1a=3m=11 print(modInverse(a, m))

## C

UsingSystem;classGFGpublicstaticvoidMain()}

## PHP

The following code: phpfunctionmodInverse(\$a,\$m) \$a = 3 and \$m = 11;echomodInverse(\$a,\$m); phpfunctionmodInverse(\$a,\$m);

## Javascript

The following script function modInverse(a, m): let a = 3; let m = 11; document.write(modInverse(a, m));/script The following script function Time Complicatency: O (m). Method No. 2 (Works when m and a are coprime) The goal is to employ Extended Euclidean algorithms, which take two numbers ‘a’ and ‘b’ and discover their gcd, as well as find ‘x’ and ‘y’ such thatax + by = gcd, to solve the problem (a, b) In order to find the multiplicative inverse of ‘a’ under’m,’ we substitute b = m in the preceding formula.

Taken together, ax + my 1 (mod m) gives usax + my 1 (mod m).

The following is an example of how the aforesaid algorithm is implemented.

## C++

The following script function modInverse(a, m): let a = 3; let m = 11; document.write(modInverse(a, m)); Complications in respect to time: O (m). The second approach is described below (Works when m and a are coprime) In this case, the idea is to useExtended Euclidean algorithms that take two integers ‘a’ and ‘b’ and find their gcd, as well as find the values of x and y such that ax + by = gcd (a, b) Put b = m in the preceding formula to determine the multiplicative inverse of the letter an under the letter m.

In the case of two integers y, we getax + my m (mod m).

(mod m) We may then deduce from the Extended Euclid Algorithm that ‘x’ is the multiplicative inverse of the variable “a.” The method described above has been implemented in the following manner:

## C

Includestdio. hintgcdExtended(inta,intb,int* x,int* y);voidmodInverse(inta,intm)intx1, y1;intgcdExtended(inta,intb,int* x,int* y)intx1, y1;intgcd = gcdExtended(b percent a, a,x1,y1);*x = ()

## PHP

≅phpfunctionmodInverse(\$a,\$m)} funcgcdExtended(\$a,\$b,\$x,\$y)\$x1;\$y1;\$gcd= gcdExtended(\$b percent \$a,\$a,\$x1,\$y1);\$x=\$y1- (int)(\$b/\$a) *\$x1;\$y=\$x1;\$gcd= gcdExtended(\$b percent \$a,\$a,\$x1,\$y1) \$a= 3;\$m= 11;modInverse(\$a,\$m); modInverse(\$a,\$m); Iterative implementation of the modular multiplicative inverse:

## C++

Includebits/stdc++.husingnamespacestd;intmod returnx;inverse(inta,intm)if(x0)x += m0;intmain ()

## C

Includestdio.hintmod returnx;inverse(inta,intm)if(x0)x += m0;intmain ()

## Java

ClassGFGif(x0)x += m0;returnx;publicstaticvoidmain(String args);publicstaticvoidmain(String args)

## Python3

The following are the results of DefmodInverse(a, m):m0=my=0x+1if(m=1):return0while(m1=1):return0 while(a1):q=a/mm=a percent yy=tt=x-q*ytif(x0):x=x+m0returnxa=3m=11 DefmodInverse(a, m):m0 = my= print(“The modular multiplicative inverse is”,modInverse(a, m)) print(“The modular multiplicative inverse is”,modInverse(a, m))

## C

UsingSystem;classGFGif(x0)x += m0;returnx;publicstaticvoidMain();classGFGif(x0)x += m0;publicstaticvoidMain();

## PHP

≅phpfunctionmodInverse(\$a,\$m) if(\$x0)\$x+=\$m0;return\$x;} \$a= 3;\$m= 11;echo”Modular multiplicative inverse isn”,modInverse(\$a,\$m);echo”Modular multiplicative inverse isn”,modInverse(\$a,\$m);echo”Modular multiplicative inverse isn”,modInverse(\$a,\$m);echo”Modular multiplicative inverse isn”,modInverse(\$a,\$m);echo”Modular multiplicative inverse is

## Javascript

Modification of the script function modInverse (a, m) returnx;let a = 3;let m = 11;document if (x0)x += m0, returnx;let m = 11;document. written as write(‘The modular multiplicative inverse is \$’);/scriptOutput The inverse of a modular multiplicative module is four. Time Complicatency: o o o o o o o o o o o o o o o o o (Log m) Method No. 3 (Works when m is prime) If we know that m is a prime number, we may also apply Fermat’s little theorem to get the inverse of the prime number. an m-1 1 an m-1 1 an m-1 1 an m-1 1 (mod m) The result is a -1 am-2 when both sides are multiplied by one (mod m) The following is an example of how the aforementioned concept was put into action.

## C++

Includeiostreamusingnamespacestd;intgcd(inta,intb);intpower(intx, unsignedinty, unsignedintm);voidmod;intgcd(inta,intb);intpower(intx, unsignedinty, unsignedintm);voidmod Inverse(inta,intm)intpower(intx, unsignedinty, unsignedintm)intgcd(inta,intb)intmain(intx, unsignedinty, unsignedintm) ()

## Java

Import java.io.*;classGFGstaticintpower(intx, inty, intm)staticintgcd(inta, intb)publicstaticvoidmain(String args)staticintgcd(inta, intb)staticintgcd(inta, intb)staticintgcd(inta, intb)staticintgcd(inta,

## Python3

DefmodInverse(a, m):g=gcd DefmodInverse(a, m) (a, m) Print the message “Inverse does not exist” if (g!=1) is true. else: The modular multiplicative inverse is “,power(a, m-2, m)). print(“Modular multiplicative inverse is “,power(a, m-2, m)). defpower(x, y, m):if(y=0):return1p=power(x, y/2, m) defpower(x, y, m) defpower(x, y, m) (x*p) percent mp=(p*p) percent mif(y percent 2=0):returnpelse:return((x*p) percent mp):return Define GCD(a, b) as follows: if (a=0): returnbreturngcd(b percent a, a)a=3m=11modInverse; defgcd(a, b) as follows: (a, m)

## C

UsingSystem;classGFG}staticintpower(intx,inty,intm)staticintgcd(inta,intb)publicstaticvoidMain()}

## PHP

≅phpfunctionmodInverse(\$a,\$m)} functionpower(\$x,\$y,\$m)functiongcd(\$a,\$b)\$a= 3;\$m= 11;modInverse(\$a,\$m); functionpower(\$x,\$y,\$m)functiongcd(\$a,\$b)\$a= 3;\$m= 11; modInverse(\$a,\$m);

## Javascript

ScriptfunctionmodInverse(a, m) ScriptfunctionmodInverse(a, m) power to perform a function (x, y, m) Let a = 3, m = 11, and functiongcd(a, b)let a = 3;modInverse(a, m);/scriptOutput The inverse of a modular multiplicative module is four. Time Occupational Complexity:O (Log m) We’ve gone through three different approaches to finding the multiplicative inverse modulo m. 1) The naive method, or, O (m) 2) The GCD algorithm of Euler was extended, and the result was O (Log m) 3) Fermat’s Little Theorem, often known as O (Log m) Computing the modular multiplicative inverse is a critical step in the RSA public-key encryption technique, which is used to protect sensitive information.

Please leave a comment if you see something that is inaccurate or if you wish to offer further information regarding the subject matter covered here.

## Modular multiplicative inverse – Wikipedia

In mathematics, and notably in the field of arithmetic, amodular multiplicative inverseof an integer is defined as an integerx such that the productaxis is congruent to 1 with regard to the modulus of the integer. According to the usual notation ofmodular arithmetic, this congruence is expressed as, which is a shorthand method of presenting the fact thatmdivides (evenly) the quantityax by the integermis 1, or, to put it another way, the residual after dividingax by the integermis 1. Assuming that there is an inverse modulom for a congruence, there exist an unlimited number of solutions to this congruence, which constitute a congruence class with regard to the modulus in which it is defined.

Assuming that the congruence class containingw is denoted by the notation, this may be written as follows:where the symbol signifies the multiplication of equivalence classes modulom, the congruence class containingw is denoted by the notation Written in this manner, the analogy with the conventional idea of amultiplicative inversein the set ofrationalorreal numbersis easily depicted, with the numbers being replaced by congruence classes and the binary operation being altered in the proper way.

In the same way that the comparable operation on the real numbers has a basic application, this operation has a fundamental use in solving linear congruences of the type It is also possible to use modular multiplicative inverses for practical purposes in the field of cryptography, for example, in public-key cryptography and the RSA method.

One advantage of computer implementation of these applications is that there is a very fast method (the extended Euclidean algorithm) that can be utilized for the computation of modular multiplicative inverses, which is advantageous for computer implementation of these applications.

## Modular arithmetic

Two numbers, a and b, that are congruent modulomifmdivides their difference for a given positive integerm are referred to as being congruent modulomifmdivides their difference. This binary relationship is symbolized by the symbol In this case, the equivalency relation is on the set of integers, Z, and the equivalence classes are referred to as congruence classes modulomorresidue classes modulom, respectively. Suppose we want to indicate the congruence class that contains the integera, then A Linear congruence is a type of modular congruence that has the shape of a line Linear congruences, in contrast to linear equations over the reals, may have zero, one, or numerous solutions, depending on the context.

1. It follows that if d is the greatest common divisor of the numbers a and m, then the linear congruenceaxb(modm)has solutions only when and only if ddividesb.
2. When an integer is multiplicatively inversed in a modular way with regard to its modulus, it is a solution of the linear congruence.
3. coprime).
4. Although this is commonly used to indicate that ax 1 (modm) has a solution, it might be considered an abuse of notation since it could be misunderstood as the reciprocal of ax.1 (modm) (which, contrary to the modular multiplicative inverse, is not an integer except whenais 1 or -1).
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### Integers modulom

Modulom is a congruence relation that divides the set of integers into mcongruence classes, which are a subset of each other. The following is an example of how to define addition and multiplication operations on thesemobjects: To either add or multiply two congruence classes, first select a representative (in any way) from each class, then perform the usual integer operation on the two representatives, and finally take the congruence class that the result of the integer operation lies in as the result of the operation on the congruence classes.

To add or multiply two congruence classes, first select a representative (in any way) from each class, then perform the usual integer operation on the two representatives, and finally take the congruence class that the result of These definitions areandrepresent the operations on congruence classes in symbols, withandrepresenting the operations on congruence classes in symbols.

Classifications of themcongruence that have these two defined operations together form what is known as the ring of integers modulom.

Traditional names for the congruence classes of the integers modulom were residue classes modulo m, which referred to the fact that all components in a congruence class have the same remainder (i.e., “residue”) whenm is divided bym.

This is demonstrated by the division methods, which demonstrate that the set of integers, when combined, create a full system of residues modulom, also known as theleast residue system modulom.

When dealing with arithmetic difficulties, it is sometimes more convenient to work with a complete system of residues and to speak in the language of congruences, while at other times it is more convenient to approach the problem from the perspective of the congruence classes of the ring.

### Multiplicative group of integers modulom

Not every element of a full residue system modulomhas a modular multiplicative inverse; for example, the number zero never has a modular multiplicative inverse. An areduced residue system is formed by deleting all of the components of a complete residue system that are not relatively prime tom and leaving only the ones that are relatively prime tom. All of the elements of a reduced residue system have modular multiplicative inverses. The number of elements in a reduced residue system is, whereis theEuler totient function, i.e., the number of positive integers smaller thanm that are relatively prime tom, and is the number of elements in a reduced residue system.

Similarly to how the product of two units is a unit, the units of a circle form a group, which is denoted by RifRis the name of the ring and is commonly denoted byRifRis the name of the circle.

It has particular significance in terms of order (size),.

In this situation, the multiplicative group of integers modulopform an acyclic group of orderp 1 and the multiplicative group of integers modulopform an acyclic group of orderp 1.

## Example

Consider the following example, which makes use of the modulus 10 to demonstrate the aforementioned definitions. Two numbers are congruent mod 10 if and only if the difference between them is divisible by 10. For example, since 10 divides 32 12 = 20, and since 10 divides 111 1 = 110, two integers are congruent mod 10. With regard to this modulus, the following are some of the ten congruence classes: It is impossible to find a solution to the linear congruence4 x 5 (mod 10)because all of the numbers that are congruent to 5 (i.e., those in) are odd, whereas4 xis always even in the case of 4 x.

• It is the case that gcd(4, 10) = 2 and that 2 doesn’t divide 5, but does divide 6.
• As a matter of fact, the number 7 satisfies this congruence (i.e., 21 1 = 20).
• In particular, every integer inwill fulfill the congruence because these integers have the form7 + 10 rfor some integerrandis definitely divisible by 10.
• This congruence contains only one congruence class of solutions, which is the solution to the congruence.
• In order to determine the product of congruence classesand, take an element from, say 25, an element from, say 2, and observe that their product (25)(2) = 50 is in the congruence class.
• The term “addition” is defined in a similar manner.
• It is possible to have a full residue system modulo 10 if the set contains each integer in a separate congruence class modulo 10.
• It is possible to have a reduced residue system modulo 10.
• This indicates that these four congruence classes are members of a group, in this instance the cyclic group of order four, which has either the number 3 or the number 7 as a generator (multiplicative).

The ring’s group of units is formed by the congruence classes that are represented. Congruence classes with modular multiplicative inverses are precisely those that fall into this category.

## Computation

With the use of the extended Euclidean technique, it is possible to find the modular multiplicative inverse of amodulom. This algorithm finds the greatest common divisor (gcd) of two numbers, saya and m, using the Euclidean principle. Because of the multiplicative inverse modulom, this gcd must be one in this case. It is possible to solve the final of numerous equations generated by the procedure for this gcd by using this method. Then, using a technique known as “back substitution,” it is possible to construct an expression that connects the original arguments and this gcd.

To put it another way, this is that has been determined, which is the modular multiplicative inverse ofa The extended Euclidean algorithm is a more efficient version of the method that, by the use of auxiliary equations, decreases the number of runs through the process from two to only one (back substitution may be thought of as traveling through the algorithm in reverse).

In large O notation, this method runs in timeO(log(m) 2), assuming|a|m, and is thought to be significantly faster and more efficient than it.

### Using Euler’s theorem

Euler’s theorem, which is an alternative to the extended Euclidean algorithm, may be used to compute modular inverses as an alternate method. When the conditional probability of aiscoprimetom is fulfilled, that is, if gcd(a,m) = 1, then whereisEuler’s totient function. Because abelongs to the multiplicative group if and only ifaiscoprimetom, this follows as a logical consequence. As a result, a modular multiplicative inverse may be obtained in a straightforward manner: The particular situation wheremis a prime and a modular inverse is provided by is known as It is often slower than the extended Euclidean algorithm, although it is occasionally employed when a modular exponentiation implementation for a certain problem is already available.

This is a particularly remarkable advantage of this approach.

### Multiple inverses

When using the Euclidean method and three multiplications for each new input, it is feasible to compute the inverses of multiple numbersa I modulo a commonm with a single execution of the algorithm and modulo a commonm. It is the essential concept to build the product of all thea I invert that, and then multiply bya jfor allji to leave just the desireda1i remaining. The algorithm is (all arithmetic is performed modulom): (all arithmetic is performed modulom)

1. Calculate the prefix products for the word alli
2. Computeb1nusing any algorithm that is currently available
3. Calculate forifromndown to 2, forifromndown to 2

In order to take advantage of parallel computing, it is possible to perform the multiplications in a tree structure rather than linearly.

## Applications

A modular multiplicative inverse may be found in many algorithms that rely on the idea of modular arithmetic, and this can be useful in many situations. Examples include the use of modular arithmetic in cryptography, where it enables some operations to be carried out more rapidly and with less storage needs, but other operations become more difficult to carry out. Both of these characteristics may be utilized to one’s benefit. Encryption and decryption of a message are performed using a pair of integers that are multiplicative inverses with respect to a modulus that has been carefully chosen in the RSA method.

Because it is thought to be computationally impossible to distinguish the concealed number from the public number, the system is able to function to protect the user’s personal information.

The following is one possible solution:

1. A modular multiplicative inverse may be found in many algorithms that rely on the idea of modular arithmetic, and finding it has numerous applications in computer science and mathematics. Examples include the use of modular arithmetic in cryptography, where it enables some operations to be completed more rapidly and with less storage needs, while other processes become more complex as a result. Taking use of both of these characteristics is a good idea. Encryption and decryption of a message are accomplished using a pair of integers that are multiplicative inverses with respect to a modulus that has been carefully chosen in the RSA method. Two numbers are utilized in this technique: one is made public and may be used in a quick encryption procedure
2. The other, which is used in a decryption procedure, is kept confidential. Because it is thought to be computationally impossible to distinguish the concealed number from the public number, the system is able to function to protect the user’s identity. A distinct type of precise division problem may be illustrated by the odd word-size number problem in computer science, where you have a list of odd word-size numbers that are individually divisible byk and you desire to divide them all byk. As an example, the following is one possible resolution:

Division is a slower operation than multiplication on many machines, particularly those lacking hardware support for division. As a result, this technique can result in a significant speedup on some devices. Although the first step is time-consuming, it only has to be completed once. The Chinese Remainder Theorem guarantees that a solution to a system of linear congruences will be found using modular multiplicative inverses. This is achieved by employing modular multiplicative inverses. The systemX 4(mod 5)X 4(mod 7)X 6(mod 11)has similar solutions because the numbers 5, 7, and 11 are pairwise coprime.

For example, X= 3 (7 11) 4 + 6 (5 11) 4 + 6 (5 7) 6 = 3504 and in its unique reduced formX= 3504 39 (mod 385)because 385 is the LCMof 5, 7, and 11 The modular multiplicative inverse also plays an important role in the formulation of the Klosterman sum, as can be seen below.