It would take quite a long time for a . Longer messages reveal the most the letter e equivalent, however, this is not necessarily so for our message. The reason is (M-1) * (M-1) = (-1) * (-1) = 1 MOD M. For example: when using an alphabet length of M = 27 and an encoding key a=26 then its decoding key is a-1 =26. So on for each letter, the final encrypted message is ZIEZQ. Why did US v. Assange skip the court of appeal? if the letter e (the most frequent letter in the English language) occurs 20 times in the plain text its replacement letter will appear 20 times in the cipher text. They are trade-offs in terms of their efficiency: the gain of not having to determine the most frequent letter in the cipher text for the brute force approach is at the cost of producing all possible cipher codes. Example3: Now, it is your turn. In formula:
u(M) = (M-1) b(M) using the above formula for the number of bad keys yields
= M-1 - (M/p -1) distributing the minus sign to the terms in the parenthesis yields
= M-1 - M/p + 1 canceling out the 1s yields
= M - M/p
This turns out to be a handy formula for the number of good keys. Our ultimate goal is not to develop a formula for the number of bad keys but rather for the number of good keys. The affine cipher is itself a special case of the Hill cipher, which uses an invertible matrix, rather than a straight-line equation, to generate the substitution . Activity Code: 65665. Example3: For M=16=24 we have u(16) = 24 - 23 = 8 which are the 8 good keys a=1,3,5,7,9,11,13,15. The letter A remains unchanged ans id always encoded A. Also, there is no general match on how to handle digits or special characters. The alphabet function sL returns the smallest index at which it occurs to a letter that is present in L. The index of the first character can be configured. For example if we use "abcdefghijklmnopqrstuvwxyz" and a multiplier of 3, gives "adgjmpsvybehknqtwzcfilorux". A summary of our explorations for the number of good keys shows:
1) u(p) = p - 1, if M is prime M=p. The plain letter c is stored as 103, however, I want the c to equal 2 in compliance with our translation a=0, b=1, c=2, etc. 3
The following table shows the calculation for the case of the separated partial alphabets L1, L2 as well as for a merged alphabet L = "0-9A-Fa-f". How to encrypt using Multiplicative cipher? Example: D = 3, so $ 3 \times 17 \mod 26 \equiv 25 $ and the letter at rank 25 is Z. The basic formula to be used in such a scenario to generate a multiplicative cipher is as follows (Alphabet Number * key)mod (total number of alphabets) The number fetched through output is mapped in the table mentioned above and the corresponding letter is taken as the encrypted letter. Lets summarize our discoveries. Divide the letters of the message into groups of two or three. Lets check this for an alphabet length of M=29. For example, Caesar cipher using a left rotation of three places, equivalent to a right shift of 23 as given below. Extracting arguments from a list of function calls. This encoding and decoding is working based on alphabet shifting & transforming the letters into numbers . Since there are 9 threes (or 9 multiples of 3) in 27 and therefore 8 threes when counting only up to 26 yielding the 8 listed bad keys. that 3 and 9 are inverse to each other because of the commutative property of the MOD-multiplication (exhibited by the diagonal as a line of reflection). It would take quite a long time for a computer to brute-force through a majority of nine million keys. The multiplicative cipher has little interest, but it is often used for learning computer science and ciphers. So, we are left with determining the decoding key a-1 knowing the original encoding key a. We will multiply MOD 26 as we are using the 26 letters of the English alphabet. A multiplicative cipher is a type of cipher that comes under a monoalphabetic cipher, in which each letter that is present in the plaintext is replaced by a corresponding letter of the ciphertext, according to a fixed multiplication key. PLAIN LETTERNATANTSecret key a=2130190131900120012Cipher letteraamaam
You can see the dilemma of this message. Moreover, you can see that the plain letter V encrypts to the cipher text letter b (=1) when using a=5 as the encoding key. You can change your choice at any time on our, Modular Multiplicative Inverse Calculator. Cryptoanalysis - Cracking the Multiplication
Cipher
Just like the Cipher Caesar Cipher, the Multiplication is not secure at all. Convert each letter in the plain text alphabet to a corresponding integer in the range of 0 to m -1; 2. unchanged so that you can detect the format of the original message easier. The 26-letter Latin alphabet allows only 11 keys: 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25 (these are coprime numbers with 26). Thus, we now go ahead and practice a bit more computer programming. If a single character is encrypted by E(C) = (c * k) % 36 then possible keys k are numbers that are coprime to 36, ie. div#home a:link {
I will couple the Multiplication Cipher with the Caesar Cipher (which produces 26 unique encryptions) to obtain a super encryption that will allow 12*26=312 possible unique encryptions. Firstly I have no idea how they derived this formula, but I think I have a general idea. or ? That is why the English alphabet in the calculator above is expanded with space, comma, and dot up to 29 symbols; 29 is a prime integer. The encryption of upper case plain letter works similarly except that I have to subtract A=65 (instead of a=101 as above) to obtain our desired plain letter number. Why does Acts not mention the deaths of Peter and Paul? Decrypt, In a Multiplicative cipher, each character of the alphabet is assigned a value (starting at a zero index [A=0, B=1, etc]) and a coprime key to the length of the alphabet is chosen. Thus, the encryption process is a Caesar cipher merged with a multiplication cipher. 12
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Moreover, we build the mathematical foundation to understand secure encryption systems such as the RSA encryption. Apr 6, 2013 at 10:46 . We know already that:
((60) = ((22*3*5)
= (22-21)*(3-1)*(5-1)((M) = ((p12* p2* p3)
= (p12- p11)*( p2-1)*( p3-1). Once we have the solution, our x is the modular multiplicative inverse of a modulo m. Rewrite the above equation like that Step 1: For decryption first we need to find the multiplication inverse of the key. Multiplicative Cipher on dCode.fr [online website], retrieved on 2023-05-02, https://www.dcode.fr/multiplicative-cipher, multiplicative,multiplication,modulo,cipher, https://www.dcode.fr/multiplicative-cipher, What is Multiplicative Cipher? Counter examples are: 45 and 18 are not relative prime since gcd(45,18)=9 and not 1. In the detailed representation of the alphabets (click on the "" -button), the alphabets can be edited in the short-write mode. Find mod of any numb. Each odd plain letter translates into 13 (=n):
a=13 odd letters
13*1 = 13 MOD 26,
13*3 = 13*2 + 13*1 = 0 + 13 = 13 MOD 26,
13*5 = 13*4 + 13*1 = 0 + 13 = 13 MOD 26,
13*7 = 13*6 + 13*1 = 0 + 13 = 13 MOD 26, etc. Multiplicative cipher encryption|Multiplicative cipher|Multiplicative cipher example|What is multiplicative cipher PLAYFAIR CIPHER WITH EXAMPLE||SUBSTITUTION TECHNIQUE||MATHEMATICS OF. The following steps take place: In the example, an overflow has occurred in the third letter, so that modulo |L| = 4 is calculated. This is important because if the key is known by an unauthorized party, they will be able to decrypt the message. The procedure to use the multiplicative inverse calculator is as follows: Step 1: Enter the values in the numerator and denominator input field Step 2: Now click the button "Solve" to get the output Step 3: The multiplicative inverse value will be displayed in the "Answer" field What is Multiplicative Inverse? How a top-ranked engineering school reimagined CS curriculum (Ep. using properties 1) and 2) yields
= (3-1)*(23-22)
= 2*4
= 8. More precisely: Out of the 25 (= p * q - 1) integers that are smaller than 26, we had 12 (=13-1) multiples of 2 {2,4,6,8,10,12,14,16,18,20,22,24} and the 1 (=2-1) multiple of 13 {13} as bad keys, so that 25-12-1=12 good keys are remaining:
a = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25
Notice that u(26) = 12 = 25-12-1 = (p*q - 1) (p-1) - (q-1)
Example2: For M=10=5*2, we obtain u(10)=4 good keys which are obtained by crossing out the 4 (=5-1) multiples of 2 and the 1 (=2-1) multiples of 5 as bad keys:
a = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Notice that again u = 4 = 9 4 1 = (p*q - 1) (p-1) (q-1)
Example3: For M=15=5*3, we obtain u(15)=8 good keys which are obtained by crossing out the 4 (=5-1) multiples of 2 and the 2 (=3-1) multiples of 5:
a = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
Notice that again u = 8 = 14 4 2 = (p*q - 1) (p-1) (q-1)
The number of good keys can always be computed by u(p*q) = (p*q - 1) - (p-1) -(q-1). These ads use cookies, but not for personalization. Alternatively, the non-alphabet letters in the key and the plain text can also be filtered out to increase the security. Therefore, no matter how he decides to crack the cipher text, it wont take long. One of the main advantages of the multiplicative cipher is its simplicity i.e. (I.e. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Affine cipher - encoder / decoder. Can I use an 11 watt LED bulb in a lamp rated for 8.6 watts maximum. In affine cipher each letter in an alphabet is mapped to its numeric equivalent, encrypted using a simple mathematical function, and converted back to a letter. In this lab, you'll learn about the multiplication cipher, a monoalphabetic cipher. Convert each group of numbers into column matrices. Exporting results as a .csv or .txt file is free by clicking on the export icon However, it is not a secure method of encryption and can be easily broken too. ((8)= ((23)=23 -22 =4 as 1,3,5,7 are relative prime to 8. However, there are some additional integers that are not prime (i.e. In order to create a n x n size matrix, keyphrase length should be square of an integer, i.e., 4, 9, 16. To do so, we have to look at the encryption equation C=a*P MOD 26 and solve it for the desired plain text letter P.
In order to solve an equation like 23=5*P for P using the rational numbers, we would divide by 5 or multiply by 1/5 to obtain the real solution P=23/5. div#home {
On the right we ended up with the explicit formula for ((M) when M consists of one prime power and two primes. Since we are performing MOD 26 arithmetic, we use the MOD-operator % that guarantees us the product (a*(pl -'a'))%26; to be between 0 and 25. Our implementation of Vigenre, Beaufort, etc. We can combine these two criteria into one easy criterion. Below is the C++ program that performs the task for us, it just finds all the factors of an entered alphabet length M by testing all the integers less than M for possible factors. If the alphabet of capital letters A-Z is used, this assignment results: Now a key between 1 and 26 is chosen. Each row that contains each integer from 0 to 25 exactly once and therefore yields a unique cipher letter will serve. 3) If the alphabet length M is a product of two prime numbers p and q
The last case we have to study is when M is a product of two primes. 14
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An alphabet[1] is an ordered set of all characters which can occur in a plaintext, a secret text, or the key. Can you? You noticed, that the multiplicative property of Eulers (-function, expressed in property 4), is used to decompose any integer M into its prime factors or prime power factors to then apply the first two properties to each prime or prime power. 0
The x values are the ones that we can choose independently, here the length of the alphabet M. Each y-value is dependent on the choice of x, i.e. E (x) = (ax + b) mod m D (x) = a -1 (x - b) mod m For more math formulas, check out our Formula Dossier What 4 concepts are covered in the Affine Cipher Calculator? Step 2: The basic formula that can be used to implement Multiplicative Cipher is: Decryption= (C * Multiplication inverse of the key) Mod 26 Here, c = ciphertext Mod = Modulo Step 3: Let's see how decryption can be done using the above formula: Ciphertext = QCCSWJUPQCCSW and multiplication inverse key = 15 If you choose to do so, dont forget to also redefine the corresponding decoding key in int a=5, ainverse=21; . Text is divided into blocks of size n, and each block forms a vector of size n. Each vector is multiplied by the key matrix of n x n. The result, vector of size n, is a block of encrypted text. For the same reason, an alphabet length of M=31 produces u=30 unique encryptions. You may see ads that are less relevant to you. Mathematically: a-1 * a = a * a-1 = 1. The key should be changed frequently to prevent cryptographic attacks. 5
A reciprocal is one of a pair of numbers that when multiplied with another number equals the number 1. It is possible to distinguish between 2 types of actions in the plain text: uppercase letters [A-Z] and digits [0-9]. If a single character is encrypted by E(C) = (c * k) % 36 then possible keys k are numbers that are coprime to 36, ie.gcd(k,36)=1.Furthermore it makes not much sense to consider numbers not between 1 and 36, because of the modulo. See the image attached below for a better understanding. The three factors in the parentheses already have the same desired format, however, the single 2 destroys it. Therefore, we just need to divide 27 by the only prime divisor 3 and subtract 1 at the end to find the number of bad keys: 8 = 27/3 1. For instance, to find the inverse of the good key a=5 we have to look at the fifth row which shows that a-1 equals 21 since the only 1 in this row is in the V- or 21-column (5 * 21 = 1 MOD 26). Which number would that be? Try to understand as much as possible first, then continue reading. Therefore, since there are no other prime divisors and thus no multiples, all integers less than M serve as good keys. where the operation of multiplication substitutes the operation of division by the modular multiplicative inverse. ((5)=_____ as 1,2,3,4 are relative prime to 5. The formula for encrypting a letter x using the affine cipher is: y = ( a x + b) mod 26 And apparently the decryption formula is x = a 1 ( y b) mod 26 Where a 1 is the multiplicative inverse of a mod 26. background-color: #620E01;
The use of several alphabets does not require the algorithms to distinguish between upper and lower case letters. There are other numbers co-prime with 26 (which are greater than 26) but they give alphabets identical to those above. However, it turns out to be indispensable when M is not the product of two primes, but say a product of a prime and a prime power. That is, . the number of unique encryptions u are dependent on the chosen alphabet length M. Since u can be expressed as a formula that involves M, namely u=M-1, we say that u is a function of M and write u(M)=M-1. The next two lines then show us that the variable false is defined as 0 and true as 1. Decoding aam can either yield NAT or ANT as the plain text. 0x95 = 1001 0101 = x 7 + x 4 + x 2 + 1 0x8A = 1000 1010 = x 7 + x 3 + x And the product of the two polynomial reduced modulo the irreductible polynomial is 1, as expected. The inverse function returns the n-th character for a number n in L. To n, the length of the list L is added or subtracted as often as necessary until the index lies in the list. And, for this to happen, we need to have a modular inverse of the key matrix in - ring of integers modulo m. If source vector B is multiplied by matrix A to get vector C, then to restore vector B from vector C (decrypt text), one needs to multiply it by the modular inverse of the matrix. , 8
Now when a=25, we have: 25*25 = 625. This formula can be simplified into the product of two factors. We then perform matrix multiplication modulo the length of the . When doing so we will discover very important mathematical encryption tools such as Eulers (-function, Eulers and Lagranges Theorem and study further examples of groups, rings and fields. If 2 would be used as key, then C=2 -> 2*2 = 4 -> so the character C is encoded as an E. Analogous, P=15 -> 15*2 = 30 -> 30 modulo 26 = 4 -> the character P would also be encoded as an E.Therefore, it is not possible to determine if an E in a ciphertext corresponds to a C or a P. In order to achieve a unique matching only keys that are co-prime to the length of the alphabet can be used. As 29 is prime, it has no divisors except for 1 and 29 and thus there are no multiples as bad keys. Which language's style guidelines should be used when writing code that is supposed to be called from another language? How does the j decode to the H, and the u decode to the E? Modified 8 years, 6 months ago. Therefore, an eavesdropper simply has to count letter frequencies to identify the most frequent cipher letter. This process repeats until M is reduced to 1 and therefore less than the smallest factor possible, 2. Except explicit open source licence (indicated Creative Commons / free), the "Multiplicative Cipher" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Multiplicative Cipher" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) Example1: M=9=32 has the only prime divisor 3 and thus b=9/3 1 = 2 bad keys which are 3 and 6 as the multiples of 3 that are less than 9. While using Caesar cipher technique, encrypting and decrypting symbols involves converting the values into numbers with a simple basic procedure of addition or subtraction. The grey rows show what would be expected for the order, and the red one shows what your text gives for the order: If we use a value which is not co-prime, such as 2, we will not get unique characters for the mapping: Bib: @misc{asecuritysite_99257, title = {Multiplication Cipher}, year={2023}, organization = {Asecuritysite.com}, author = {Buchanan, William J}, url = {https://asecuritysite.com/coding/mult}, note={Accessed: May 02, 2023}, howpublished={\url{https://asecuritysite.com/coding/mult}} }. Thus, among those numbers that occur twice in the cipher code, 14, 17 and 20, we can eliminate the odd 17. Certainly, it might be a double encoded message that has to be decoded twice, possibly using two different keys or even two different ciphers. If the modular multiplicative inverse of a modulo m exists, the operation of division by a modulo m can be defined as multiplying by the inverse. Affordable solution to train a team and make them project ready. }
Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. Calculate the value of each letter as follows (where a and b are the keys of the password): E (x)= (ax + b) mod m 3. Calculator Use Multiplication of positive or negative whole numbers or decimal numbers as the multiplicand and multiplier to calculate the product using long multiplication. This is not a useful encryption system since it may yield ambiguous messages. How to encrypt using Multiplicative cipher? Now, how do you decrypt the above message? Thus our decoding function P = a-1*C MOD 26 tells us to simply multiply each cipher letter by the inverse of the encoding key a=5, namely by the decoding key a-1=21 MOD 26 and we can eventually decode:
Cipher textanromrjukahhouh013171412179201007714207
0131981819742017178417PLAIN TEXTANTISTHECARRIER
For example, multiplying the cipher letter r=17 by a-1 = 21 decodes the r to T=19 since 21*17 = 357 = 19 MOD 26. Notice in the last row that all we need to know are the prime factors p of M without knowing how often they occur. 9. 13
Notice that we found the good keys indirectly. Are there any canonical examples of the Prime Directive being broken that aren't shown on screen? As 36=2*2*3*3, the possible keys are basically all numbers not multiples of 2 and/or 3. 15
Example: Encrypt DCODE with the key $ k = 17 $ and the 26-letter alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ. Instead of adding a number as we did in the Caesar Cipher, we will now multiply each plain letter by an integer a, our secret encoding key. 21 is an inverse to 5 MOD 26, therefore 5 is inverse to 21 and the two 1s are mirrored over the diagonal line. Modular inverse of a matrix. Example2: M=81=34 has again 3 as the only prime divisor and thus b = 81/3 1 = 34/3 1 = 33 1 = 26 bad keys. Each character is multiplied with this key and the corresponding letter is substituted. 10
Simply by looking at the table, we find that the following keys (whose rows are bold) produce a unique encryption and therefore call them the good keys:
a = 1,3,5,7,9,11,15,17,19,21,23,25
Why those and what do they have in common? This principle of finding the number of bad keys holds true for any alphabet length that is a prime power: There are M/p multiples of p less or equal to M, and therefore M/p - 1 many less than M. And we are only interested in those integers less than M since we are calculating MOD M which involves the integers 0 to M-1. Before considering such encoding techniques, we go ahead and check if the other frequent number, 20, is the cipher E.
Checking the E column, we can see that the possible two keys are the bad one a=18 and the good one a=5. That means:
Because a=2 is a bad key all the multiples of a must be bad keys aswell. Lets simply test all possible keys of the multiplication ciphers MOD 26:
PLAIN LETTER
0000000000000000000000000 a ABCDEFGHIJKLMNOPQRSTUVWXYZ00000000000000000000000000010123456789101112131415161718192021222324252024681012141618202224024681012141618202224303691215182124147101316192225258111417202340481216202426101418220481216202426101418225051015202549141924381318232712172216111621606121824410162228142006121824410162228142070714212916234111825613201815223101724512198081624614224122021018081624614224122021018909181101921120312214132251423615247162581710010204142481821222616010204142481821222616110112271831425102161721324920516112238194151201224102282061841621401224102282061841621413013013013013013013013013013013013013013140142164186208221024120142164186208221024121501541982312116520924132176211025143187221116016622122188241442010016622122188241442010170178251672415623145221342112320112191011891801810220124221462416801810220124221462416819019125241710322158120136251811423169221147200201482221610424181260201482221610424181262102116116122171272231813832419149425201510522022181410622420161284022181410622420161284230232017141185225221916131074124211815129632402422201816141210864202422201816141210864225025242322212019181716151413121110987654321
We learned already that the key a=2 (as can be seen in the 3rd row) does not produce a unique encryption. I accomplish this. Therefore,
Formula for the number of good keys if M is a prime power:
If M = pn , the number of good keys is u(M) = pn - pn-1. Say you first want to encode the letter c then you have to enter e when asked. As some of them fail to produce a unique encryption, we will discover an easy criterion for keys that produce the desired unique encryptions (the good keys) and apply it to different alphabet lengths. 28 equals 2*2*7 so that all the keys that are multiples of 2 or 7 do not and all non-multiples of 2 or 7 do produce a unique encryption: Z28* = {1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27} allowing only 12 different unique encryptions. 3 * 9 = 9 * 3 =27) the MOD- multiplication is commutative (3 * 9 = 9 * 3 = 1 MOD 26). Of course, you dont want to receive any more ambiguous messages. Affine cipher - Modular multiplicative inverse. This calculator uses Hill cipher to encrypt/decrypt a block of text. Example1: When using fractions,
5-1=1/5 is the inverse number to 5,
3-1=1/3 is the inverse number to 3,
3/2 is the inverse number to 2/3. Therefore, all the keys that are multiples of 5 such as a=10,15,20,25,30 will also translate the H into 0(=a). Say M=26=2*13=n*m. Since n and m are two distinct primes, they certainly are relative prime, so that the condition for property 4) is fulfilled. where c is the modular multiplicative inverse of a. I first subtract 65 =A and then multiply that difference by the good key a=5 yielding 10 again. Here is the C++ Code for the encryption and decryption of the multiplication cipher:
//Multiplication Cipher using the good key a=5
//Author: Nils Hahnfeld, 9/22/99
#include
#include
void main()
{
char cl,pl,ans;
int a=5, ainverse=21; //as a-1*a=21*5=105=1 MOD 26
clrscr();
do
{
cout << "Multiplication Cipher: (e)ncode or (d)ecode or (~) to exit:" ;
cin >> ans;
if (ans=='e')
{
cout<< "Enter plain text: "<< endl;
cin >> pl;
while(pl!='~')
{
if ((pl>='a') && (pl<='z'))
cl='a' + (a*(pl -'a'))%26;
else if ((pl>='A') && (pl<='Z'))
cl='A' + (a*(pl -'A'))%26;
else cl=pl;
cout << cl;
cin >> pl;
}
}
else if (ans=='d')
{
cout << "Enter cipher text: " << endl;
cin >> cl;
while(cl!='~')
{
if ((cl>='a') && (cl<='z'))
pl='a' + (ainverse*(cl -'a'))%26;
else if ((cl>='A') && (cl<='Z'))
pl='A' + (ainverse*(cl -'A'))%26;
else pl=cl;
cout << pl;
cin >> cl;
}
}
}
while(ans!='~');
}
Programmers Remarks:
Can you understand the code yourself? In this chapter we will study the Multiplicative Cipher. 6*3=18. The following table shows the numbers relative prime to M for the first 21 integers. Therefore, a simple prime check program would be sufficient to find the divisors p of M. We then set up the factors of the form (1- 1/p), multiply them and eventually multiply that answer by M.
Example1:
Say M=180, then a prime check program yields the prime factors 2,3 and 5, so that
((180) = 180 * (1-1/2) * (1-1/3) * (1-1/5)
= 180 * (1/2) * (2/3) * (4/5)
= 90 * (2/3) * (4/5)
= 60 * (4/5)
= 48
Example2:
Say M=360, since 360=2*180 the prime factors are again 2,3 and 5, so that
((360) = 360 * (1-1/2) * (1-1/3) * (1-1/5)
= 360 * (1/2) * (2/3) * (4/5)
= 180 * (2/3) * (4/5)
= 120 * (4/5)
= 96
Example3 is for you:
Say M=90, since 90=____ the prime factors are _______, so that
((90) = 90 * (1-1/__) * (1-1/__) * (1-1/__)
= 90 * ____________________
= _______________
= _______________
= ___
Of course, I could have computed the answers in the above examples right away but I wanted to give you the chance of brushing up on your skills to multiply fractions. The given examples show you the calculation process. The theory can be found after the calculator.
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