Verify this now! The message is an alphabetical substitution, the frequency analysis should make it possible to find the most common letters. 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. This weirdness is not really weird. 11 7 Except for 2 and 13, all prime numbers less than 26 are among the keys (why do they have to?). Step 2: First of all we will require an alphabet table with numeric values attached to each alphabet so that we can do the encryption process fastly. To have the solution, the right part of the linear diophantine equation should be a multiple of the . That is, . Thank you! 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? We denote 5-1 the inverse of 5. This process repeats until M is reduced to 1 and therefore less than the smallest factor possible, 2. Try to explain this, than continue reading! While you still can simply enter an integer number to calculate its remainder of Euclidean division by a given modulus, this modulo calculator can do much more. Which language's style guidelines should be used when writing code that is supposed to be called from another language? The formula MOD(E$2*$B4,26) computes the number of the plain letter T, namely 19. Builds the Affine Cipher Translation Algorithm from a string given an a and b value This calculator has 3 inputs. Connect and share knowledge within a single location that is structured and easy to search. Affine Cipher is the combination of Multiplicative Cipher and Caesar Cipher algorithm. Divide the letters of the message into groups of two or three. If we had a video livestream of a clock being sent to Mars, what would we see? The following C++ program firstly determines the factors for an entered alphabet length M and secondly their multiples, the bad keys. 565), Improving the copy in the close modal and post notices - 2023 edition, New blog post from our CEO Prashanth: Community is the future of AI, Link between Cipher suites and certificate key. Is there such a thing as "right to be heard" by the authorities? From now on we will use a handy Notation for the set of possible and good keys: 1) All the possible keys for an alphabet length of 26 are clearly all the numbers between 1 and 26, denoted as Z26. How many multiples of 3 will not produce a unique encryption? From property 1) we know that ((2)=1 and ((13)=12, and consequently, ((2*13) = ((2)*((13) = 1*12 = 12 which is exactly property 3). Subsequently, that difference is multiplied by the good key a=5 which I defined as such in int a=5. To find the inverse for each good key a, you just need to look back at the 26 by 26 encryption table. 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. 7 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. } Therefore, no matter how he decides to crack the cipher text, it wont take long. 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. Extracting arguments from a list of function calls. In our example, after subtracting 101 from the plain letter c we get the desired 2 that is now multiplied by a=5 yielding 10. Thirdly, listing the good keys would be best done using C++ vectors or even C-style arrays which you might know. Here is how: u = (p*q - 1) - (p-1) (q 1) getting rid of the first two parentheses yields = p*q -1 - p + 1 (q 1) the two 1s cancel each other out yielding = p*q p (q 1) factoring the p yields = p*(q-1) (q 1) (q-1) in both terms can be factored yielding = (q-1) * (p 1) which can also be written as = (p-1) * (q 1) Formula for the number of good keys if M is the product of two primes: The number of good keys is u(M) = u(p*q) = (p-1)*(q-1). Each character in the message is multiplied with this key. Try to understand as much as possible first, then continue reading. In the detailed representation of the alphabets (click on the "" -button), the alphabets can be edited in the short-write mode. 9. In order to have a modular multiplicative inverse, determinant and modulo (length of the alphabet) should be coprime integers, refer to Modular Multiplicative Inverse Calculator. Alternatively, the non-alphabet letters in the key and the plain text can also be filtered out to increase the security. In order to create a n x n size matrix, keyphrase length should be square of an integer, i.e., 4, 9, 16. So in our case, it was GEEKSFORGEEKS, so it will become: Multiplicative Cipher text = QCCSWJUPQCCSW. Solution of Multipilicative Inverse of 7. The use of several alphabets does not require the algorithms to distinguish between upper and lower case letters. The mono-alphabetic substitution cipher provides the simplest form of cryptography, where the cipher alphabet is simply a rearrangement of the plaintext alphabet. Other frequent letters such as T, A, O and N occurring with about (8%) might be of further help to crack the cipher text. He investigated these number properties and was the first one to come up with a function, Eulers (-function, also called Eulers Totient function, that determines the number of integers that are relative prime to a given integer M. It is a function that is in the heart of Cryptography and used i.e. 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.) 1) This program both encodes and decodes. I want to show you an example where we used it already. What is the difference between "cipher" and "encryption"? 2. If so please go ahead and modify the following program. Credit goes to the Swiss Mathematician Leonard Euler (pronounced Oiler, 1707-1783). Let us consider the above cipher text i.e. Example: Encrypt DCODE with the key k= 17 k = 17 and the 26-letter alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ See the image attached below for a better understanding. If multiplication is used to convert to cipher text, it is called a wrap-around situation. Instead of performing a transformation before encryption, this implementation allows several alphabets to be specified (see below), thereby accomplishing the same within the encryption process. You can verify this as follows: out of the __ integers that are less than 65, we first cross out all the ___ multiples of __ and then cross out the __ multiples of __ resulting in ______ = 48 good keys. (I.e. If we extract those rows with the good keys a = 1,3,5,7,9,11,15,17,19,21,23,25 and their corresponding columns, we obtain: 13579111517192123251135791115171921232533915211719255111723551525919323717111217721923112511531751999119113215231572517111173252117951231915151519231591721253711171725715235213111919191951731512511239217212111117723319925155232317115251971211593252523211917151197531 This reduced table shows i.e. Can we increase the number of unique encryptions by further extending our alphabet? Step 4: So, once the calculation part is done now you can easily encrypt your given plain text. One of the major goals of current Mathematics research is to design faster factoring algorithms as todays are fairly slow. each occurring exactly twice. Here, it reduces the number of possible good keys to two. We get the following encoding and decoding table. So, lets understand why the bad keys a = 2,4,6,8,10,12,13,14,16,18,20,22,24 dont produce a unique encryption. In fact, the security of i.e. Find mod of any numb. First, symbols of the used alphabet (alphabet as a set of symbols, for example, the alphabet in the above calculator includes space, comma, and dot symbols) are encoded with digits, for example, symbol's order number in the set. Thus they have the following restrictions: Introduction to Monotonic Stack - Data Structure and Algorithm Tutorials. The encrypted text is the smallest digit of an addition of plaintext and key when both are hexadecimal digits. Why does Acts not mention the deaths of Peter and Paul? Try it! Now when a=25, we have: 25*25 = 625. 3. Method 2: Merged: In the alphabet, mod 22 is calculated because the alphabet contains 22 elements. 2 Convert each group into a string of numbers by assigning a number to each letter of the message. In the process you'll become comfortable with modular arithmetic and begin to understand its importance to modern cryptography. An extreme example would be when a=0: all plain letters are translated into 0s which are all as so that no decryption is possible. The ultimate trick to yet produce the same format is factoring: from each parentheses we factor the first integer (which is a divisor of M) and obtain: ((60) = 22*(1 -1/2) * 3*(1 -1/3) * 5 * (1 -1/5)((M) = p12 * (1 -1/ p1) * p2*(1 -1/ p2) * p3 * (1 -1/ p3) = 22*3*5*(1 -1/2)*(1 -1/3)*(1 -1/5) = p12* p2* p3*(1 -1/ p1)*(1 -1/ p2) * (1 -1/ p3) = 60*(1 -1/2)*(1 -1/3)*(1 -1/5) = M * (1 -1/ p1) * (1 -1/ p2) * (1 -1/ p3). The Multiplicative Cipher is an Affine cipher (ax+b) with the value b null (equal to 0), so a multiplication by a a. You can try the sample button which uses a multiplication of 3, and a message of "knowledgeispower" gives enqohmjsmyctqomz. We can also calculate all the possible keys for the Affine Cipher. ((21)=________________________ as 1,2,4,5,8,10,11,13,16,17,19,20 are relative prime to 21. Example2: For M=9=32 we have u(9) = 32 - 31 = 9 3 = 6 which are the 6 good keys a=1,2,4,5,7,8. dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!A suggestion ? 4 The only disadvantage is that the minus sign itself has to be written as "---", so as not to be confused as a range operator. Finding the decoding keys for each good key a in the same manner, we obtain the following key pairs: Good Encoding key aIts decoding key a-111395217159311191571723191121523172525 Three important observations: All decoding keys a-1 in the right column are among the set of all encoding keys a. using properties 1) and 2) yields = (3-1)*(23-22) = 2*4 = 8. 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. Which number would that be? 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. The key should be changed frequently to prevent cryptographic attacks. A summary of our explorations for the number of good keys shows: 1) u(p) = p - 1, if M is prime M=p. That is: I leave the translation from an upper case plain letter to a lower case cipher letter as an easy exercise for you. Cipher textanromrjukahhouh013171412179201007714207 He finds the cipher letter h to be most frequent. To use this worksheet, you must supply: a modulus N, and either: rev2023.5.1.43405. color: #ffffff; However, there are some additional integers that are not prime (i.e. Fraction calculator - subtracting fractions step by step with explanation With the Fractions Calculator, you can subtract any two mixed numbers or proper and improper fractions. The Affine Cipher uses modulo arithmetic to perform a calculation on the numerical value of a letter to create the ciphertext. where the operation of multiplication substitutes the operation of division by the modular multiplicative inverse. Cipher textanromrjukahhouha=1ANROMRJUKAHHOUHa=3ANXWEXDYMALLWYLa=5ANTISTHECARRIERa=7ANVCYVFOUABBCOBa=9ANZQKZBIEAVVQIVa=11ANLGULPQIADDGQDa=15ANPUGPLKSAXXUKXa=17ANBKQBZSWAFFKSFa=19ANFYCFVMGAZZYMZa=21ANHSIHTWYAJJSWJa=23ANDEWDXCOAPPECPa=25ANJMOJRGQATTMGT MS Excel as a simple encryption and decryption tool: I created the following table in MS Excel with the CHAR and the MOD function: Cipher textanromrjukahhouhaa-101317141217920100771420739ANXWEXDYMALLWYL521ANTISTHECARRIER715ANVCYVFOUABBCOB93ANZQKZBIEAVVQIV1119ANLGULPQIADDGQD157ANPUGPLKSAXXUKX1723ANBKQBZSWAFFKSF1911ANFYCFVMGAZZYMZ215ANHSIHTWYAJJSWJ2317ANDEWDXCOAPPECP2525ANJMOJRGQATTMGT For example, I created the T in the row a=5 using the Excel-formula =CHAR(65+MOD(E$2*$B4,26)) where the cell E$2 contains 17 and the cell $B4 contains 21 as the decoding key a-1. For the fraction a/b, the multiplicative inverse is b/a. 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. Affordable solution to train a team and make them project ready. In conclusion, we can say that multiplicative cipher is a simple encryption technique that can be easily implemented. Since one can be divided without remainder only by one, the equation above has the solution only if . The trick is now that if we enter more than one letter all but the first entered letter are buffered (which means temporarily stored in the computers RAM) until read in in cin >> pl; inside the while loop. Multiplicative encryption uses a key $ k $ (an integer) and an alphabet. color: #ffffff; While deriving the formula for M=60=22*3*5 in the left column I will deduce simultaneously the explicit formula for M=p12*p2*p3 with p1 being the first prime factor 2, p2 being the second prime factor 3 and p3 being the third prime factor 5 in the right column. The same alphabet is used to generate the encrypted text. The decryption process is simply the reverse of the encryption process, i.e., by dividing the numerical value of each letter in the ciphertext by key and then taking the result modulo of the key. 15 Of course, you dont want to receive any more ambiguous messages. color: #ffffff; a bug ? ((3)=3-1=2 as 1 and 2 are relative prime to 3. Before we conclude this section with the highlight of creating a sole formula for ((M) from these four properties, we will consider 2 examples for each of the 4 properties of Eulers (-function. This brute force approach will work fast enough for integers M that have 10 digits or less. div#home a:link { 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. 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. Although the function is well-defined when a letter occurs more than once, this makes little sense in encryption algorithms, since the reversibility suffers. 2) If M is a prime power, M=pn: Now lets look back at M=27 as an example where we only have the one prime factor p=3, such that M=33. How to encrypt using Multiplicative cipher? The following steps take place: In the example, an overflow has occurred in the third letter, so that modulo |L| = 4 is calculated. For M=31 we have u(31)=30. Wonderful, that is all we need to solve our encryption function C= a*P MOD 26 for the plain letter P in order to then decode the encrypted message: Multiplying both sides of our encryption equation the equation yields a-1*C = a-1*(a*P) (1) = (a-1*a)*P (2) = 1*P (3) = P MOD 26 (4) Remark: Solving this equation required the 4 group properties: the existence of an inverse and the closure in (1), the associative property in (2), the inverse property in (3) and the unit element property in (4). for the RSA encryption. 3 We have explored the first three properties already, however, the 4th property is new - but not totally new. All we need to know are the prime divisors of M, but we dont even need to know how often a prime number divides M. We obtain ((2*13) = ((2) *((13). Lets consider two options: Option 1: Cracking the cipher code using letter frequencies If plain letters are replaced by cipher letters the underlying letter frequencies remain unchanged. width: max-content; By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. ((5)=_____ as 1,2,3,4 are relative prime to 5. Each letter is associated with its rank $ c $ in the alphabet (starting from 0). Lets investigate this in the following section. I'm learning and will appreciate any help. For larger integers, however, dividing by every integer less than M slows the program down enormously. 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). 15 We also turn the plaintext into digraphs (or trigraphs) and each of these into a column vector. Back to the virus carrier message. The key should be relatively prime to the length of the alphabet. WAP to implement Additive cipher(key=20), Multiplicative cipher(key=15)and affine cipher(key=15,20). Can you? 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. To ensure that no two letters are mapped to the same letter, a and m must be coprime. Zero has no modular multiplicative inverse. This is just what we wanted except that the answer 10 does not equal the desired cipher letter k on the computer. 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). Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. This means that the key should be a large, random number that is difficult to guess or factor. The determinant of the matrix should not be equal to zero, and, additionally, the determinant of the matrix should have a modular multiplicative inverse. For classical methods, the alphabet often consists only of the uppercase letters (A-Z). Each character of the used alphabet is assigned to a value. 11 padding-right: 20px; This inverse modulo calculator calculates the modular multiplicative inverse of a given integer a modulo m. First of all, there is a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x, and it is not the same as modular multiplicative inverse. That was trial and error and might take quite a while. Example the letter M (12th letter in this zero indexed alphabet) and key 3 would be 12 * 3 = 36. Write to dCode! Modular Multiplicative Inverse a -1. Similarly, the multiples of a=7 will translate an F (=5) into an 0 (=a) because 7 does so. Copyright 1998 - 2023 CrypTool Contributors. Just as the regular multiplication of two integers is commutative (i.e. This table shows the occurances of the letters in the text (ignoring the case of the letters): This table shows how the text matches a normal probability to text (where 'E' has the highest level of occurance and 'Z' has the least). How do we deal with non-letters? v l X X X