13 This is not a useful encryption system since it may yield ambiguous messages. I.e. In an additive cipher, the cipher alphabet is a shift of the plaintext alphabet. 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. A function that performs this is called an alphabet function. Equivalently stated, 105 divided by 26 leaves a remainder of 1. Also, there is no general match on how to handle digits or special characters. color: #aaaaaa; The Affine Cipher uses modulo arithmetic to perform a calculation on the numerical value of a letter to create the ciphertext. Here, it reduces the number of possible good keys to two. ((5)=_____ as 1,2,3,4 are relative prime to 5. This is not very useful. Affine Cipher - Crypto Corner 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. This is very likely in English texts and virtually certain in the German language where on average every 5th letter is an E. Even if an eavesdropper decides to produce all 12 possible plain texts, they can be generated with the help of a computer within a few seconds. The Caesar cipher is a special case of the Affine cipher where A is 1 and B is the shift/offest. This program is an extension of the previous simple factoring program. Mathematically: a-1 * a = a * a-1 = 1. Example5: If M=65=5*13=p*q, then the formula yields u(65) = (5-1)*(13-1) = 4*12 = 48. Affine Cipher is the combination of Multiplicative Cipher and Caesar Cipher algorithm. If a=1 is used as a key, each cipher letter equals its plain letter which shows that it does produce a unique encryption. After finding each factor of M, I just print them out in for (j=1;j #include #include #include void main() { int M, m, j, factor, factor2; bool prime; clrscr(); cout << "This program finds the 'bad' keys for an entered alphabet length M." << endl; cout << "===========================================================================" << endl; do { cout << "Enter the alphabet length or 0 to exit: M="; cin >> M; m=M; factor=2; prime=0; //initialization while(factor <= m) { if (m%factor==0) { if (factor!=M) { cout << "Divisor of "<< M << " =" << setw(3) <. 2) The setwidth command setw() assigns as many spaces as entered in the parentheses for a numerical output in order to have a well-formatted output. 23 An affine cipher is a cipher belonging to the group of monoalphabetic substitution ciphers. A longer alphabet produces less unique encryptions. 3) u(p*q) = (p-1)*(q-1), if M is a product of two primes M=p*q. Moreover, we build the mathematical foundation to understand secure encryption systems such as the RSA encryption. 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. Our implementation of Vigenre, Beaufort, etc. It is easy to implement and easy to understand, and it does not require any large amount of computational power. Learn more. Equivalently stated: what product of a-1 and 5 equals 1 more than a multiple of 26 such as 27, 53, 79, 105, etc? The o =14 decodes to I = 8 since 21*14 = 224 = 8 MOD 26, the m =12 decodes to S=18 since 21*12 = 252 = 18 MOD 26. 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). QCCSWJUPQCCSW as an example to perform decryption using the multiplicative cipher. Hey, this shows a great way to produce more unique encryptions which of course makes life harder for an eavesdropper: Recommendation for more security: Choose the alphabet length M to be a prime number to make cracking the cipher text more difficult. Online calculator: Substitution cipher decoder - PLANETCALC How to make sense of the decryption formula for Affine ciphers? Does the increase of our alphabet length by 1 increase the number of unique encryptions obtained? That would additionally require that the good keys form a commutative group with respect to addition. An alphabet[1] is an ordered set of all characters which can occur in a plaintext, a secret text, or the key. 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. (Identification), How to decipher Multiplicative cipher without key? Activity Code: 65665. div#home a:link { Which number would that be? Again, we just have to find the cipher numbers in the 5th row and then go up that column to the very top to find the corresponding plain letter. 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). dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!A suggestion ? b ^ ^ ^ 8 ^ a G n n n n n R R R f h h h h h h $ u R ` R R R n n n n 7 R j n n n f R f k \ ^ % n n `d P ^ v$ .$ r % T 0 G $ r 2 % n n n n Chapter 2 Multiplicative Cipher In this chapter we will study the Multiplicative Cipher. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. However, we dont need to consider keys that are greater than 26 since each of them has an equivalent key less than 26 that yields the same encryption: the even multiples of 13 (i.e. color: #ffffff; PLAIN LETTERNATANTSecret key a=2130190131900120012Cipher letteraamaam You can see the dilemma of this message. I leave the translation from an upper case plain letter to a lower case cipher letter as an easy exercise for you. 7 There are several way to implement the inversion and the affine transformation described in the AES to get the final SBox. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. It surely acquires this simple form for any number of primes or prime powers. gcd(k,36)=1. DOC Chapter 2 : Multiplicative Cipher - TI89 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. 15 A corresponding warning is displayed. background-color: #620E01; For now, lets focus on the lower case letters. The Multiplicative Cipher is an Affine cipher (ax+b) with the value b null (equal to 0), so a multiplication by $ a $. 5 For classical methods, the alphabet often consists only of the uppercase letters (A-Z). 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. Affine cipher - online encoder / decoder - Calcoolator.eu Encrypt and decrypt any cipher created in a Playfair cipher. Cite as source (bibliography): 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. If you are able to invent a fast factoring algorithm, you will not have to worry about a future job. div#home { (Attacks). In some secret manner, the sender and the recipient had to agree on the encoding key a. Long Multiplication Calculator Since 36 is greater than the length of the used alphabet, 36 modulo 26 = 10 is calculated. will translate the H (=7) into a (=0), because 5*7 = 35 = 0 MOD 35. Here is a non-calculator way to understand why 25 is inverse to itself: Since 25 = -1 MOD 26, it follows 25 * 25 = (-1) * (-1) = 1 MOD 26. 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. That means the key should not have any common factors with the alphabet or plaintext except for 1. color: #ffffff; v l X X X If so please go ahead and modify the following program. Which was the first Sci-Fi story to predict obnoxious "robo calls"? They are very special primes as they must consist of 100 digits or more. Lets investigate this in the following section. Learn how PLANETCALC and our partners collect and use data. Lets check why: 1*1=1 MOD 26 which explains a = a-1 = 1 (Big deal!). The given examples show you the calculation process. 39, 65, 91, ) have its equivalent key in a=13, another bad key, since 39=65=91=13 MOD 26. Therefore, the set of all encoding keys must equal the set of all decoding keys. Exporting results as a .csv or .txt file is free by clicking on the export icon Although the function is well-defined when a letter occurs more than once, this makes little sense in encryption algorithms, since the reversibility suffers. Say, we want to encrypt the plain letter C=67. 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. Of course, you dont want to receive any more ambiguous messages. To find the inverse for each good key a, you just need to look back at the 26 by 26 encryption table. We wont have to do it that way again since there is a much more straightforward method. The next two lines then show us that the variable false is defined as 0 and true as 1. Step 1: For decryption first we need to find the multiplication inverse of the key. or ? 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. what are prime divisors of 178247 or of 56272839 ?). 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. 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. div#home a:visited { 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). RSA Express Encryption/Decryption Calculator This worksheet is provided for message encryption/decryption with the RSA Public Key scheme. 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. 2.5 Counting the Number of Good Keys for various Alphabet Lengths M An Introduction to the Euler Function. No, it is not. Implementation of Affine Cipher - GeeksforGeeks GitHub - Mehul2205/Cryptography-Assignments: These are the lab Each character is multiplied with this key and the corresponding letter is substituted. Contributed by: Shawna Martell (March 2011) Open content licensed under CC BY-NC-SA Snapshots 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. 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. 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 1 What is Multiplicative Cipher in Cryptography? - GeeksforGeeks The number fetched through output is mapped in the table mentioned above and the corresponding letter is taken as the encrypted letter. 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. Since, for the standard alphabet, there are 12 numbers less than 26 which are . Subsequently, that difference is multiplied by the good key a=5 which I defined as such in int a=5. Just as the regular multiplication of two integers is commutative (i.e. 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. Please enable JavaScript to use all functions of this website. 3. Note the difference in 'D' and 'd': The index value is the same, but the 'd' is. Solution of Multipilicative Inverse of 7. (I can not list those here as they depend on the alphabet length M.) We are now able to summarize how to encrypt a message using the multiplication cipher: To encrypt a plain letter P to the cipher letter C using the Multiplication Cipher, we use the encryption function: f : P ( C=(a*P) MOD 26. I will complete the first ones and leave the second ones for you as exercises. Example4: If M=39=3*13=p*q, then the formula yields u(39) = (3-1)*(13-1) = 2*12 = 24. Each letter is enciphered with the function (ax + b) mod 26. Extracting arguments from a list of function calls. 3.0.4224.0. 17 14 You are asked to enter your plain letter in cin >> pl; As long as you dont enter ~ the while-condition while(pl!='~') is fulfilled and the entered plain letter (=pl) is being encoded. Try it for yourself! Two MacBook Pro with same model number (A1286) but different year. This, however, limits readability. Notice in all three equations that because a=2 turns the 13 (=N) into 0 in 2*13 = 0, all the multiples of a=2 translate the N into 0 (=a). In fact, the security of i.e. The solution can be found with the Extended Euclidean algorithm. The first character G corresponds to the six. 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? Coincidence? acknowledge that you have read and understood our, Data Structure & Algorithm Classes (Live), Data Structures & Algorithms in JavaScript, Data Structure & Algorithm-Self Paced(C++/JAVA), Full Stack Development with React & Node JS(Live), Android App Development with Kotlin(Live), Python Backend Development with Django(Live), DevOps Engineering - Planning to Production, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Interview Preparation For Software Developers. 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. the commonly used RSA Cipher is based on the relative slowness of such factoring programs. We can combine these two criteria into one easy criterion. I accomplish this. It thus gives a great example that we are only guaranteed to solve this equation for numbers that form a group with respect to multiplication MOD 26. 4 We will check in the Abstract Algebra section at the end of this chapter that the set of good keys MOD 26, Z26* = {1,3,5,7,9,11,15,17,19,21,23,25}, does form a multiplicative group. In order to create unique cipher characters, we must use a multiplier which is co-prime (the values do not share any factors when dividing - see Try GCD of 5) in relation to the size of the alphabet (26), so you should use either 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 or 25. =CODE("a") yields 97). Just as 5*1/5 yields 1, 5 * 5-1 shall equal 1 MOD 26. 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? Let s be such a reversible function. Online calculator: Modular arithmetic - PLANETCALC for the RSA encryption. 7 a feedback ? Online calculator: Modular Multiplicative Inverse Calculator - PLANETCALC This is important because if the key shares a factor with the plaintext, it can be easily broken by factoring in the key. The same alphabet is used to generate the encrypted text. Modular Multiplicative Inverse a -1. }. 18 You now understand why cryptographers have an affection for prime numbers. To use this worksheet, you must supply: a modulus N, and either: or . That are those that dont have a common divisor with 26. ((15)=((3*5)=(3-1)*(5-1)=2*4=8 as 1,2,4,7,8,11,13,14 are relative prime to 15. They seem to not follow any apparent pattern. Modular inverse of a matrix. Since the number of unique encryptions u is a function of the alphabet length M, we may write in function notation: u(M) to denote the number of unique encryptions (which equals the number of good keys) as a function of M. I.e. 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. The only two keys that are inverse to themselves are 1 and 25, which means that the encoding key equals its decoding key. We, therefore, name the good keys as follows: Definition of numbers that are relative prime: Two integers are called relative prime if their greatest common divisor equals 1. //Author: Nils Hahnfeld 10/15/99 //Factoring program #include #include #include void main() { int M, factor ; clrscr(); do { cout << "Enter the integer that you want to factor or 0 to exit: M="; cin >> M; factor=2; while(factor <= M) { if (M%factor==0) //check all integers less than M as factors { cout << factor << endl; M/=factor; factor=1; } factor++; } }while(M!=0); } Programmers remarks: Starting with 2, this program checks the integers from 2 to M-1 as potential factors of M in if (M%factor==0). Longer messages reveal the most the letter e equivalent, however, this is not necessarily so for our message. 11 Affine Cipher - Online Decryption, Decoder, Encoder, Calculator As you can see on the wiki, decryption function for affine cipher for the following encrytption function: E (input) = a*input + b mod m is defined as: D (enc) = a^-1 * (enc - b) mod m The only possible problem here can be computation of a^-1, which is modular multiplicative inverse. 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. Its numerical equivalent reveals the row and therefore the key a as follows: PLAIN LETTER 0000000000000000000000000 ABCDEFGHIJKLMNOPQRSTUVWXYZ101234202468303691240481216505101520254914192438131823271217221611162160612182470714212808162469091811010010204141101122718120122410221301301301401421641501541981601662212170178251618018102201901912524200201482210211611622022181410230232017141185225221916131074124211815129632402422201825025242322 After intercepting the cipher text, an eavesdropper simply finds the most frequent letter of this rather brief message. For example, Caesar cipher using a left rotation of three places, equivalent to a right shift of 23 as given below. As 36=2*2*3*3, the possible keys are basically all numbers not multiples of 2 and/or 3. ((24) = ((23 *3) = ((23)*((3) = (23-22)*(3-1) = 4*2 = 8 as 1,5,7,11,13,17,19,23 are relative prime to 24. 15 Each character from the plaintext is always mapped to the same character in the ciphertext as in the Caesar cipher. Note: This cipher is closely related to the. The multiplicative cipher has little interest, but it is often used for learning computer science and ciphers. How to calculate the modular multiplicative inverse for the Affine Cipher Multiplicative - CrypTool Portal This corresponds to the K. If "GEHEIMNIS" would be completely encoded by this procedure, the ciphertext would be: "SMVMYKNYC". Subsequently, ( is computed by property 1) if such factors are primes or by property 2) if they are prime powers.