16 Write to dCode! Does that even mean that the good keys form a field? This is important because if the key shares a factor with the plaintext, it can be easily broken by factoring in the key. Apr 6, 2013 at 10:46 . To ensure that no two letters are mapped to the same letter, a and m must be coprime. Secondly, we would translate every upper case plain letter into a lower case cipher letter so that we dont reveal information about the beginning of a sentence. gcd(k,36)=1. He obtains: Cipher textanromrjukahhouh013171412179201007714207 013116711232140151519215PLAIN TEXTANLGHLXCOAPPTCP That message does not reveal a virus carrier. 2) u(pn)= pn - pn-1, if M is a power of a prime M= pn. div#home a:link { 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. 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. Subsequently, that difference is multiplied by the good key a=5 which I defined as such in int a=5. A reciprocal is one of a pair of numbers that when multiplied with another number equals the number 1. Try it for yourself! The following C++ program firstly determines the factors for an entered alphabet length M and secondly their multiples, the bad keys. Since any plain letter fulfills the condition in while(cl!='~') The loop is reentered and the next cipher letter is displayed in cout << cl; We can then end this while loop by entering ~ and then choose to either encode, decode or exit the program. It is possible to distinguish between 2 types of actions in the plain text: uppercase letters [A-Z] and digits [0-9]. 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. dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!A suggestion ? Which language's style guidelines should be used when writing code that is supposed to be called from another language? In fact, any character is stored as a number: i.e. I accomplish this. Convert each group of numbers into column matrices. In an additive cipher, the cipher alphabet is a shift of the plaintext alphabet. Step 2: The basic formula that can be used to implement Multiplicative Cipher is: Decryption= (C * Multiplication inverse of the key) Mod 26. If you dont know, exercise your patience, later in this chapter I will present a more elegant program that uses the Euclidean Algorithm to determine the good keys. 3 This is not very useful. Zero has no modular multiplicative inverse. To verify this: 262 = 676 =1 MOD 27. Test it yourself. We know already that: ((60) = ((22*3*5) = (22-21)*(3-1)*(5-1)((M) = ((p12* p2* p3) = (p12- p11)*( p2-1)*( p3-1). 5 1. 11 It would take quite a long time for a computer to brute-force through a majority of nine million keys. 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. 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. 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. The MOD 26 calculation leaves the 10 unchanged. Therefore, the set of all encoding keys must equal the set of all decoding keys. In linear algebra, an n-by-n (square) matrix A is called invertible if there exists an n-by-n matrix such that. Ask Question Asked 9 years, 11 months ago. If the plaintext is made of both letters (a to z) and digits (0 to 9), how do you find the key domain of the multiplication cipher? 1 The only disadvantage is that the minus sign itself has to be written as "---", so as not to be confused as a range operator. 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. Those are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75 and 78 as the multiples of 3 that are less than 81. Here, it reduces the number of possible good keys to two. WAP to find the solutions of equations: a.14x=12mod 18 b.3x+4=6 mod 132. Tool to decrypt/encrypt with multiplicative encryption, a substitution cipher based on a multiplication operation. unchanged so that you can detect the format of the original message easier. 17 The modular multiplicative inverse of an integer a modulo m is an integer b such that Finally, I have to add the usual 65 = A (why?) It describes the multiplicative property of (. The key should be changed frequently to prevent cryptographic attacks. (I.e. Are these quarters notes or just eighth notes? 3. For the fraction a/b, the multiplicative inverse is b/a. 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. 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. 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". Let s be such a reversible function. Moreover, since a=13 is a bad key its multiples 26, 39, must also be bad keys. rev2023.5.1.43405. 15 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. color: #aaaaaa; 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. In the next chapter, I will show you one principle of increasing the safety of a cipher code. Thus, among those numbers that occur twice in the cipher code, 14, 17 and 20, we can eliminate the odd 17. 2.5 Counting the Number of Good Keys for various Alphabet Lengths M An Introduction to the Euler Function. The multiplicative cipher has little interest, but it is often used for learning computer science and ciphers. Lab 2. (Definition). To have the solution, the right part of the linear diophantine equation should be a multiple of the . Playfair cipher online encoder and decoder. Additionally, you will learn that the RSA Cipher uses prime numbers as well. It is suitable for small-scale applications but not recommended for practical purposes. For the encryption to be reversible (so that the message can be decrypted), the key must be a coprime number with 26 (where 26 is the number of letters of the alphabet). Examples for property 1): 3 and 5 are two primes. How to encrypt using Multiplicative cipher? For letters that do not occur in L, the alphabet function sL is undefined. This modulo calculator performs arithmetic operations modulo p over a given math expression. For the purpose of setting up an explicit formula for ((M), we now try to give the three factors (in parentheses) the same format. I'm learning and will appreciate any help. 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 Generally: The good keys are those as that are relative prime to M and are denoted as ZM*. 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 example, if we have the number 7, the multiplicative inverse, or reciprocal, would be 1/7 because when you multiply 7 and 1/7 together, you get 1. The determinant of the matrix should not be equal to zero, and, additionally, the determinant of the matrix should have a modular multiplicative inverse. Each character from the plaintext is always mapped to the same character in the ciphertext as in the Caesar cipher. 10 background-color: #620E01; Cryptoanalysis - Cracking the Multiplication Cipher Just like the Cipher Caesar Cipher, the Multiplication is not secure at all. The same alphabet is used to generate the encrypted text. 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. If a is a good key, that is if a is relatively prime to 26, then f produces a one-to-one relationship between plain and cipher letters, which therefore permits a unique encryption. This process repeats until M is reduced to 1 and therefore less than the smallest factor possible, 2. See the image attached below for a better understanding. So on for each letter, the final encrypted message is ZIEZQ. We first found the bad keys as the multiples of the prime divisors of the alphabet length M. Consequently, the good keys are the remaining integers less than M. Again a perfect task for a computer, especially when we have to find the prime divisors of bigger integers. Say you first want to encode the letter c then you have to enter e when asked. This website would like to use cookies for Google Analytics. Agree Find mod of any numb. In conclusion, we can say that multiplicative cipher is a simple encryption technique that can be easily implemented. 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. Step 4: So, once the calculation part is done now you can easily encrypt your given plain text. However, there is no 7 the numerical equivalent of letter h - in the E column. 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). Affine cipher - Modular multiplicative inverse. In this lab, you'll learn about the multiplication cipher, a monoalphabetic cipher. Example: If we use the encoding key a=3, we find that the decoding key a-1 is 9 as the 1 occurs in the J- or 9-column telling us additionally that the plain letter J (=9) encrypts to the cipher letter b (=1). What would you do? 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? 0 Remember to assign letters to blank spaces. Lets check why: 1*1=1 MOD 26 which explains a = a-1 = 1 (Big deal!). While using Caesar cipher technique, encrypting and decrypting symbols involves converting the values into numbers with a simple basic procedure of addition or subtraction. Note that you may need to run it several times to find completely accurate solution. In this video u will learn how to encrypt the message using multiplicative cipher technique.Plain text to cipher text.Calculator tricks. Connect and share knowledge within a single location that is structured and easy to search. Since 625=24*26+1 which means that 625 leaves a remainder of 1 when divided by 26, we have 625 = 1 MOD 26 and altogether 25 * 25 = 625 = 1 MOD 26. 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. Which number would that be? padding-right: 20px; Encrypted text: The quick brown fox jumps over the lazy dog. If you choose to do so, dont forget to also redefine the corresponding decoding key in int a=5, ainverse=21; . 18 However, it is not a secure method of encryption and can be easily broken too. Example4: For M= 34 =81, we get u(81) = 34 - 33 = 81 27 = 54. Try to answer it for yourself. margin-bottom: 16px; In, this way you can implement Encrypt a plain text and Decrypt a cipher text for Multiplicative cipher in cryptography. Aha, that realization helps a lot, since that also means that prime Ms produce M-1 unique encryptions. Alphabets (yes, there may be several: more below) can be described by a list L of letters. The number fetched through output is mapped in the table mentioned above and the corresponding letter is taken as the encrypted letter. 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 an alphabet length of 26 this corresponds to 12 keys: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25. However, when using MOD arithmetic and solving 23=5*P MOD 26, we dont deal with fractions but only integers. the commonly used RSA Cipher is based on the relative slowness of such factoring programs. 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. The calculator logic is explained below the calculator. In fact, the cipher E can only be an even cipher letter as only even numbers appear in the E-column. Moreover, multiplying any two good keys yields again a good key. Even products of 3 primes or prime powers like 30 or 60 can now be dealt with due to the 4th property: Example2: If M=30=2*3*5, then ((30) = ((2*3*5) using property 4) yields = ((2)*((3*5) again property 4) yields = ((2)*((3)*((5) now using property 1) yields = 1*2*4 = 8. Ubuntu won't accept my choice of password. A=65, B=66, C=67, .., Z=100, a=101, b=102, c=103, z=125. background-image: none; I first subtract 65 =A and then multiply that difference by the good key a=5 yielding 10 again.