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. For the English alphabet, where m = 26, this means a cannot be 2, 4, 6, 8 (any even number) or 13. CRITERION FOR GOOD KEYS A key a produces a unique encryption, if the greatest common divisor of 26 and a equals 1, which we write as: gcd(26, a)=1 Convince yourself that 26 has a greatest common divisor equal to 1 with each of these good keys a = 1,3,5,7,9,11,15,17,19,21,23,25. Step 2: The basic formula that can be used to implement Multiplicative Cipher is: Decryption= (C * Multiplication inverse of the key) Mod 26. For illustration purposes we use the message "GEHEIMNIS" and the key 3. 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. In such case, divide M by that factor: M/=factor; and start checking M/factor for factors less than M/factoretc. gcd(k,36)=1. Multiplicative Cipher 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. That is, they mustn't have any common divisors. Options regulate the case when a letter does not appear in any alphabet: it is not encrypted, but transferred directly to the output. Modified 8 years, 6 months ago. Firstly I have no idea how they derived this formula, but I think I have a general idea. However, there are some additional integers that are not prime (i.e. 13 Example: Encrypt DCODE with the key $ k = 17 $ and the 26-letter alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ. div#home { We can also calculate all the possible keys for the Affine Cipher. Multiply It! ~=.., $=.. etc. To ensure that no two letters are mapped to the same letter, a and m must be coprime. It has to be placed after the cout command as in: cout << setw(2) << j*factor. Since, for the standard alphabet, there are 12 numbers less than 26 which are . A key a does not produce a unique encryption, if 1) a divides 26 evenly or if 2) a is a multiple of such divisors. Equivalently stated: what product of a-1 and 5 equals 1 more than a multiple of 26 such as 27, 53, 79, 105, etc? ((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. 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. This process repeats until M is reduced to 1 and therefore less than the smallest factor possible, 2. I.e. So the cipher text symbol will be w for the letter a in this case. The solution shows the work for the Standard Algorithm. 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 How to recognize a Multiplicative ciphertext? The same alphabet is used to generate the encrypted text. Can you? 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. 2.2 Decryption of the Multiplication Cipher Now that the virus carrier message was encoded in a unique manner how can it be decoded? using properties 1) and 2) yields = (3-1)*(23-22) = 2*4 = 8. 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. Therefore, we first have to add 65 to the 19 in order to translate the 84 eventually into the desired T using =CHAR(65+MOD(E$2*$B4,26)). Determining the bad keys for a given alphabet length M is a perfect task for a computer. This requires additional meta-information of the letters that must be recorded before encryption. 5 2) u(pn)= pn - pn-1, if M is a power of a prime M= pn. 18 Implementation of Affine Cipher - GeeksforGeeks In fact, I always have to subtract 101 from each entered lower case plain letter to get its corresponding number. They seem to not follow any apparent pattern. (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. In fact, the sets of the encoding and decoding keys are identical. Example: D = 3, so $ 3 \times 17 \mod 26 \equiv 25 $ and the letter at rank 25 is Z. This program is an extension of the previous simple factoring program. We have to understand why multiplying by a bad key a MOD 26 yields some integers more than once and others not at all. 25, Encrypt What Should Be the Length of the Symmetric Key in Cryptography? Equivalently stated, 105 divided by 26 leaves a remainder of 1. Can we increase the number of unique encryptions by further extending our alphabet? color: #aaaaaa; } 343 and 14 are not relative prime since gcd(343,14)=7. The letter A remains unchanged ans id always encoded A. However, it is not a secure method of encryption and can be easily broken too. Example 1: For M=27=33: Inserting 3 for p and 3 for n in pn - pn-1 yields u(27) = 33 - 32 = 27-9 = 18 which is just what we wanted. A corresponding warning is displayed. Therefore, no matter how he decides to crack the cipher text, it wont take long. Ask Question Asked 9 years, 11 months ago. However, subtracting the number of bad keys from the number of all possible keys (=M-1) yields the number of good keys. block cipher - Multiplicative Inverse in AES - Cryptography Stack Exchange 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. WAP in python to find out the additive and multiplicative inverse of an integer b using extended Euclidean algorithmof set Zn. Try to understand as much as possible first, then continue reading. 24 So on for each letter, the final encrypted message is ZIEZQ. For instance, we encoded with a=3, then the encryption function was C=3*P mod 26. How could it be broken? 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. 17 As an attentive reader, we realize that the MOD multiplication of the keys is closed (recall the group properties in the previous chapter). Finally I understand how to calculate the modular multiplicative inverse :) $\endgroup$ - np00. (Attacks). By using this website, you agree with our Cookies Policy. dCode retains ownership of the "Multiplicative Cipher" source code. Before Conversion: ABCDEFGHIJKLMNOPQRSTUVWXYZ After Conversion: XYZABCDEFGHIJKLMNOPQRSTUVW Age Calculators You may see ads that are less relevant to you. Substitution cipher decoder. We make use of First and third party cookies to improve our user experience. Aha, that realization helps a lot, since that also means that prime Ms produce M-1 unique encryptions. For now, lets focus on the lower case letters. Convert each group of numbers into column matrices. affine cipher Multiplicative encryption uses a key $ k $ (an integer) and an alphabet. 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. It only takes a minute to sign up. In order to increase the probability of this, the alphabet is expanded, so its length becomes the prime integer. "<. This is important because even if a key is secure when it is first chosen, it can become less secure over time as technology and methods for breaking encryption increase. PLAIN LETTERNATANTSecret key a=2130190131900120012Cipher letteraamaam You can see the dilemma of this message. 2. That means: Because a=2 is a bad key all the multiples of a must be bad keys aswell. The formula MOD(E$2*$B4,26) computes the number of the plain letter T, namely 19. where the operation of multiplication substitutes the operation of division by the modular multiplicative inverse. div#home a:link { I will complete the first ones and leave the second ones for you as exercises. This encoding and decoding is working based on alphabet shifting & transforming the letters into numbers . 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. Moreover, since a=13 is a bad key its multiples 26, 39, must also be bad keys. We just had to multiply each cipher letter by a-1. 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. Also, each B and each M turn into 2 (=c) since 2*1 = 2 MOD 26 and 2*14 = 28 = 2 MOD 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. Among the 12 good keys we pick a=5 to encode the virus carrier message as follows: PLAIN TEXTANTISTHECARRIER0131981819742017178417 013171412179201007714207Cipher textanromrjukahhouh Exercise1: Encrypt the same plain text using the key a=7. 3. Why are players required to record the moves in World Championship Classical games? 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. Example: Encrypt DCODE with the key k= 17 k = 17 and the 26-letter alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ Which keys now yield a unique encryption? Online calculator: Modular inverse of a matrix - PLANETCALC Does that even mean that the good keys form a field? unchanged so that you can detect the format of the original message easier. Say, we want to encrypt the plain letter C=67. Apr 6, 2013 at 10:46 . Multiplicative Cipher : Encryption Decryption Method | Mono-alphabetic Substitution Cryptography Quick Trixx 5.13K subscribers Subscribe 38K views 5 years ago Cryptography and Network Security. 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. 23 Identify blue/translucent jelly-like animal on beach. If a=1 is used as a key, each cipher letter equals its plain letter which shows that it does produce a unique encryption. The three factors in the parentheses already have the same desired format, however, the single 2 destroys it. I will answer it at the end of this chapter in the Abstract Algebra section. 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. That is Encrypt and decrypt any cipher created in a Playfair cipher. Tool to decrypt/encrypt with multiplicative encryption, a substitution cipher based on a multiplication operation. We saw that an alphabet length of M=26 produces 12 unique encryptions, since the even numbers as multiples of 2 and the 13 are the 13 bad keys. 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. Modulus m. In order to be able to use the command setw() we have to include the iomanip.h library in #include . 1 22 div#home a:hover { 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. Boolean algebra of the lattice of subspaces of a vector space? rev2023.5.1.43405. The Multiplicative Cipher is an Affine cipher (ax+b) with the value b null (equal to 0), so a multiplication by $ a $. 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". what are prime divisors of 178247 or of 56272839 ?). You could also define a to be a different good key. No, 13 is missing. The multiplicative inverse of a modulo m exists if and only if a and m are coprime (i.e., if gcd(a, m) = 1). Multiplicative Cipher - Tutorial - scanftree Now the cipher letter cl equals k and we can end the lower case encoding. To show this, let's look at this equation: This is a linear diophantine equation with two unknowns; refer to Linear Diophantine Equations Solver. The basic implementation of affine cipher is as shown in the image below In this chapter, we will implement affine cipher by creating its corresponding class that includes two basic functions for encryption and decryption. This is important because if the key shares a factor with the plaintext, it can be easily broken by factoring in the key. 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. or . Now, how do you decrypt the above message? color: #ffffff; where the operation of multiplication substitutes the operation of division by the modular multiplicative inverse. Definition of an inverse number: A number a-1 that yields 1 when multiplied by a is called the inverse of a. 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). 21 is an inverse to 5 MOD 26, therefore 5 is inverse to 21 and the two 1s are mirrored over the diagonal line. It is not difficult to understand that the length of such numbers requires the usage of computers. That are those that dont have a common divisor with 26. It would take quite a long time for a computer to brute-force through a majority of nine million keys. Try it for yourself! Therefore, a translation must take place, which can on the one hand transform letters in numbers and, conversely, re-generate letters again. For the same reason, an alphabet length of M=31 produces u=30 unique encryptions. 15 Which ones are those? This weirdness is not really weird. What is the symbol (which looks similar to an equals sign) called? Multiplication Cipher For each character of the plain message, apply the following calculation: ($ 26 $ being the number of letters in the alphabet). Try it! We then perform matrix multiplication modulo the length of the . Modular Multiplicative Inverse a -1. Calculates a modular multiplicative inverse of an integer a, which is an integer x such that the product ax is congruent to 1 with respect to the modulus m. ax = 1 (mod m) ax aa1 1 (mod m) a x a a 1 1 ( mod m) Integer a. In some secret manner, the sender and the recipient had to agree on the encoding key a. Our alphabet length of 28 now yields how many unique encryptions? 2.4 Varying the Alphabet Length varies the Number of Good Keys Using an alphabet length of M=27: Say for legibility reasons we add a blank symbol as our 27th plain letter. As 36=2*2*3*3, the possible keys are basically all numbers not multiples of 2 and/or 3. He decodes all the other cipher letters by finding their corresponding number in the 23rd row (see above) and then goes up that column to find the original plain letter. The solution can be found with the Extended Euclidean algorithm. For a check: the eight integers 1,5,7,11,13,17,19,23 are relative prime to 24 and thus the good keys for M=24. Two MacBook Pro with same model number (A1286) but different year. Once we have the solution, our x is the modular multiplicative inverse of a modulo m. Rewrite the above equation like that As 29 is prime, it has no divisors except for 1 and 29 and thus there are no multiples as bad keys. We denote 5-1 the inverse of 5. Since the bool.h library is very short I want to show you its contents: typedef int bool; const int false = 0; const int true = 1; In the first line the new data type bool is defined of type int so that the (two) bool-variables are just regular integers. Our good-key-criterion declares those integers to be good keys that are relative prime to 27. 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. margin-bottom: 16px; It was encoded MOD 26. The reason for that is that a prime number has per definition no prime divisor except for 1 and itself. Multiplicative cipher encryption|Multiplicative cipher|Multiplicative cipher example|What is multiplicative cipher PLAYFAIR CIPHER WITH EXAMPLE||SUBSTITUTION TECHNIQUE||MATHEMATICS OF. It describes the multiplicative property of (. 25 No provisions are made for high precision arithmetic, nor have the algorithms been encoded for efficiency when dealing with large numbers. Options: Multiplier: filter whitespace characters group 5 characters filter non-alphabet characters convert to first alphabet These ads use cookies, but not for personalization. 7 Multiplicative - CrypTool Portal Calculator Use Multiplication of positive or negative whole numbers or decimal numbers as the multiplicand and multiplier to calculate the product using long multiplication. } Solution of Multipilicative Inverse of 7. } Similarly, the multiples of a=7 will translate an F (=5) into an 0 (=a) because 7 does so. Examples are: 4 and 5 are relatively prime because gcd(4,5)=1. Apr 6, 2013 at 10:02 $\begingroup$ Well done!${}{}$ $\endgroup$ - Jyrki Lahtonen. ((5)=_____ as 1,2,3,4 are relative prime to 5. The following C++ program firstly determines the factors for an entered alphabet length M and secondly their multiples, the bad keys. 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).
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