Aha, that realization helps a lot, since that also means that prime Ms produce M-1 unique encryptions. color: #ffffff; You should have realized that decoding means to undo the original multiplication. In order to create a n x n size matrix, keyphrase length should be square of an integer, i.e., 4, 9, 16. The determinant of the matrix should not be equal to zero, and, additionally, the determinant of the matrix should have a modular multiplicative inverse. The 18th character in the used alphabet corresponds to the S. The first character in the ciphertext therefore would be S. The remaining characters are encoded in the same way. 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. 12 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). Consider the letters and the associated numbers to be used as shown below , The numbers will be used for multiplication procedure and the associated key is 7. 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. Which language's style guidelines should be used when writing code that is supposed to be called from another language? How to encrypt using Multiplicative cipher? The key should not be easily guessable or should not be easily cracked. By using our site, you So in our case, it was GEEKSFORGEEKS, so it will become: Multiplicative Cipher text = QCCSWJUPQCCSW. The only disadvantage is that the minus sign itself has to be written as "---", so as not to be confused as a range operator. 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. Online calculator: Substitution cipher decoder - PLANETCALC Contributed by: Shawna Martell (March 2011) Open content licensed under CC BY-NC-SA Snapshots Modulus m. The mono-alphabetic substitution cipher provides the simplest form of cryptography, where the cipher alphabet is simply a rearrangement of the plaintext alphabet. affine cipher Thus, dividing is performed slightly different: instead of dividing by 5 or multiplying by 1/5, we first write 5-1 (instead of 1/5) where 5-1 now equals an integer and multiply both sides by that integer 5-1. Let s be such a reversible function. So are 2 and 3, 2 and 5, 3 and 10, 26 and 27, 45 and 16. Our alphabet length of 28 now yields how many unique encryptions? If a single character is encrypted by E(C) = (c * k) % 36 then possible keys k are numbers that are coprime to 36, ie. Each letter is associated with its rank $ c $ in the alphabet (starting from 0). What would you do? Note The advantage with a multiplicative cipher is that it can work with very large keys like 8,953,851. Parts of Long Multiplication 2 5 6 Multiplicand 3 2 Multiplier + 5 1 2 Partial Product + 7 6 8 Partial Product = 8 1 9 This shows that when using an encoding key that is one less than the alphabet length M, namely a = M-1, then the decoding key must also equal M-1, a-1 = M-1. Which ones are those? An affine cipher is a cipher belonging to the group of monoalphabetic substitution ciphers. 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)). In fact, I always have to subtract 101 from each entered lower case plain letter to get its corresponding number. 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. =CODE("a") yields 97). In general we have the: Formula for the number of good keys if M is a prime If the alphabet length M=p is prime, the number of good keys is u(p) = p-1. The first time the loop passes the line cout << cl; the translated plain letter pl that was read in as cin >> pl; before the while loop is output as its cipher letter cl. So on for each letter, the final encrypted message is ZIEZQ. The next two lines then show us that the variable false is defined as 0 and true as 1. That means: Because a=2 is a bad key all the multiples of a must be bad keys aswell. Lets investigate this in the following section. Online calculator: Hill cipher - PLANETCALC The only two keys that are inverse to themselves are 1 and 25, which means that the encoding key equals its decoding key. So there is an infinite number of possible keys, but many will give identical messages, because for a $ k $ key, then the $ k + 26 $ key gives an identical cipher. To have the solution, the right part of the linear diophantine equation should be a multiple of the . How many multiples of 3 will not produce a unique 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. Apr 6, 2013 at 10:02 $\begingroup$ Well done!${}{}$ $\endgroup$ - Jyrki Lahtonen. What 1 formula is used for the Affine Cipher Calculator? Variant Beaufort cipher Base32 Hash function Morse code to text Z-Base-32 View Affine cipher Slope / a Step Down Step Up I do not think any special calculator is needed in each of these cases. We will multiply MOD 26 as we are using the 26 letters of the English alphabet. Notice in the last row that all we need to know are the prime factors p of M without knowing how often they occur. Lets write down the Formula for the number of bad keys if M is a prime power b(M) = number of bad keys = M/p - 1. A corresponding warning is displayed. But the modular multiplicative inverse is a different thing, that's why you can see our inverse modulo calculator below. 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. background-color: #620E01; Simply: Z26* = {1,3,5,7,9,11,15,17,19,21,23,25}. Why is that? A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. does not work internally with letters, but with numbers. Affine Cipher is the combination of Multiplicative Cipher and Caesar Cipher algorithm. 3) ((p*q) = (p-1)*(q-1) for two distinct primes p and q. 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. ((3)=3-1=2 as 1 and 2 are relative prime to 3. 0 } In order to increase the probability of this, the alphabet is expanded, so its length becomes the prime integer. Our ultimate goal is not to develop a formula for the number of bad keys but rather for the number of good keys. Copyright 1998 - 2023 CrypTool Contributors. 39, 65, 91, ) have its equivalent key in a=13, another bad key, since 39=65=91=13 MOD 26. . The Affine Cipher uses modulo arithmetic to perform a calculation on the numerical value of a letter to create the ciphertext. I leave the translation from an upper case plain letter to a lower case cipher letter as an easy exercise for you. The letter A remains unchanged ans id always encoded A. Since each plain letter turns into 0 for a=0 and remains unchanged for a=1, we start with a=2. Coincidence? 9,15,21 and 25). Just as the regular multiplication of two integers is commutative (i.e. 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. 4 We obtain ((2*13) = ((2) *((13). 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. 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. 22 In linear algebra, an n-by-n (square) matrix A is called invertible if there exists an n-by-n matrix such that. Hill Cipher - Crypto Corner Please enable JavaScript to use all functions of this website. Decoding aam can either yield NAT or ANT as the plain text. In such case, divide M by that factor: M/=factor; and start checking M/factor for factors less than M/factoretc. Thus, x indeed is the modular multiplicative inverse of a modulo m. Everyone who receives the link will be able to view this calculation, Copyright PlanetCalc Version: These are valuable information for an eavesdropper that help cracking the message. By using this website, you agree with our Cookies Policy. 23 If you choose to do so, dont forget to also redefine the corresponding decoding key in int a=5, ainverse=21; . What really matters is not the alphabet length M but rather the number of multiples of the prime factors of M that are less than M: the less multiples of prime factors (as for the alphabet length of 27), the more as produce a unique encryption and vice versa. 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. 26, 52, 78, ) have its equivalent key in a=0, a very bad key, since 26=52=78=0 MOD 26. We get the following encoding and decoding table. This weirdness is not really weird. Consequently, the longer a cipher text, the easier the cipher E can be detected. 9 In fact, the sets of the encoding and decoding keys are identical. 8 Then we choose a matrix of n x n size, which will be the cipher's key. 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). GitHub - Mehul2205/Cryptography-Assignments: These are the lab We wont have to do it that way again since there is a much more straightforward method. Multiplicative Cipher - Online Decoder, Encoder RSA Calculator - College of Computing & Informatics 2) u(pn)= pn - pn-1, if M is a power of a prime M= pn. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. RSA Express Encryption/Decryption Calculator This worksheet is provided for message encryption/decryption with the RSA Public Key scheme. div#home a { Which number would that be? 5 This calculator uses Hill cipher to encrypt/decrypt a block of text. The three factors in the parentheses already have the same desired format, however, the single 2 destroys it. The command const is used as a safety feature in C++: both variables are constant and can never be modified in any program. color: #ffffff; Remember that a function, per definition, assigns to each x-value one particular y-value. ((15)=((3*5)=(3-1)*(5-1)=2*4=8 as 1,2,4,7,8,11,13,14 are relative prime to 15. Vice versa, the cost of detecting the most frequent cipher letter in the first approach is at the gain of producing only one plain text provided that the most frequent cipher letter turns out to be unique. What are the variants of the Multiplicative cipher. Moreover, we build the mathematical foundation to understand secure encryption systems such as the RSA encryption. 9 I will answer it at the end of this chapter in the Abstract Algebra section. Mathematically: a-1 * a = a * a-1 = 1. Or can we even increase the mere 12 unique encryptions for the Multiplication Cipher by varying the alphabet length? 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. Now, lets come to the highlight of this section: I will show you in a few steps how to compute ((M) for any M from one equation instead of combining the four properties? That means the key should not have any common factors with the alphabet or plaintext except for 1. The message is an alphabetical substitution, the frequency analysis should make it possible to find the most common letters. Multiplicative encryption uses a key $ k $ (an integer) and an alphabet. This is not a useful encryption system since it may yield ambiguous messages. To find a multiplicative inverse We need to find a number x such that: If we find the number x such that the equation is true, then x is the inverse of a, and we call it a^-1. Before Conversion: ABCDEFGHIJKLMNOPQRSTUVWXYZ After Conversion: XYZABCDEFGHIJKLMNOPQRSTUVW Age Calculators which we used in our virus carrier example. Note: This cipher is closely related to the. A reciprocal is one of a pair of numbers that when multiplied with another number equals the number 1. 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. See the image attached below for a better understanding. Step 4: So, once the calculation part is done now you can easily encrypt your given plain text. Therefore, we just have to add a number in order to get k=111. Or are they possibly the primes between 1 and 25? In this chapter we will study the Multiplicative Cipher. (Attacks). C = (a * P) mod 26 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. Remember to assign letters to blank spaces. They seem to not follow any apparent pattern. 2.1 Encryption using the Multiplication Cipher Instead of encoding by adding a constant number, we multiply each plain letter by our secret key a. 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. The first character G corresponds to the six. 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. We can therefore always find a-1 for a given good key a. or . Additionally, you will learn that the RSA Cipher uses prime numbers as well. Example5: Try it yourself! Equivalently stated, 105 divided by 26 leaves a remainder of 1. If M=60=22*3*5, then ((60) = ((22*3*5) using property __ yields = ((22)*((3*5) using property __ yields = ((22)*((3)*((5) using properties __ and __ yields = (22 21)*2*4 = 2*2*4 = 16. dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!A suggestion ? div#home a:link { 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. Example: Encrypt DCODE with the key k= 17 k = 17 and the 26-letter alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ 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. 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. However, it yields the original text. Equivalently stated: what product of a-1 and 5 equals 1 more than a multiple of 26 such as 27, 53, 79, 105, etc? All symbols to be encrypted must belong to alphabet, Everyone who receives the link will be able to view this calculation, Copyright PlanetCalc Version: 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). Alphabets (yes, there may be several: more below) can be described by a list L of letters. The key should be kept secret and only shared with authorized parties. Each character in the message is multiplied with this key. Modified 8 years, 6 months ago. Therefore, I need to subtract 101 from the 103 to get the desired 2, similarly, I again would have to subtract 101 from any plain letter b=102 to get the desired 1. However, converting 19 to its character does not yield the desired T. The T is stored as 84 which you could see by entering the Excel formula =CODE("T"). Take a moment now to verify the Rule for finding the decoding key a-1: 1) For a given good key a, find the unique 1 in the a-row, 2) From that 1 go all the way up that column, 3) The letters numerical equivalent that you hit on the very top is the inverse of a. 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. 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. Please, check our dCode Discord community for help requests!NB: for encrypted messages, test our automatic cipher identifier! background-image: none; The solution shows the work for the Standard Algorithm. Affine cipher: Encode and decode - cryptii 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. For an alphabet length of 26 this corresponds to 12 keys: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25. where c is the modular multiplicative inverse of a. The encryption process is done by multiplying the numerical value of each letter in the plaintext by the key and then taking the result modulo the key. Thus, among those numbers that occur twice in the cipher code, 14, 17 and 20, we can eliminate the odd 17. Except for 2 and 13, all prime numbers less than 26 are among the keys (why do they have to?). So the cipher text symbol will be w for the letter a in this case. That is, they mustn't have any common divisors. Moreover, since a=13 is a bad key its multiples 26, 39, must also be bad keys. Ask Question Asked 9 years, 11 months ago. You have 36 possible "characters" here. CacheSleuth - Multiplicative Cipher Subsequently, that difference is multiplied by the good key a=5 which I defined as such in int a=5. We just had to multiply each cipher letter by a-1. I found a-1 = 2 by simply testing the integers in Z5*={1,2,3,4}. You can observe this order-doesnt-matter rule in the original 26x26 multiplication table: The diagonal line from the top left to the bottom right forms a reflection line. Then the if-condition if (ans=='e') is fulfilled so that we enter the encoding part of the program. 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. For example, Caesar cipher using a left rotation of three places, equivalent to a right shift of 23 as given below. ((28) = _____________________________ as 1,3,5,9,11,13,15,17,19,23,25,27 are relative prime to 28. 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. a=4 is inverse to itself modulo 5 since a * a-1 = 4 * 4 = 16 = 1 MOD 5. Furthermore it makes not much sense to consider numbers not between 1 and 36, because of the modulo. I.e. } Even though this cipher seems to be more complex than the Caesar cipher, it is not more secure. Alternatively, the non-alphabet letters in the key and the plain text can also be filtered out to increase the security. or ? Examples are: 4 and 5 are relatively prime because gcd(4,5)=1. 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?