For letters that do not occur in L, the alphabet function sL is undefined. How to recognize a Multiplicative ciphertext? color: #ffffff; 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. Multiply It! It was encoded MOD 26. Examples are: 4 and 5 are relatively prime because gcd(4,5)=1. Calculator Use Multiplication of positive or negative whole numbers or decimal numbers as the multiplicand and multiplier to calculate the product using long multiplication. 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. This is just what we wanted except that the answer 10 does not equal the desired cipher letter k on the computer. > m o ` a b c d e f g h i j k l 7 9 bjbjUU (> 7| 7| ). 23 This inverse modulo calculator calculates the modular multiplicative inverse of a given integer a modulo m. Multiplicative inverse vs. Modular multiplicative inverse warning 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. Does the increase of our alphabet length by 1 increase the number of unique encryptions obtained? 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). This is the reason why a=2 yields an ambiguous decryption. Write to dCode! Each letter is associated with its rank $ c $ in the alphabet (starting from 0). Definition of an inverse number: A number a-1 that yields 1 when multiplied by a is called the inverse of a. For an alphabet length of 26 this corresponds to 12 keys: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25. Method 2: Merged: In the alphabet, mod 22 is calculated because the alphabet contains 22 elements. Method 1: Separated: In each sub-alphabet, mod 16 is calculated (hex addition), since each sub-alphabet contains 16 elements, and it remains in the same partial alphabet from which the plaintext letter originates. Let us consider the above cipher text i.e. Lets summarize our discoveries. 1 The basic modulation function of a multiplicative cipher in Python is as follows . You could also define a to be a different good key. The multiplicative inverse of a modulo m exists if and only if a and m are coprime (i.e., if gcd(a, m) = 1). This calculator uses an adjugate matrix to find the inverse, which is inefficient for large matrices due to its recursion, but perfectly suits us. or . Since 36 is greater than the length of the used alphabet, 36 modulo 26 = 10 is calculated. color: #aaaaaa; How to pick a symmetric cipher for a given cipher text size? We just had to multiply each cipher letter by a-1. 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". 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. Note the difference in 'D' and 'd': The index value is the same, but the 'd' is. 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. Alternatively, the non-alphabet letters in the key and the plain text can also be filtered out to increase the security. ((15)=((3*5)=(3-1)*(5-1)=2*4=8 as 1,2,4,7,8,11,13,14 are relative prime to 15. The final formula uses determinant and the transpose of . 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? In order to be able to use the command setw() we have to include the iomanip.h library in #include . The odd multiples of 13 (i.e. This process repeats until M is reduced to 1 and therefore less than the smallest factor possible, 2. But the modular multiplicative inverse is a different thing, that's why you can see our inverse modulo calculator below. Copyright 1998 - 2023 CrypTool Contributors. 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. If we dont want to give an eavesdropper any additional information about our secret message, we would firstly either not use such characters at all or we would expand our alphabet length and encode them just like the other plain letters. 343 and 14 are not relative prime since gcd(343,14)=7. This online calculator tries to decode substitution cipher without knowing the key. Variant Beaufort cipher Base32 Hash function Morse code to text Z-Base-32 View Affine cipher Slope / a Step Down Step Up Example1: When using fractions, 5-1=1/5 is the inverse number to 5, 3-1=1/3 is the inverse number to 3, 3/2 is the inverse number to 2/3. 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. If you are able to invent a fast factoring algorithm, you will not have to worry about a future job. Modified 8 years, 6 months ago. The grey rows show what would be expected for the order, and the red one shows what your text gives for the order: If we use a value which is not co-prime, such as 2, we will not get unique characters for the mapping: Bib: @misc{asecuritysite_99257, title = {Multiplication Cipher}, year={2023}, organization = {Asecuritysite.com}, author = {Buchanan, William J}, url = {https://asecuritysite.com/coding/mult}, note={Accessed: May 02, 2023}, howpublished={\url{https://asecuritysite.com/coding/mult}} }. 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). The modular multiplicative inverse of an integer a modulo m is an integer b such that Example4: What is the inverse of 3 MOD 11? We can therefore always find a-1 for a given good key a. 5 7 11 13 17 19 23 25 29 31 35 The given examples show you the calculation process. 25, Encrypt We factor p1=2 yielding = 2*(2-1)*(3-1)*(5-1) = p1* (p1- 1)*( p2-1)*( p3-1). The formula for encrypting a letter x using the affine cipher is: y = ( a x + b) mod 26 And apparently the decryption formula is x = a 1 ( y b) mod 26 Where a 1 is the multiplicative inverse of a mod 26. Example5: If M=65=5*13=p*q, then the formula yields u(65) = (5-1)*(13-1) = 4*12 = 48. Each character is multiplied with this key and the corresponding letter is substituted. The key should be kept secret and only shared with authorized parties. How a top-ranked engineering school reimagined CS curriculum (Ep. The formula MOD(E$2*$B4,26) computes the number of the plain letter T, namely 19. Are these quarters notes or just eighth notes? Feedback and suggestions are welcome so that dCode offers the best 'Multiplicative Cipher' tool for free! Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. 14 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. Simply: Z26 = {0,1,2,3,, 24,25}. Lab 2. 3. For larger integers, however, dividing by every integer less than M slows the program down enormously. I found a-1 = 2 by simply testing the integers in Z5*={1,2,3,4}. 3.0.4224.0. . 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. This weirdness is not really weird. background-image: none; Just as the regular multiplication of two integers is commutative (i.e. Remember that a function, per definition, assigns to each x-value one particular y-value. 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"). The Multiplicative Cipher is an Affine cipher (ax+b) with the value b null (equal to 0), so a multiplication by a a. 5 2.5 Counting the Number of Good Keys for various Alphabet Lengths M An Introduction to the Euler Function. Thus, among those numbers that occur twice in the cipher code, 14, 17 and 20, we can eliminate the odd 17. 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. If so please go ahead and modify the following program. and all data download, script, or API access for "Multiplicative Cipher" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app! However, there are some additional integers that are not prime (i.e. However, it yields the original text. The answer is a simple No: Only those encryption systems that withstand all possible attacks are secure and thus useful. gcd(k,36)=1. Examples for property 1): 3 and 5 are two primes. What is the difference between "cipher" and "encryption"? Are they the odd numbers between 1 and 25? Identify blue/translucent jelly-like animal on beach. 21 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. Mathematically: a-1 * a = a * a-1 = 1. Coincidence? Therefore, each integer less than 29 is a good key MOD 29: Z29* = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28}. 1. Examples for property 3): 15 and 21 are products of two primes. What are the variants of the Multiplicative cipher. 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 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. 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. WAP in python to find out the additive and multiplicative inverse of an integer b using extended Euclidean algorithmof set Zn. 2. As a small example we consider Vigenre with the following two alphabets: In both cases, both the plaintext and the key should both consist of the text "0123456789abcdABCD". This is important because if the key is known by an unauthorized party, they will be able to decrypt the message. All symbols to be encrypted must belong to alphabet, Everyone who receives the link will be able to view this calculation, Copyright PlanetCalc Version: N (=13) translates into a for any even key a aswell because even keys N 4*13 = 2*(2*13) = 2*0 = 0 MOD 26, 6*13 = 3*(2*13) = 3*0 = 0 MOD 26, 8*13 = 4*(2*13) = 4*0 = 0 MOD 26, etc. Is there such a thing as "right to be heard" by the authorities? This allows us to force results to belong to the same alphabet. 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. 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. 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. 17 If a single character is encrypted by E(C) = (c * k) % 36 then possible keys k are numbers that are coprime to 36, ie. In order to create a n x n size matrix, keyphrase length should be square of an integer, i.e., 4, 9, 16. } Learn how PLANETCALC and our partners collect and use data. That is: It only takes a minute to sign up. Example the letter M (12th letter in this zero indexed alphabet) and key 3 would be 12 * 3 = 36. In the detailed representation of the alphabets (click on the "" -button), the alphabets can be edited in the short-write mode. >def unshift (key, ch): offset = ord (ch) - ASC_A return chr ( ( (key [0] * (offset + key [1])) % WIDTH) + ASC_A) Note The advantage with a multiplicative cipher is that it can work with very large keys like 8,953,851. Thus, being prime is not quite the reason for a good key, but almost. This means that the key should be a large, random number that is difficult to guess or factor. Can you? 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. The reason for that is that a prime number has per definition no prime divisor except for 1 and itself. QCCSWJUPQCCSW as an example to perform decryption using the multiplicative cipher. In some secret manner, the sender and the recipient had to agree on the encoding key a. For the M, 12*3=36 would result. A=65, B=66, C=67, .., Z=100, a=101, b=102, c=103, z=125. 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: In this chapter we will study the Multiplicative Cipher. ((28) = _____________________________ as 1,3,5,9,11,13,15,17,19,23,25,27 are relative prime to 28. Lets check why: 1*1=1 MOD 26 which explains a = a-1 = 1 (Big deal!). 28 equals 2*2*7 so that all the keys that are multiples of 2 or 7 do not and all non-multiples of 2 or 7 do produce a unique encryption: Z28* = {1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27} allowing only 12 different unique encryptions. Ask Question Asked 9 years, 11 months ago. Since we calculate MOD 26, thus dealing with integers from 0 to 25, we now have to find an integer a-1 among those integers that yields 1 MOD 26 . Therefore, we just have to add a number in order to get k=111. That would additionally require that the good keys form a commutative group with respect to addition. Examples for property 4): 24 and 28 are products of primes and prime powers. Moreover, since a=13 is a bad key its multiples 26, 39, must also be bad keys. where the operation of multiplication substitutes the operation of division by the modular multiplicative inverse. 23 A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. This is also the case when the letter is in the key. See the image attached below for a better understanding. One of the main advantages of the multiplicative cipher is its simplicity i.e. 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. However, it turns out to be indispensable when M is not the product of two primes, but say a product of a prime and a prime power. 21 is an inverse to 5 MOD 26, therefore 5 is inverse to 21 and the two 1s are mirrored over the diagonal line. Let s be such a reversible function. These ads use cookies, but not for personalization. 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. Multiplying such answers yields the number of good keys for any given alphabet length. 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)). It may be denoted as , where the fact that the inversion is m-modular is implicit. Thirdly, listing the good keys would be best done using C++ vectors or even C-style arrays which you might know. I want to show you an example where we used it already. Find mod of any numb. Thus they have the following restrictions: 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. 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. 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. Thank you! In fact, all the upper case letters on Excel are 65 numbers higher than those we are using, the lower case letters on Excel are 97 numbers above ours (i.e. Connect and share knowledge within a single location that is structured and easy to search. Multiplicative Simplified variant of the affine cipher Cipher Description Security About alphabets Plaintext: The quick brown fox jumps over the lazy dog. 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. 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). 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. 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. To verify this: 262 = 676 =1 MOD 27. For a check: the same eight integers 1,5,7,11,13,17,19,23 are relative prime to 30 and are thus the good keys for M=30. This is not very useful. 7 } The explanation of cipher, which is below the calculator, assumes an elementary knowledge of matrices. Affine cipher - encoder / decoder. What is the inverse of 7 MOD 11? dCode retains ownership of the "Multiplicative Cipher" source code. Firstly I have no idea how they derived this formula, but I think I have a general idea. 25 The solution shows the work for the Standard Algorithm. Therefore, a translation must take place, which can on the one hand transform letters in numbers and, conversely, re-generate letters again. This is not a useful encryption system since it may yield ambiguous messages. For example, if the key is 7 and the plaintext letter is s, the numerical value of s is 18, and so the ciphertext letter is (18*7) mod 26 = 22. div#home a:link { It uses genetic algorithm over text fitness function to break the encoded text. Therefore, an eavesdropper simply has to count letter frequencies to identify the most frequent cipher letter. I do not think any special calculator is needed in each of these cases. Example4: If M=39=3*13=p*q, then the formula yields u(39) = (3-1)*(13-1) = 2*12 = 24. Except for 2 and 13, all prime numbers less than 26 are among the keys (why do they have to?). ((25)=____________ as all integers from 1 to 24 except for 5,10,15,20 are relatively prime to 25. 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 Which cracking method should a code cracker use. If 2 would be used as key, then C=2 -> 2*2 = 4 -> so the character C is encoded as an E. Analogous, P=15 -> 15*2 = 30 -> 30 modulo 26 = 4 -> the character P would also be encoded as an E.Therefore, it is not possible to determine if an E in a ciphertext corresponds to a C or a P. In order to achieve a unique matching only keys that are co-prime to the length of the alphabet can be used. 2) Lastly, I want to explain the trick how I manage to encode not only a letter but a whole word or sentence if necessary. An alphabet[1] is an ordered set of all characters which can occur in a plaintext, a secret text, or the key. So the cipher text symbol will be w for the letter a in this case. Modular arithmetic is used; that is, all operations (addition, subtraction, and multiplication) are done in the ring of integers, where the modulus is m - the length of the alphabet. Cryptoanalysis - Cracking the Multiplication Cipher Just like the Cipher Caesar Cipher, the Multiplication is not secure at all. No provisions are made for high precision arithmetic, nor have the algorithms been encoded for efficiency when dealing with large numbers. You can verify this as follows: out of the 38 (=p*q-1) integers that are less than 39, we first cross out all the 12 (=13-1) multiples of 3 {3,6,9,12,15,18,21,24,27,30,33,36} and then cross out the 2 (=3-1) multiples of 13 {13,26} resulting in 38 12 2 = 24 good keys. No, it is not. This principle of finding the number of bad keys holds true for any alphabet length that is a prime power: There are M/p multiples of p less or equal to M, and therefore M/p - 1 many less than M. And we are only interested in those integers less than M since we are calculating MOD M which involves the integers 0 to M-1. Now that we have explored the criteria for unique encryptions and the number of good keys for certain alphabet lengths, it is the nature of Mathematics to generalize the observations and to set up an explicit formula for the number of unique encryptions based on the alphabet length M. For that purpose we have to consider 3 different cases of the alphabet length M 1) If M is a prime number: We observed in the previous section that the prime alphabet length M=29 yields u=28 unique encryptions. will translate the H (=7) into a (=0), because 5*7 = 35 = 0 MOD 35. Equivalently stated, 105 divided by 26 leaves a remainder of 1. 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. "<. 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. Note that you may need to run it several times to find completely accurate solution. 2) Learn how to compute and use the modular inverse to decode. 6 According to the definition in wikipedia, in classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. Back to the virus carrier message. color: #ffffff; 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. 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). You can try the sample button which uses a multiplication of 3, and a message of "knowledgeispower" gives enqohmjsmyctqomz. color: #ffffff; 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). Can I use an 11 watt LED bulb in a lamp rated for 8.6 watts maximum. The MOD 26 calculation leaves the 10 unchanged. Reciprocal (or) Multiplicative Inverse is: I.e. 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.

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