Convert each group of numbers into column matrices. 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. Example: Encrypt DCODE with the key $ k = 17 $ and the 26-letter alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ. Which keys now yield a unique encryption? Simply by looking at the table, we find that the following keys (whose rows are bold) produce a unique encryption and therefore call them the good keys: a = 1,3,5,7,9,11,15,17,19,21,23,25 Why those and what do they have in common? However, when using MOD arithmetic and solving 23=5*P MOD 26, we dont deal with fractions but only integers. } 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. 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. Example2: M=81=34 has again 3 as the only prime divisor and thus b = 81/3 1 = 34/3 1 = 33 1 = 26 bad keys. Key is the matrix; however, it is convenient to use the key phrase, which is transformed into the digit representation and matrix. 5 The explanation of cipher, which is below the calculator, assumes an elementary knowledge of matrices. Since 36 is greater than the length of the used alphabet, 36 modulo 26 = 10 is calculated. 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. This is also the case when the letter is in the key. 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). Therefore, no matter how he decides to crack the cipher text, it wont take long. Aha, that realization helps a lot, since that also means that prime Ms produce M-1 unique encryptions. Options: Multiplier: filter whitespace characters group 5 characters filter non-alphabet characters convert to first alphabet Does the increase of our alphabet length by 1 increase the number of unique encryptions obtained? In conclusion, we can say that multiplicative cipher is a simple encryption technique that can be easily implemented. 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. 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) <. This encoding and decoding is working based on alphabet shifting & transforming the letters into numbers . Decoding aam can either yield NAT or ANT as the plain text. 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"). 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. 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. For the M, 12*3=36 would result. To find the multiplicative inverse of a real number, simply divide 1 by that number. 8 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. Each letter is associated with its rank $ c $ in the alphabet (starting from 0). Finally, I have to add the usual 65 = A (why?) For example if we use "abcdefghijklmnopqrstuvwxyz" and a multiplier of 3, gives "adgjmpsvybehknqtwzcfilorux". The final formula uses determinant and the transpose of . 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. Here both approaches are treated: for separate partial alphabets and for a memorized alphabet. A longer alphabet produces less unique encryptions. In linear algebra, an n-by-n (square) matrix A is called invertible if there exists an n-by-n matrix such that. The encryption of upper case plain letter works similarly except that I have to subtract A=65 (instead of a=101 as above) to obtain our desired plain letter number. Extracting arguments from a list of function calls. Identify blue/translucent jelly-like animal on beach. The use of several alphabets does not require the algorithms to distinguish between upper and lower case letters. For an alphabet length of 26 this corresponds to 12 keys: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25. Try it for yourself. WAP in python to find out the additive and multiplicative inverse of an integer b using extended Euclidean algorithmof set Zn. Affine cipher - Modular multiplicative inverse. A corresponding warning is displayed. Finally I understand how to calculate the modular multiplicative inverse :) $\endgroup$ - np00. Find mod of any numb. 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. In fact, I always have to subtract 101 from each entered lower case plain letter to get its corresponding number. 23 } 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. Example4: For M= 34 =81, we get u(81) = 34 - 33 = 81 27 = 54. While deriving the formula for M=60=22*3*5 in the left column I will deduce simultaneously the explicit formula for M=p12*p2*p3 with p1 being the first prime factor 2, p2 being the second prime factor 3 and p3 being the third prime factor 5 in the right column. For illustration purposes we use the message "GEHEIMNIS" and the key 3. I do not think any special calculator is needed in each of these cases. 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 basic formula to be used in such a scenario to generate a multiplicative cipher is as follows . No provisions are made for high precision arithmetic, nor have the algorithms been encoded for efficiency when dealing with large numbers. Step 4: So, once the calculation part is done now you can easily encrypt your given plain text. Right, we have to add 101 to the 10 which we do by adding a=101 in cl='a' + (a*(pl -'a'))%26. 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. In an additive cipher, the cipher alphabet is a shift of the plaintext alphabet. The inverse function returns the n-th character for a number n in L. To n, the length of the list L is added or subtracted as often as necessary until the index lies in the list. Which language's style guidelines should be used when writing code that is supposed to be called from another language? Consider an alphabet length of M=35: the bad key a=5 (why?) . Do they? 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. 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". Furthermore it makes not much sense to consider numbers not between 1 and 36, because of the modulo. That is why the English alphabet in the calculator above is expanded with space, comma, and dot up to 29 symbols; 29 is a prime integer. I first subtract 65 =A and then multiply that difference by the good key a=5 yielding 10 again. 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. 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. 2. Lets consider two options: Option 1: Cracking the cipher code using letter frequencies If plain letters are replaced by cipher letters the underlying letter frequencies remain unchanged. Thus, the number of bad keys can simply be found by dividing the alphabet length M by the only prime divisor p and subtracting 1 from that fraction: M/p - 1. For M=31 we have u(31)=30. I.e. 26, 52, 78, ) have its equivalent key in a=0, a very bad key, since 26=52=78=0 MOD 26. background-color: #620E01; So on for each letter, the final encrypted message is ZIEZQ. Back to the virus carrier message. 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. Remember that a function, per definition, assigns to each x-value one particular y-value. Let us consider the above cipher text i.e. div#home a:hover { For example, take the list L = "ABCD", whose length is 4. Now, how do you decrypt the above message? A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. 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. Why does Acts not mention the deaths of Peter and Paul? 3) ((p*q) = (p-1)*(q-1) for two distinct primes p and q. does not work internally with letters, but with numbers. 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). where the operation of multiplication substitutes the operation of division by the modular multiplicative inverse. 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). } Also, each B and each M turn into 2 (=c) since 2*1 = 2 MOD 26 and 2*14 = 28 = 2 MOD 26. What is the symbol (which looks similar to an equals sign) called? Multiplicative Simplified variant of the affine cipher Cipher Description Security About alphabets Plaintext: The quick brown fox jumps over the lazy dog. 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. Example3: Doing arithmetic MOD 7, the inverse of a=3 is a-1 = 5 because a * a-1 = 3*5 = 15 = 1 MOD 7. If you are able to invent a fast factoring algorithm, you will not have to worry about a future job. That means: Because a=2 is a bad key all the multiples of a must be bad keys aswell. Similarly, the multiples of a=7 will translate an F (=5) into an 0 (=a) because 7 does so. 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? Before considering such encoding techniques, we go ahead and check if the other frequent number, 20, is the cipher E. Checking the E column, we can see that the possible two keys are the bad one a=18 and the good one a=5. Read about it on wiki, I will provide only example. a=13 yields an ambiguous message since each even plain letter is translated into a (=0): a=13 even letters 13*0 = 0 MOD 26, 13*2 = 0 MOD 26, 13*4 = (13*2) * 2 = 0 * 2 = 0 MOD 26, 13*6 = (13*2) * 3 = 0 * 3 = 0 MOD 26, etc. 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. PLAIN LETTERNATANTSecret key a=2130190131900120012Cipher letteraamaam You can see the dilemma of this message. Affordable solution to train a team and make them project ready. Equivalently stated: what product of a-1 and 5 equals 1 more than a multiple of 26 such as 27, 53, 79, 105, etc? This formula can be simplified into the product of two factors. 21 We wont have to do it that way again since there is a much more straightforward method. 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. Our good-key-criterion declares those integers to be good keys that are relative prime to 27. Firstly I have no idea how they derived this formula, but I think I have a general idea. However, it can be simplified further using the fact that we are considering here alphabets of length M that are powers of a prime p: M=pn for some positive integer n. Thus, our formula simplifies to: u(M) = pn pn/p which simplifies further to = pn - pn-1. 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. Are there any canonical examples of the Prime Directive being broken that aren't shown on screen? or ? Now every row contains exactly one 1 revealing that there exists an inverse for each a which is precisely the reason why those as are the good keys. In fact, any character is stored as a number: i.e. (I.e. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Mathematically, calculate the modular inverse $ k^{-1} $ of the key modulo 26 and apply the calculation for each letter: Example: The key $ 17 $ has the inverse modulo 26 of the value $ 23 $ so Z (index 25) becomes $ 25 \times 23 \mod 26 \equiv 3 $ and 3 corresponds to D in the alphabet. Say you first want to encode the letter c then you have to enter e when asked. Try to explain this, than continue reading! An alphabet[1] is an ordered set of all characters which can occur in a plaintext, a secret text, or the key. 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. 6*3=18. 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. 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. To ensure that no two letters are mapped to the same letter, a and m must be coprime. The basic modulation function of a multiplicative cipher in Python is as follows . Each character is multiplied with this key and the corresponding letter is substituted. 3.0.4224.0, The greatest common divisor of two integers, The greatest common divisor and the least common multiple of two integers, Solution of nonhomogeneous system of linear equations using matrix inverse. Example5: If M=65=5*13=p*q, then the formula yields u(65) = (5-1)*(13-1) = 4*12 = 48. 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. Ask Question Asked 9 years, 11 months ago. In this chapter we will study the Multiplicative Cipher. Thus, safer encryptions are necessary. For letters that do not occur in L, the alphabet function sL is undefined. where c is the modular multiplicative inverse of a. 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. Consequently, the longer a cipher text, the easier the cipher E can be detected. Additional restrictions to the key are imposed by the need to decrypt encrypted text :). 2) Learn how to compute and use the modular inverse to decode. But the modular multiplicative inverse is a different thing, that's why you can see our inverse modulo calculator below. 24 Multiplying such answers yields the number of good keys for any given alphabet length. 3.0.4224.0. 0 Alphabets (yes, there may be several: more below) can be described by a list L of letters. One of the main advantages of the multiplicative cipher is its simplicity i.e. 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)). As an example, lets encode and decode NAT and ANT. For instance, we encoded with a=3, then the encryption function was C=3*P mod 26. The basic task behind the multiplicative cipher is to use a large prime number as a multiplication key, and then use the modular arithmetic of the integers modulo, the key to encode and decode the plaintext. Reminder : dCode is free to use. Builds the Affine Cipher Translation Algorithm from a string given an a and b value This calculator has 3 inputs. Each character from the plaintext is always mapped to the same character in the ciphertext as in the Caesar cipher. If a = 1, the Affine cipher is equivalent of a Caesar cipher. The solution can be found with the Extended Euclidean algorithm. Except for 2 and 13, all prime numbers less than 26 are among the keys (why do they have to?). div#home { Please enable JavaScript to use all functions of this website. 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. Or are they possibly the primes between 1 and 25? It uses genetic algorithm over text fitness function to break the encoded text. 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. Except explicit open source licence (indicated Creative Commons / free), the "Multiplicative Cipher" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Multiplicative Cipher" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) ((5)=_____ as 1,2,3,4 are relative prime to 5. However, there is no 7 the numerical equivalent of letter h - in the E column. Now the cipher letter cl equals k and we can end the lower case encoding. 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. What is the inverse of 7 MOD 11? Ok, lets continue with the encoding part. You now understand why cryptographers have an affection for prime numbers. The answer is a simple No: Only those encryption systems that withstand all possible attacks are secure and thus useful. Notice that we found the good keys indirectly. The encrypted text is the smallest digit of an addition of plaintext and key when both are hexadecimal digits. It would take quite a long time for a computer to brute-force through a majority of nine million keys. dCode retains ownership of the "Multiplicative Cipher" source code. Tool to decrypt/encrypt with multiplicative encryption, a substitution cipher based on a multiplication operation. Well, I leave all the entered non-letters such as ! background-image: none; If a single character is encrypted by E(C) = (c * k) % 36 then possible keys k are numbers that are coprime to 36, ie.gcd(k,36)=1.Furthermore it makes not much sense to consider numbers not between 1 and 36, because of the modulo. This is important because if the key is known by an unauthorized party, they will be able to decrypt the message. 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. Why did US v. Assange skip the court of appeal? Viewed 4k times . I.e., for M=27 we just need to know that 3 is a prime divisor of 27 but not how often it divides 27. 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). 12 Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. 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. div#home a:visited { Parts of Long Multiplication 2 5 6 Multiplicand 3 2 Multiplier + 5 1 2 Partial Product + 7 6 8 Partial Product = 8 1 9 3 * 9 = 9 * 3 =27) the MOD- multiplication is commutative (3 * 9 = 9 * 3 = 1 MOD 26). Options regulate the case when a letter does not appear in any alphabet: it is not encrypted, but transferred directly to the output. Example3: Now, it is your turn. If we had a video livestream of a clock being sent to Mars, what would we see? I'm learning and will appreciate any help. color: #ffffff; for M=29 we have u(29)=28. We get the following encoding and decoding table. It is easy to implement and easy to understand, and it does not require any large amount of computational power. =CODE("a") yields 97). Cite as source (bibliography):
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