multiplicative cipher calculator multiplicative cipher calculator

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. Therefore, a translation must take place, which can on the one hand transform letters in numbers and, conversely, re-generate letters again. 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. 3) u(p*q) = (p-1)*(q-1), if M is a product of two primes M=p*q. 21 Determining the bad keys for a given alphabet length M is a perfect task for a computer. How would anyone ever break even this basic, amateurish cipher/encryption scheme? Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. 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 when multiplied by 5: a-1 * 5 = 1 MOD 26. Each letter is enciphered with the function (ax + b) mod 26. background-image: none; acknowledge that you have read and understood our, Data Structure & Algorithm Classes (Live), Data Structures & Algorithms in JavaScript, Data Structure & Algorithm-Self Paced(C++/JAVA), Full Stack Development with React & Node JS(Live), Android App Development with Kotlin(Live), Python Backend Development with Django(Live), DevOps Engineering - Planning to Production, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Interview Preparation For Software Developers. This weirdness is not really weird. Try it for yourself! 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. Which ones are those? 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. It is easy to implement and easy to understand, and it does not require any large amount of computational power. 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). I'm learning and will appreciate any help. This requires additional meta-information of the letters that must be recorded before encryption. 1) Learn how to decode the Multiplication Cipher. 3 color: #ffffff; 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. 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. The encryption function looks like this: f (x)= ax+b mod . That is The multiplicative cipher has little interest, but it is often used for learning computer science and ciphers. There are several way to implement the inversion and the affine transformation described in the AES to get the final SBox. 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. Try it! Options regulate the case when a letter does not appear in any alphabet: it is not encrypted, but transferred directly to the output. dCode retains ownership of the "Multiplicative Cipher" source code. or ? It converts to the plain letter number 26 so that we now have to encrypt MOD 27. //Author: Nils Hahnfeld 10-16-99 //Program to determine ((M)using M*(1-1/p1)*(1-1/p2)* #include #include void main() { int factor, M, m; float phi; clrscr(); cout << "This programs uses M*(1-1/p1)*(1-1/p2)* to calculate phi(M). The command const is used as a safety feature in C++: both variables are constant and can never be modified in any program. For classical methods, the alphabet often consists only of the uppercase letters (A-Z). 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. color: #ffffff; The basic modulation function of a multiplicative cipher in Python is as follows . Then we perform the reverse operations performed by the encryption algorithm. Generally: An alphabet of length M has the keys: ZM = {0,1,2,3,, M-2,M-1} 2) Now, the good keys are the ones that are relative prime to 26 as listed above and are denoted as Z26*. The basic modulation function of a multiplicative cipher in Python is as follows . div#home a:hover { This program is an extension of the previous simple factoring program. How do we deal with non-letters? The index of coincidence is unchanged from plain text. It is a-1=4 since 3*4 = 12 = 1 MOD 11. That means: Because a=2 is a bad key all the multiples of a must be bad keys aswell. In order to be able to use the command setw() we have to include the iomanip.h library in #include . Learn more. The following steps take place: In the example, an overflow has occurred in the third letter, so that modulo |L| = 4 is calculated. Thus, the encryption process is a Caesar cipher merged with a multiplication cipher. Note: This cipher is closely related to the. Convert each letter in the plain text alphabet to a corresponding integer in the range of 0 to m -1; 2. It uses genetic algorithm over text fitness function to break the encoded text. By using this website, you agree with our Cookies Policy. M23456789101112131415161718192021( (M)12242648121041268816618812 Similar to our notation, the properties of Eulers (-function that computes the number of integers that are relatively prime to M and wrote similarly to our notation: Eulers (-function: 1) ((p) = p-1 for a prime p. 2) ((pn) = pn - pn-1 for a prime power pn. To have the solution, the right part of the linear diophantine equation should be a multiple of the . We obtain ((2*13) = ((2) *((13). Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 24 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. First of all, you need to know which one of the 12 good keys was used. 5 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)). Let us understand this by implementing a simple example using the Multiplicative Cipher. 1 20 Longer messages reveal the most the letter e equivalent, however, this is not necessarily so for our message. Although the function is well-defined when a letter occurs more than once, this makes little sense in encryption algorithms, since the reversibility suffers. To show this, let's look at this equation: This is a linear diophantine equation with two unknowns; refer to Linear Diophantine Equations Solver. Those are the 8 integers 3, 6, 9, 12, 15, 18, 21, 24. 2) If M is a prime power, M=pn: Now lets look back at M=27 as an example where we only have the one prime factor p=3, such that M=33. The Affine Cipher uses modulo arithmetic to perform a calculation on the numerical value of a letter to create the ciphertext. Here both approaches are treated: for separate partial alphabets and for a memorized alphabet. Modified 8 years, 6 months ago. To decode the above virus carrier message we found the inverse of a=5 through a clever check of the products of a and a-1 that produced one more than multiples of 26. 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 This is not a useful encryption system since it may yield ambiguous messages. 1. However, it yields the original text. So it will look like this after calculation. Affine Cipher is the combination of Multiplicative Cipher and Caesar Cipher algorithm. Apr 6, 2013 at 10:46 . Therefore, no matter how he decides to crack the cipher text, it wont take long. Learn how PLANETCALC and our partners collect and use data. This, however, limits readability. The MOD 26 calculation leaves the 10 unchanged. 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. Why did US v. Assange skip the court of appeal? The encrypted text is the smallest digit of an addition of plaintext and key when both are hexadecimal digits. Our implementation of Vigenre, Beaufort, etc. 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. 1 How a top-ranked engineering school reimagined CS curriculum (Ep. QCCSWJUPQCCSW as an example to perform decryption using the multiplicative cipher. Copyright 1998 - 2023 CrypTool Contributors. Notice in the last row that all we need to know are the prime factors p of M without knowing how often they occur. 25, Encrypt 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. Let s be such a reversible function. Examples for property 2): 8 and 25 are prime powers. background-color: #620E01; div#home a:link { Back to the virus carrier message. This brute force approach will work fast enough for integers M that have 10 digits or less. 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. Code Other frequent letters such as T, A, O and N occurring with about (8%) might be of further help to crack the cipher text. The 14 as the possible cipher E then tells him to test the keys a=10 and a=23. "Ordered" means that sorting is possible and we can speak of the n-th character of an alphabet. In case you wonder why the discussion of cracking codes is made public; why is it not kept secret to maintain the security of ciphers? 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. 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. It is suitable for small-scale applications but not recommended for practical purposes. I will explain the usage of letter frequency as a very important means to crack cipher codes in the next chapter. 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.e., for M=27 we just need to know that 3 is a prime divisor of 27 but not how often it divides 27. 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). Combining this fact with the fact that each key a possesses a decoding key a-1, the set of the good keys forms a commutative group with the unit element 1. Key is the matrix; however, it is convenient to use the key phrase, which is transformed into the digit representation and matrix. (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. For the fraction a/b, the multiplicative inverse is b/a. 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. Lets check this for an alphabet length of M=29. Notice, that all we need to find are the different primes, say p1, p2,, pn, as our explicit formula for the number of unique encryptions appears to be: Formula for the number of good keys for any alphabet length M: For an alphabet length M, there are ((M) = M * (1- 1/p1) * (1- 1/p2) ** (1- 1/pn) good keys where each pi is a prime divisor of M. It is really enjoyable to use this simple formula as we just need to find all prime divisors of M and dont have to worry about how often they occur. v l X X X These calculations were correct but almost required a calculator. A corresponding warning is displayed. Example5: If M=65=5*13=p*q, then the formula yields u(65) = (5-1)*(13-1) = 4*12 = 48. This website would like to use cookies for Google Analytics. It describes the multiplicative property of (. The only two keys that are inverse to themselves are 1 and 25, which means that the encoding key equals its decoding key. Viewed 4k times . 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. That means the key should not have any common factors with the alphabet or plaintext except for 1. Lets check why: 1*1=1 MOD 26 which explains a = a-1 = 1 (Big deal!). 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. Also, there is no general match on how to handle digits or special characters. The multiplicative cipher is a simplification of the Affine cipher. The given examples show you the calculation process. This yields the correct plain text: Cipher textanromrjukahhouh013171412179201007714207 0131981819742017178417PLAIN TEXTANTISTHECARRIER As you can see, detecting the most frequent cipher letter is of enormous help in cryptography. In the next chapter, I will show you one principle of increasing the safety of a cipher code. As 29 is prime, it has no divisors except for 1 and 29 and thus there are no multiples as bad keys. Counter examples are: 45 and 18 are not relative prime since gcd(45,18)=9 and not 1. Invented by Lester S. Hill in 1929, it was the first polygraphic cipher in which it was practical (though barely) to operate on more than three symbols at once. 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. The reciprocal of a number x is a number, which, when multiplied by the original x, yields 1, called the multiplicative identity. To do so, we have to look at the encryption equation C=a*P MOD 26 and solve it for the desired plain text letter P. In order to solve an equation like 23=5*P for P using the rational numbers, we would divide by 5 or multiply by 1/5 to obtain the real solution P=23/5. 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. A=65, B=66, C=67, .., Z=100, a=101, b=102, c=103, z=125. ~=.., $=.. etc. The trick is now that if we enter more than one letter all but the first entered letter are buffered (which means temporarily stored in the computers RAM) until read in in cin >> pl; inside the while loop. 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. We will multiply MOD 26 as we are using the 26 letters of the English alphabet. 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. Examples are: 4 and 5 are relatively prime because gcd(4,5)=1. Information Security Stack Exchange is a question and answer site for information security professionals. ((21)=________________________ as 1,2,4,5,8,10,11,13,16,17,19,20 are relative prime to 21. Simply: Z26* = {1,3,5,7,9,11,15,17,19,21,23,25}. So the cipher text symbol will be w for the letter a in this case. 17 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}} }. where the operation of multiplication substitutes the operation of division by the modular multiplicative inverse. Mathematically: a-1 * a = a * a-1 = 1. Alphabets (yes, there may be several: more below) can be described by a list L of letters. Lets summarize our discoveries. The answer is a simple No: Only those encryption systems that withstand all possible attacks are secure and thus useful. 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. 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. Multiplicative Simplified variant of the affine cipher Cipher Description Security About alphabets Plaintext: The quick brown fox jumps over the lazy dog. Now when a=25, we have: 25*25 = 625. The statements while(cl!='~') and cout << cl; cin >> pl; are in charge of it. 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 modular multiplicative inverse of a modulo m can be found with the Extended Euclidean algorithm. For an alphabet length of 26 this corresponds to 12 keys: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25. Our alphabet length of 28 now yields how many unique encryptions? Does the increase of our alphabet length by 1 increase the number of unique encryptions obtained? So on for each letter, the final encrypted message is ZIEZQ. You now understand why cryptographers have an affection for prime numbers. 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). In fact, any character is stored as a number: i.e. Certainly, it might be a double encoded message that has to be decoded twice, possibly using two different keys or even two different ciphers. h2 { Our good-key-criterion declares those integers to be good keys that are relative prime to 27. Subsequently, ( is computed by property 1) if such factors are primes or by property 2) if they are prime powers. Hey, this shows a great way to produce more unique encryptions which of course makes life harder for an eavesdropper: Recommendation for more security: Choose the alphabet length M to be a prime number to make cracking the cipher text more difficult. 3. In, this way you can implement Encrypt a plain text and Decrypt a cipher text for Multiplicative cipher in cryptography. If we extract those rows with the good keys a = 1,3,5,7,9,11,15,17,19,21,23,25 and their corresponding columns, we obtain: 13579111517192123251135791115171921232533915211719255111723551525919323717111217721923112511531751999119113215231572517111173252117951231915151519231591721253711171725715235213111919191951731512511239217212111117723319925155232317115251971211593252523211917151197531 This reduced table shows i.e. Not every key phrase is qualified to be the key; however, there are still more than enough. 15 This formula can be simplified into the product of two factors.