multiplicative cipher calculator multiplicative cipher calculator
} Back to the virus carrier message. Well, I leave all the entered non-letters such as ! What is the symbol (which looks similar to an equals sign) called? So, lets understand why the bad keys a = 2,4,6,8,10,12,13,14,16,18,20,22,24 dont produce a unique encryption. 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. We wont have to do it that way again since there is a much more straightforward method. Say you first want to encode the letter c then you have to enter e when asked. In order to simplify the representation of the alphabets, the following abbreviation has been introduced: The minus sign in the following letter 1-letter 2 is extended to all the letters between the two flanking letters. Apr 6, 2013 at 10:46 . 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. The reciprocal of a number x is a number, which, when multiplied by the original x, yields 1, called the multiplicative identity. This calculator uses an adjugate matrix to find the inverse, which is inefficient for large matrices due to its recursion, but perfectly suits us. width: max-content; ((3)=3-1=2 as 1 and 2 are relative prime to 3. Try it for yourself. 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 7 11 13 17 19 23 25 29 31 35 So, we are left with determining the decoding key a-1 knowing the original encoding key a. Each character from the plaintext is always mapped to the same character in the ciphertext as in the Caesar cipher. color: #ffffff; 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. Convert each letter in the plain text alphabet to a corresponding integer in the range of 0 to m -1; 2. 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). 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. How to recognize a Multiplicative ciphertext? Do they? The 14 as the possible cipher E then tells him to test the keys a=10 and a=23. Ask Question Asked 9 years, 11 months ago. Let us understand this by implementing a simple example using the Multiplicative Cipher. Are the used 12 unique encryptions a set number? 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. 4 The bad key a=2 yields an ambiguous message as we saw in the introductory example: each A turns into 0 (=a) since 2*0 = 0 MOD 26 just as each N turns into 0 since 2*13 = 26 = 0 MOD 26. Multiplicative encryption uses a key $ k $ (an integer) and an alphabet. However, there are some additional integers that are not prime (i.e. Why did US v. Assange skip the court of appeal? We can see in the table that an A will always translate into 0 (=a) since the product of any such key a with 0 (=A) yields 0. By substitution, in fact, during encryption each letter is associated with only one other, by calculating all the possible associations (by encrypting the 26 letters of the alphabet) then it is possible to deduce an alphabet substitution that will serve as a decryption table. So it will look like this after calculation. 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. See the image attached below for a better understanding. Here is how: u = (p*q - 1) - (p-1) (q 1) getting rid of the first two parentheses yields = p*q -1 - p + 1 (q 1) the two 1s cancel each other out yielding = p*q p (q 1) factoring the p yields = p*(q-1) (q 1) (q-1) in both terms can be factored yielding = (q-1) * (p 1) which can also be written as = (p-1) * (q 1) Formula for the number of good keys if M is the product of two primes: The number of good keys is u(M) = u(p*q) = (p-1)*(q-1). 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)). rev2023.5.1.43405. Can you? Which was the first Sci-Fi story to predict obnoxious "robo calls"? Finally I understand how to calculate the modular multiplicative inverse :) $\endgroup$ - np00. Text is divided into blocks of size n, and each block forms a vector of size n. Each vector is multiplied by the key matrix of n x n. The result, vector of size n, is a block of encrypted text. Note: This cipher is closely related to the. color: #aaaaaa; It surely acquires this simple form for any number of primes or prime powers. Lets add a dot to our alphabet to denote the end of a sentence in the original message. 23 Extracting arguments from a list of function calls. This is important because if the key is known by an unauthorized party, they will be able to decrypt the message. Other frequent letters such as T, A, O and N occurring with about (8%) might be of further help to crack the cipher text. Alphabets (yes, there may be several: more below) can be described by a list L of letters. Since one can be divided without remainder only by one, the equation above has the solution only if . We obtain ((2*13) = ((2) *((13). It converts to the plain letter number 26 so that we now have to encrypt MOD 27. For separate partial alphabets the following results: For a merged alphabets, the encrypted text is "02468ACEacACEae024". 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. This is not a useful encryption system since it may yield ambiguous messages. 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 explanation of cipher, which is below the calculator, assumes an elementary knowledge of matrices. Therefore, the set of all encoding keys must equal the set of all decoding keys. The following steps take place: In the example, an overflow has occurred in the third letter, so that modulo |L| = 4 is calculated. Let s be such a reversible function. 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! We can combine these two criteria into one easy criterion. 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). ~=.., $=.. etc. How many multiples of 3 will not produce a unique encryption? In fact, the sets of the encoding and decoding keys are identical. In fact, any character is stored as a number: i.e. According to the definition in wikipedia, in classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. Information Security Stack Exchange is a question and answer site for information security professionals. Remember that the first 3 ciphers are meant to familiarize you with basic encryption systems. You have 36 possible "characters" here. It is actually less secure than the Caesar cipher because the number of possible keys is smaller. 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. Examples for property 3): 15 and 21 are products of two primes. 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. Thank you! 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. This modulo calculator performs arithmetic operations modulo p over a given math expression. We can therefore always find a-1 for a given good key a. An affine cipher is a cipher belonging to the group of monoalphabetic substitution ciphers. That is: 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. The modular multiplicative inverse of an integer a modulo m is an integer b such that , A corresponding warning is displayed. Therefore, no matter how he decides to crack the cipher text, it wont take long. The modular multiplicative inverse of a modulo m can be found with the Extended Euclidean algorithm. div#home a:visited { Even though this cipher seems to be more complex than the Caesar cipher, it is not more secure. It uses genetic algorithm over text fitness function to break the encoded text. ((24) = ((23 *3) = ((23)*((3) = (23-22)*(3-1) = 4*2 = 8 as 1,5,7,11,13,17,19,23 are relative prime to 24. The three factors in the parentheses already have the same desired format, however, the single 2 destroys it. 2.2 Decryption of the Multiplication Cipher Now that the virus carrier message was encoded in a unique manner how can it be decoded? b ^ ^ ^ 8 ^ a G n n n n n R R R f h h h h h h $ u R ` R R R n n n n 7 R j n n n f R f k \ ^ % n n `d P ^ v$ .$ r % T 0 G $ r 2 % n n n n Chapter 2 Multiplicative Cipher In this chapter we will study the Multiplicative Cipher. Network Security: Multiplicative InverseTopics discussed:1) Explanation on the basics of Multiplicative Inverse for a given number.2) Explanation on the basi. 2) Learn how to compute and use the modular inverse to decode. I do not think any special calculator is needed in each of these cases. No, 13 is missing. Why is that? However, there is no 7 the numerical equivalent of letter h - in the E column. an idea ? So which ones do? A function that performs this is called an alphabet function. (Attacks). Modulo Arithmetic & Ciphers. unchanged so that you can detect the format of the original message easier. Note The advantage with a multiplicative cipher is that it can work with very large keys like 8,953,851. Ok, lets continue with the encoding part. While using Caesar cipher technique, encrypting and decrypting symbols involves converting the values into numbers with a simple basic procedure of addition or subtraction. a feedback ? dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!A suggestion ? Technically 1 too, but this would be no change from plaintext. 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. This website would like to use cookies for Google Analytics. 6 I first subtract 65 =A and then multiply that difference by the good key a=5 yielding 10 again. The theory can be found after the calculator. Example 1: For M=27=33: Inserting 3 for p and 3 for n in pn - pn-1 yields u(27) = 33 - 32 = 27-9 = 18 which is just what we wanted. ((5)=_____ as 1,2,3,4 are relative prime to 5. 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: Exporting results as a .csv or .txt file is free by clicking on the export icon The conversion to letters takes place modulo to the alphabet length: If a 1 is added to the last character, the result of the sum is the first character of the alphabet. Equivalently stated: what product of a-1 and 5 equals 1 more than a multiple of 26 such as 27, 53, 79, 105, etc? However, subtracting the number of bad keys from the number of all possible keys (=M-1) yields the number of good keys. 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 . Firstly I have no idea how they derived this formula, but I think I have a general idea. 11 For example if we use "abcdefghijklmnopqrstuvwxyz" and a multiplier of 3, gives "adgjmpsvybehknqtwzcfilorux". Moreover, multiplying any two good keys yields again a good key. This corresponds to the K. If "GEHEIMNIS" would be completely encoded by this procedure, the ciphertext would be: "SMVMYKNYC". Since, for the standard alphabet, there are 12 numbers less than 26 which are . If the modular multiplicative inverse of a modulo m exists, the operation of division by a modulo m can be defined as multiplying by the inverse. PLAIN LETTERNATANTSecret key a=2130190131900120012Cipher letteraamaam You can see the dilemma of this message. 3) u(p*q) = (p-1)*(q-1), if M is a product of two primes M=p*q. Can we do even better with M=28 ? It has to be placed after the cout command as in: cout << setw(2) << j*factor. 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. I leave the translation from an upper case plain letter to a lower case cipher letter as an easy exercise for you. Thus, property 4) yields nothing new if our alphabet length is the product of two primes. Coincidence? 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. where the operation of multiplication substitutes the operation of division by the modular multiplicative inverse. allowing a total of 28 different unique encryptions. Options regulate the case when a letter does not appear in any alphabet: it is not encrypted, but transferred directly to the output. Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. 2) u(pn)= pn - pn-1, if M is a power of a prime M= pn. Which keys now yield a unique encryption? ((25)=____________ as all integers from 1 to 24 except for 5,10,15,20 are relatively prime to 25. For an alphabet length of 26 this corresponds to 12 keys: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25. This is very likely in English texts and virtually certain in the German language where on average every 5th letter is an E. Even if an eavesdropper decides to produce all 12 possible plain texts, they can be generated with the help of a computer within a few seconds. For the same reason, an alphabet length of M=31 produces u=30 unique encryptions. We first found the bad keys as the multiples of the prime divisors of the alphabet length M. Consequently, the good keys are the remaining integers less than M. Again a perfect task for a computer, especially when we have to find the prime divisors of bigger integers. 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. It may be denoted as , where the fact that the inversion is m-modular is implicit. This, however, limits readability. 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. First we need to calculate the modular multiplicative inverse of keyA. In fact, the security of i.e. https://de.wikipedia.org/wiki/Alphabet_(Kryptologie). 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. Cite as source (bibliography): In order to decrypt the message we need a combination of a Caesar and a multiplication cipher decryption. Thirdly, listing the good keys would be best done using C++ vectors or even C-style arrays which you might know. An extreme example would be when a=0: all plain letters are translated into 0s which are all as so that no decryption is possible. The multiplicative cipher is a special case of the Affine cipher where B is 0. 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. 36 modulo 26 = 10 so the letter K would be chosen. Affine Cipher is the combination of Multiplicative Cipher and Caesar Cipher algorithm. } 21 is an inverse to 5 MOD 26, therefore 5 is inverse to 21 and the two 1s are mirrored over the diagonal line. Here is the C++ Code for the encryption and decryption of the multiplication cipher: //Multiplication Cipher using the good key a=5 //Author: Nils Hahnfeld, 9/22/99 #include #include void main() { char cl,pl,ans; int a=5, ainverse=21; //as a-1*a=21*5=105=1 MOD 26 clrscr(); do { cout << "Multiplication Cipher: (e)ncode or (d)ecode or (~) to exit:" ; cin >> ans; if (ans=='e') { cout<< "Enter plain text: "<< endl; cin >> pl; while(pl!='~') { if ((pl>='a') && (pl<='z')) cl='a' + (a*(pl -'a'))%26; else if ((pl>='A') && (pl<='Z')) cl='A' + (a*(pl -'A'))%26; else cl=pl; cout << cl; cin >> pl; } } else if (ans=='d') { cout << "Enter cipher text: " << endl; cin >> cl; while(cl!='~') { if ((cl>='a') && (cl<='z')) pl='a' + (ainverse*(cl -'a'))%26; else if ((cl>='A') && (cl<='Z')) pl='A' + (ainverse*(cl -'A'))%26; else pl=cl; cout << pl; cin >> cl; } } } while(ans!='~'); } Programmers Remarks: Can you understand the code yourself? Step 3: Now, apply the formula which is mentioned above. 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. The message "ACDC" should be encrypted with the key "ABBA" according to the Vigenre method. 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 will explain the usage of letter frequency as a very important means to crack cipher codes in the next chapter. It is suitable for small-scale applications but not recommended for practical purposes. Say a=5 was chosen. Multiplicative encryption uses a key k k (an integer) and an alphabet. Options: Multiplier: filter whitespace characters group 5 characters filter non-alphabet characters convert to first alphabet Substitution cipher decoder. gcd(k,36)=1. 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. Alternatively, the non-alphabet letters in the key and the plain text can also be filtered out to increase the security. 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. No provisions are made for high precision arithmetic, nor have the algorithms been encoded for efficiency when dealing with large numbers.
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