multiplicative cipher calculator

Example1: If M=24=3*8=3*23, then ((24) = ((3*23) using property 4) yields = ((3)*((23). 10 Each character from the plaintext is always mapped to the same character in the ciphertext as in the Caesar cipher. 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). This is also the case when the letter is in the key. It surely acquires this simple form for any number of primes or prime powers. Find mod of any numb. Step 2: The basic formula that can be used to implement Multiplicative Cipher is: Decryption= (C * Multiplication inverse of the key) Mod 26. So in our above example, the key is 7. Longer messages reveal the most the letter e equivalent, however, this is not necessarily so for our message. ((21)=________________________ as 1,2,4,5,8,10,11,13,16,17,19,20 are relative prime to 21. Aha, that realization helps a lot, since that also means that prime Ms produce M-1 unique encryptions. 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. The answer is a simple No: Only those encryption systems that withstand all possible attacks are secure and thus useful. Before Conversion: ABCDEFGHIJKLMNOPQRSTUVWXYZ After Conversion: XYZABCDEFGHIJKLMNOPQRSTUVW Age Calculators For the fraction a/b, the multiplicative inverse is b/a. Note The advantage with a multiplicative cipher is that it can work with very large keys like 8,953,851. It converts to the plain letter number 26 so that we now have to encrypt MOD 27. 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. In order to decrypt the message we need a combination of a Caesar and a multiplication cipher decryption. Determining the bad keys for a given alphabet length M is a perfect task for a computer. Ubuntu won't accept my choice of password. The Affine Cipher uses modulo arithmetic to perform a calculation on the numerical value of a letter to create the ciphertext. This shows that when using an encoding key that is one less than the alphabet length M, namely a = M-1, then the decoding key must also equal M-1, a-1 = M-1. 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. Try to understand as much as possible first, then continue reading. When a letter occurs in several alphabets, the first of these alphabets is used. Convert each group into a string of numbers by assigning a number to each letter of the message. 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. I will explain the usage of letter frequency as a very important means to crack cipher codes in the next chapter. Also, there is no general match on how to handle digits or special characters. The index of coincidence is unchanged from plain text. Let s be such a reversible function. 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. 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. "Ordered" means that sorting is possible and we can speak of the n-th character of an alphabet. 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. For the same reason, an alphabet length of M=31 produces u=30 unique encryptions. ~=.., $=.. etc. where the operation of multiplication substitutes the operation of division by the modular multiplicative inverse. Say you first want to encode the letter c then you have to enter e when asked. The affine cipher is itself a special case of the Hill cipher, which uses an invertible matrix, rather than a straight-line equation, to generate the substitution . We then perform matrix multiplication modulo the length of the . Example: D = 3, so $ 3 \times 17 \mod 26 \equiv 25 $ and the letter at rank 25 is Z. 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}} }. In formula: u(M) = (M-1) b(M) using the above formula for the number of bad keys yields = M-1 - (M/p -1) distributing the minus sign to the terms in the parenthesis yields = M-1 - M/p + 1 canceling out the 1s yields = M - M/p This turns out to be a handy formula for the number of good keys. To find the multiplicative inverse of a real number, simply divide 1 by that number. As 36=2*2*3*3, the possible keys are basically all numbers not multiples of 2 and/or 3. 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. Certainly, it might be a double encoded message that has to be decoded twice, possibly using two different keys or even two different ciphers. What is the symbol (which looks similar to an equals sign) called? 15 This formula can be simplified into the product of two factors. Note the difference in 'D' and 'd': The index value is the same, but the 'd' is. 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. Divide the letters of the message into groups of two or three. 2) u(pn)= pn - pn-1, if M is a power of a prime M= pn. Which language's style guidelines should be used when writing code that is supposed to be called from another language? ((8)= ((23)=23 -22 =4 as 1,3,5,7 are relative prime to 8. 11 An easier way to determine the decoding key a-1 Decoding a message turns out to be really easy once we know the decoding key a-1. Code PLAIN LETTER:ABCDEFGHIJKLMNOPQRSTUVWXYZ Secret key: a=2012345678910111213141516171819202122232425 024681012141618202224024681012141618202224 Cipher letter:acegikmoqsuwyacegikmoqsuwy Notice, that only every other cipher letter appears, and that exactly twice. The copy-paste of the page "Multiplicative Cipher" or any of its results, is allowed as long as you cite dCode! Then we perform the reverse operations performed by the encryption algorithm. You can try the sample button which uses a multiplication of 3, and a message of "knowledgeispower" gives enqohmjsmyctqomz. However, there is no 7 the numerical equivalent of letter h - in the E column. In this lab, you'll learn about the multiplication cipher, a monoalphabetic cipher. 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. Also, each B and each M turn into 2 (=c) since 2*1 = 2 MOD 26 and 2*14 = 28 = 2 MOD 26. 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? 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). To do so, I distinguish between upper and lower case letters since they are encoded slightly different. Which cracking method should a code cracker use. 6*3=18. First we need to calculate the modular multiplicative inverse of keyA. 3) u(p*q) = (p-1)*(q-1), if M is a product of two primes M=p*q. 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, this way you can implement Encrypt a plain text and Decrypt a cipher text for Multiplicative cipher in cryptography. Method 2: Merged: In the alphabet, mod 22 is calculated because the alphabet contains 22 elements. ((28) = _____________________________ as 1,3,5,9,11,13,15,17,19,23,25,27 are relative prime to 28. It would take quite a long time for a computer to brute-force through a majority of nine million keys. 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. An affine cipher is a cipher belonging to the group of monoalphabetic substitution ciphers. Moreover, we build the mathematical foundation to understand secure encryption systems such as the RSA encryption. Furthermore it makes not much sense to consider numbers not between 1 and 36, because of the modulo. 21 is an inverse to 5 MOD 26, therefore 5 is inverse to 21 and the two 1s are mirrored over the diagonal line. 26, 52, 78, ) have its equivalent key in a=0, a very bad key, since 26=52=78=0 MOD 26. The determinant of the matrix should not be equal to zero, and, additionally, the determinant of the matrix should have a modular multiplicative inverse. It thus gives a great example that we are only guaranteed to solve this equation for numbers that form a group with respect to multiplication MOD 26. For a given alphabet, there are only a few possible keys. For each character of the plain message, apply the following calculation: ($ 26 $ being the number of letters in the alphabet). Modular inverse of a matrix. 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. We will multiply MOD 26 as we are using the 26 letters of the English alphabet. The 18th character in the used alphabet corresponds to the S. The first character in the ciphertext therefore would be S. The remaining characters are encoded in the same way. Network Security: Multiplicative InverseTopics discussed:1) Explanation on the basics of Multiplicative Inverse for a given number.2) Explanation on the basi. For classical methods, the alphabet often consists only of the uppercase letters (A-Z). Each letter is associated with its rank $ c $ in the alphabet (starting from 0). Say a=5 was chosen. 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. 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. If so please go ahead and modify the following program. I will complete the first ones and leave the second ones for you as exercises. Credit goes to the Swiss Mathematician Leonard Euler (pronounced Oiler, 1707-1783). What is the inverse of 5 MOD 11? 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. 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. Cipher textanromrjukahhouh013171412179201007714207 He finds the cipher letter h to be most frequent. Examples for property 3): 15 and 21 are products of two primes. a bug ? The formula MOD(E$2*$B4,26) computes the number of the plain letter T, namely 19. Example the letter M (12th letter in this zero indexed alphabet) and key 3 would be 12 * 3 = 36. The same alphabet is used to generate the encrypted text. a=4 is inverse to itself modulo 5 since a * a-1 = 4 * 4 = 16 = 1 MOD 5. 1 color: #ffffff; Which was the first Sci-Fi story to predict obnoxious "robo calls"? 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. 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. Example5: If M=65=5*13=p*q, then the formula yields u(65) = (5-1)*(13-1) = 4*12 = 48. Try it for yourself. So are 2 and 3, 2 and 5, 3 and 10, 26 and 27, 45 and 16. 7 36 modulo 26 = 10 so the letter K would be chosen. To use this worksheet, you must supply: a modulus N, and either: This inverse modulo calculator calculates the modular multiplicative inverse of a given integer a modulo m. 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. In this video u will learn how to encrypt the message using multiplicative cipher technique.Plain text to cipher text.Calculator tricks. Again, we just have to find the cipher numbers in the 5th row and then go up that column to the very top to find the corresponding plain letter. This eventually enables us to calculate the number of integers that are relative prime to these primes and prime powers. Introduction to Monotonic Stack - Data Structure and Algorithm Tutorials. 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. I'm learning and will appreciate any help. 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. Each character is multiplied with this key and the corresponding letter is substituted. Does the increase of our alphabet length by 1 increase the number of unique encryptions obtained? Thus, safer encryptions are necessary. In the process you'll become comfortable with modular arithmetic and begin to understand its importance to modern cryptography. How to encrypt using Multiplicative cipher? Or can we even increase the mere 12 unique encryptions for the Multiplication Cipher by varying the alphabet length? }. Or are they possibly the primes between 1 and 25? The letter A remains unchanged ans id always encoded A. I do not think any special calculator is needed in each of these cases. https://de.wikipedia.org/wiki/Alphabet_(Kryptologie). The following steps take place: In the example, an overflow has occurred in the third letter, so that modulo |L| = 4 is calculated. That is, . background-color: #620E01; Since a=10 is a bad key he checks the good key a=23. We get the following encoding and decoding table. If the plaintext is made of both letters (a to z) and digits (0 to 9), how do you find the key domain of the multiplication cipher? 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. By using our site, you 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. . 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. } As an attentive reader, we realize that the MOD multiplication of the keys is closed (recall the group properties in the previous chapter). 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. Our good-key-criterion declares those integers to be good keys that are relative prime to 27. 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. That is weird! Combining our three formulas for the number of good keys, we will then be able to develop a general formula for the number of good keys for any given alphabet length M. Lets start with Example1: M=26=p*q=2*13. Are they the odd numbers between 1 and 25? 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. If the alphabet of capital letters A-Z is used, this assignment results: Now a key between 1 and 26 is chosen. which we used in our virus carrier example. What is the inverse of 7 MOD 11? 17 Characters not belonging to the alphabet are not encrypted or allowed as keys. 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"). color: #aaaaaa; Please, check our dCode Discord community for help requests!NB: for encrypted messages, test our automatic cipher identifier! As some of them fail to produce a unique encryption, we will discover an easy criterion for keys that produce the desired unique encryptions (the good keys) and apply it to different alphabet lengths. In such case, divide M by that factor: M/=factor; and start checking M/factor for factors less than M/factoretc. 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. We will check in the Abstract Algebra section at the end of this chapter that the set of good keys MOD 26, Z26* = {1,3,5,7,9,11,15,17,19,21,23,25}, does form a multiplicative group. C = (a * P) mod 26 In order to create unique cipher characters, we must use a multiplier which is co-prime (the values do not share any factors when dividing - see Try GCD of 5) in relation to the size of the alphabet (26), so you should use either 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 or 25. Try to explain this, than continue reading! In fact, the cipher E can only be an even cipher letter as only even numbers appear in the E-column. In linear algebra, an n-by-n (square) matrix A is called invertible if there exists an n-by-n matrix such that. You are asked to enter your plain letter in cin >> pl; As long as you dont enter ~ the while-condition while(pl!='~') is fulfilled and the entered plain letter (=pl) is being encoded. Not every key phrase is qualified to be the key; however, there are still more than enough. The best answers are voted up and rise to the top, Not the answer you're looking for? Equivalently stated, 105 divided by 26 leaves a remainder of 1. color: #ffffff; The solution shows the work for the Standard Algorithm. For the purpose of setting up an explicit formula for ((M), we now try to give the three factors (in parentheses) the same format. } If a=4,6,8,,24, we encounter the same dilemma as for a=2. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. That is: The theory can be found after the calculator. This weirdness is not really weird. As an example, lets encode and decode NAT and ANT. All symbols to be encrypted must belong to alphabet, Everyone who receives the link will be able to view this calculation, Copyright PlanetCalc Version: Say M=26=2*13=n*m. Since n and m are two distinct primes, they certainly are relative prime, so that the condition for property 4) is fulfilled. Except for 2 and 13, all prime numbers less than 26 are among the keys (why do they have to?). Modulo Arithmetic & Ciphers. All we need to know are the prime divisors of M, but we dont even need to know how often a prime number divides M. Example: Encrypt DCODE with the key $ k = 17 $ and the 26-letter alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ. 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. Convert each letter in the plain text alphabet to a corresponding integer in the range of 0 to m -1; 2. 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. This modulo calculator performs arithmetic operations modulo p over a given math expression. //Author: Nils Hahnfeld 10/15/99 //Factoring program #include #include #include void main() { int M, factor ; clrscr(); do { cout << "Enter the integer that you want to factor or 0 to exit: M="; cin >> M; factor=2; while(factor <= M) { if (M%factor==0) //check all integers less than M as factors { cout << factor << endl; M/=factor; factor=1; } factor++; } }while(M!=0); } Programmers remarks: Starting with 2, this program checks the integers from 2 to M-1 as potential factors of M in if (M%factor==0). How to recognize a Multiplicative ciphertext? This is just what we wanted except that the answer 10 does not equal the desired cipher letter k on the computer. This means that the key should be a large, random number that is difficult to guess or factor. Mathematically: a-1 * a = a * a-1 = 1. Reciprocal (or) Multiplicative Inverse is: 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. 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. Pinehurst Ga Obituaries, Articles M