CacheSleuth - Multiplicative Cipher Learn more. where the operation of multiplication substitutes the operation of division by the modular multiplicative inverse. Key is the matrix; however, it is convenient to use the key phrase, which is transformed into the digit representation and matrix. The basic modulation function of a multiplicative cipher in Python is as follows . color: #ffffff; (I.e. Part 2: Classic Encryption Algorithms - DEV Community Each row that contains each integer from 0 to 25 exactly once and therefore yields a unique cipher letter will serve. 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? Therefore, we just have to add a number in order to get k=111. Identify blue/translucent jelly-like animal on beach. You can find the reciprocal quite easily. the commonly used RSA Cipher is based on the relative slowness of such factoring programs. Multiplicative Cipher on dCode.fr [online website], retrieved on 2023-05-02, https://www.dcode.fr/multiplicative-cipher, multiplicative,multiplication,modulo,cipher, https://www.dcode.fr/multiplicative-cipher, What is Multiplicative Cipher? For an alphabet length of 26 this corresponds to 12 keys: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25. What are the variants of the Multiplicative cipher. Example: Encrypt DCODE with the key k= 17 k = 17 and the 26-letter alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ 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? The Caesar cipher is a special case of the Affine cipher where A is 1 and B is the shift/offest. How to encrypt using Multiplicative cipher? If you choose to do so, dont forget to also redefine the corresponding decoding key in int a=5, ainverse=21; . These calculations were correct but almost required a calculator. Cite as source (bibliography): As 36=2*2*3*3, the possible keys are basically all numbers not multiples of 2 and/or 3. When you study the a=2 row precisely, you will see that the original 26 plain letters are converted into 13 even cipher letters (the even cipher letters are those whose numerical equivalent is an even number.) The calculator logic is explained below the calculator. Substitution cipher decoder. Then the if-condition if (ans=='e') is fulfilled so that we enter the encoding part of the program. 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). The multiplicative cipher is a special case of the Affine cipher where B is 0. Thus, dividing is performed slightly different: instead of dividing by 5 or multiplying by 1/5, we first write 5-1 (instead of 1/5) where 5-1 now equals an integer and multiply both sides by that integer 5-1. Caesar Cipher Encryption Decryption Converter - MYMATHTABLES.COM 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. So which ones do? A little computer program turns out to be again very valuable as the number of good keys can be easily determined by first finding all prime factors of M to then use the above explicit formula. 2) Learn how to compute and use the modular inverse to decode. Since one can be divided without remainder only by one, the equation above has the solution only if . 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. 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. 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. 9,15,21 and 25). That is, they mustn't have any common divisors. They seem to not follow any apparent pattern. I want to show you an example where we used it already. 36 modulo 26 = 10 so the letter K would be chosen. 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: How could it be broken? To use this worksheet, you must supply: a modulus N, and either: You may see ads that are less relevant to you. 17 (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. Why did US v. Assange skip the court of appeal? 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. I will explain the usage of letter frequency as a very important means to crack cipher codes in the next chapter. To find the inverse for each good key a, you just need to look back at the 26 by 26 encryption table. 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. Alphabets (yes, there may be several: more below) can be described by a list L of letters. ~=.., $=.. etc. Aha, that realization helps a lot, since that also means that prime Ms produce M-1 unique encryptions. 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 encrypted text is the smallest digit of an addition of plaintext and key when both are hexadecimal digits. 8 Please enable JavaScript to use all functions of this website. 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. > m o ` a b c d e f g h i j k l 7 9 bjbjUU (> 7| 7| ). He investigated these number properties and was the first one to come up with a function, Eulers (-function, also called Eulers Totient function, that determines the number of integers that are relative prime to a given integer M. It is a function that is in the heart of Cryptography and used i.e. 25 Additionally, you will learn that the RSA Cipher uses prime numbers as well. 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. Affine cipher - 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. This is the reason why a=2 yields an ambiguous decryption. In fact, the sets of the encoding and decoding keys are identical. Divide the letters of the message into groups of two or three. We wont have to do it that way again since there is a much more straightforward method. However, we dont need to consider keys that are greater than 26 since each of them has an equivalent key less than 26 that yields the same encryption: the even multiples of 13 (i.e. The first time the loop passes the line cout << cl; the translated plain letter pl that was read in as cin >> pl; before the while loop is output as its cipher letter cl. Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. 7 Modular Multiplicative Inverse a -1. Each character of the used alphabet is assigned to a value. Can you? So it will look like this after calculation. In the process you'll become comfortable with modular arithmetic and begin to understand its importance to modern cryptography. Thus they have the following restrictions: Lets check why: 1*1=1 MOD 26 which explains a = a-1 = 1 (Big deal!). Convert each group into a string of numbers by assigning a number to each letter of the message. Thus, we now go ahead and practice a bit more computer programming. 26, 52, 78, ) have its equivalent key in a=0, a very bad key, since 26=52=78=0 MOD 26. Simply: Z26 = {0,1,2,3,, 24,25}. For example, take the list L = "ABCD", whose length is 4. "Ordered" means that sorting is possible and we can speak of the n-th character of an alphabet. These are valuable information for an eavesdropper that help cracking the message. We factor p1=2 yielding = 2*(2-1)*(3-1)*(5-1) = p1* (p1- 1)*( p2-1)*( p3-1). By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
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