9,15,21 and 25). We just had to multiply each cipher letter by a-1. what are prime divisors of 178247 or of 56272839 ?). Not every key phrase is qualified to be the key; however, there are still more than enough. That is, . 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. for the RSA encryption. As an attentive reader, we realize that the MOD multiplication of the keys is closed (recall the group properties in the previous chapter). 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. , Since we are performing MOD 26 arithmetic, we use the MOD-operator % that guarantees us the product (a*(pl -'a'))%26; to be between 0 and 25. 14 Say, we want to encrypt the plain letter C=67. . 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. Therefore, Formula for the number of good keys if M is a prime power: If M = pn , the number of good keys is u(M) = pn - pn-1. Multiplicative encryption uses a key $ k $ (an integer) and an alphabet. Alphabets (yes, there may be several: more below) can be described by a list L of letters. The answer is a simple No: Only those encryption systems that withstand all possible attacks are secure and thus useful. In this chapter we will study the Multiplicative Cipher. Counter examples are: 45 and 18 are not relative prime since gcd(45,18)=9 and not 1. Modular Multiplicative Inverse a -1. 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.) Consequently, the longer a cipher text, the easier the cipher E can be detected. Instead of performing a transformation before encryption, this implementation allows several alphabets to be specified (see below), thereby accomplishing the same within the encryption process. 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). It has to be placed after the cout command as in: cout << setw(2) << j*factor. How do we deal with non-letters? The odd multiples of 13 (i.e. In the process you'll become comfortable with modular arithmetic and begin to understand its importance to modern cryptography. Now when a=25, we have: 25*25 = 625. The message "ACDC" should be encrypted with the key "ABBA" according to the Vigenre method. When doing so we will discover very important mathematical encryption tools such as Eulers (-function, Eulers and Lagranges Theorem and study further examples of groups, rings and fields. The three factors in the parentheses already have the same desired format, however, the single 2 destroys it. 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. So are 2 and 3, 2 and 5, 3 and 10, 26 and 27, 45 and 16. In the detailed representation of the alphabets (click on the "" -button), the alphabets can be edited in the short-write mode. We also turn the plaintext into digraphs (or trigraphs) and each of these into a column vector. 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. Agree 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. I.e. gcd(k,36)=1. 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. This is the reason why a=2 yields an ambiguous decryption. Although the function is well-defined when a letter occurs more than once, this makes little sense in encryption algorithms, since the reversibility suffers. Divide the letters of the message into groups of two or three. 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". Code Since, for the standard alphabet, there are 12 numbers less than 26 which are . Equivalently stated, 105 divided by 26 leaves a remainder of 1. each occurring exactly twice. 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. Options regulate the case when a letter does not appear in any alphabet: it is not encrypted, but transferred directly to the output. In fact, the security of i.e. 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). Why did US v. Assange skip the court of appeal? ((5)=_____ as 1,2,3,4 are relative prime to 5. The statements while(cl!='~') and cout << cl; cin >> pl; are in charge of it. We have explored the first three properties already, however, the 4th property is new - but not totally new. It is a-1=4 since 3*4 = 12 = 1 MOD 11. 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. Since each plain letter turns into 0 for a=0 and remains unchanged for a=1, we start with a=2. The only two keys that are inverse to themselves are 1 and 25, which means that the encoding key equals its decoding key. 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. 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. Feedback and suggestions are welcome so that dCode offers the best 'Multiplicative Cipher' tool for free! The first character G corresponds to the six. This is important because even if a key is secure when it is first chosen, it can become less secure over time as technology and methods for breaking encryption increase. As a small example we consider Vigenre with the following two alphabets: In both cases, both the plaintext and the key should both consist of the text "0123456789abcdABCD". This means that the key should be a large, random number that is difficult to guess or factor. Step 3: Lets see how decryption can be done using the above formula: Ciphertext = QCCSWJUPQCCSW and multiplication inverse key = 15, Ciphertext: Q > 16 Decryption: (16*15) mod 26 Plaintext: 6 > G, Ciphertext: C > 2 Decryption: (2*15) mod 26 Plaintext: 4 > E, Ciphertext: S > 18 Decryption: (18*15) mod 26 Plaintext: 10 > K, Ciphertext: W > 22 Decryption: (22*15) mod 26 Plaintext: 18 > S, Ciphertext: J > 9 Decryption: (9*15) mod 26 Plaintext: 5 > F, Ciphertext: U > 20 Decryption: (20*15) mod 26 Plaintext: 14 > O, Ciphertext: P > 15 Decryption: (15*15) mod 26 Plaintext: 17 > R, After decryption the plain text = GEEKSFORGEEKS. affine cipher For the fraction a/b, the multiplicative inverse is b/a. The basic formula to be used in such a scenario to generate a multiplicative cipher is as follows (Alphabet Number * key)mod (total number of alphabets) The number fetched through output is mapped in the table mentioned above and the corresponding letter is taken as the encrypted letter. Furthermore it makes not much sense to consider numbers not between 1 and 36, because of the modulo. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. 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. Determining the bad keys for a given alphabet length M is a perfect task for a computer. It is possible to distinguish between 2 types of actions in the plain text: uppercase letters [A-Z] and digits [0-9]. 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. 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. Modulo Arithmetic & Ciphers. I want to show you an example where we used it already. 1 7 25 Cipher textanromrjukahhouh013171412179201007714207 He finds the cipher letter h to be most frequent. Therefore, a translation must take place, which can on the one hand transform letters in numbers and, conversely, re-generate letters again. In this video u will learn how to encrypt the message using multiplicative cipher technique.Plain text to cipher text.Calculator tricks. Mathematically: a-1 * a = a * a-1 = 1. Therefore, I need to subtract 101 from the 103 to get the desired 2, similarly, I again would have to subtract 101 from any plain letter b=102 to get the desired 1. 19 The easiest way to solve this equation is to search each of the numbers 1 to 25, and see which one satisfies the equation. Calculate the value of each letter as follows (where a and b are the keys of the password): E (x)= (ax + b) mod m 3. If a=1 is used as a key, each cipher letter equals its plain letter which shows that it does produce a unique encryption. I found a-1 = 2 by simply testing the integers in Z5*={1,2,3,4}. 11 Thus, among those numbers that occur twice in the cipher code, 14, 17 and 20, we can eliminate the odd 17. Finding the decoding keys for each good key a in the same manner, we obtain the following key pairs: Good Encoding key aIts decoding key a-111395217159311191571723191121523172525 Three important observations: All decoding keys a-1 in the right column are among the set of all encoding keys a. ((8)= ((23)=23 -22 =4 as 1,3,5,7 are relative prime to 8. To verify this: 262 = 676 =1 MOD 27. Coincidence? 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. div#home a:active { (Identification), How to decipher Multiplicative cipher without key? The final formula uses determinant and the transpose of . Network Security: Multiplicative InverseTopics discussed:1) Explanation on the basics of Multiplicative Inverse for a given number.2) Explanation on the basi. Verify this now! an idea ? 12 However, there is no 7 the numerical equivalent of letter h - in the E column. What is the inverse of 7 MOD 11? 19 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. The reciprocal of a number x is a number, which, when multiplied by the original x, yields 1, called the multiplicative identity. The Multiplicative Cipher is an Affine cipher (ax+b) with the value b null (equal to 0), so a multiplication by $ a $. Extracting arguments from a list of function calls. Are these quarters notes or just eighth notes? Example the letter M (12th letter in this zero indexed alphabet) and key 3 would be 12 * 3 = 36. "Signpost" puzzle from Tatham's collection, xcolor: How to get the complementary color. In this lab, you'll learn about the multiplication cipher, a monoalphabetic cipher. Each letter is enciphered with the function (ax + b) mod 26. 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. We know already that: ((60) = ((22*3*5) = (22-21)*(3-1)*(5-1)((M) = ((p12* p2* p3) = (p12- p11)*( p2-1)*( p3-1). 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. 3. div#home a { On the right we ended up with the explicit formula for ((M) when M consists of one prime power and two primes. Identify blue/translucent jelly-like animal on beach. //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). For letters that do not occur in L, the alphabet function sL is undefined. One of the main advantages of the multiplicative cipher is its simplicity i.e. 2 That was trial and error and might take quite a while. margin-bottom: 16px; The basic modulation function of a multiplicative cipher in Python is as follows . The key should be relatively prime to the length of the alphabet. The Affine Cipher uses modulo arithmetic to perform a calculation on the numerical value of a letter to create the ciphertext. WAP to implement Additive cipher(key=20), Multiplicative cipher(key=15)and affine cipher(key=15,20). We get the following encoding and decoding table. Now, lets look at examples for MOD arithmetic: Example2: The inverse of a=3 is a-1 = 2 MOD 5 because a * a-1 = 3*2 = 6 = 1 MOD 5. unchanged so that you can detect the format of the original message easier. Reminder : dCode is free to use. If we had a video livestream of a clock being sent to Mars, what would we see? WAP in python to find out the additive and multiplicative inverse of an integer b using extended Euclidean algorithmof set Zn. Aha, that realization helps a lot, since that also means that prime Ms produce M-1 unique encryptions. 3. }. Moreover, we build the mathematical foundation to understand secure encryption systems such as the RSA encryption. The decryption process is simply the reverse of the encryption process, i.e., by dividing the numerical value of each letter in the ciphertext by key and then taking the result modulo of the key. 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. 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. h2 { Certainly, it might be a double encoded message that has to be decoded twice, possibly using two different keys or even two different ciphers. Lab 2. If you are able to invent a fast factoring algorithm, you will not have to worry about a future job. 11 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).