In this chapter we will study the Multiplicative Cipher. This eventually enables us to calculate the number of integers that are relative prime to these primes and prime powers. How to calculate the modular multiplicative inverse for the Affine Cipher Lets check why: 1*1=1 MOD 26 which explains a = a-1 = 1 (Big deal!). 17 Why did US v. Assange skip the court of appeal? Lets investigate this in the following section. It was encoded MOD 26. Cryptoanalysis - Cracking the Multiplication Cipher Just like the Cipher Caesar Cipher, the Multiplication is not secure at all. A reciprocal is one of a pair of numbers that when multiplied with another number equals the number 1. They are very special primes as they must consist of 100 digits or more. 1. Affine cipher - online encoder / decoder - Calcoolator.eu The number obtained indicates the rank in the alphabet of the corresponding numbered letter. 6 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. ((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 determinant of the matrix should not be equal to zero, and, additionally, the determinant of the matrix should have a modular multiplicative inverse. Example: Encrypt DCODE with the key $ k = 17 $ and the 26-letter alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ. a=13 yields an ambiguous message since each even plain letter is translated into a (=0): a=13 even letters 13*0 = 0 MOD 26, 13*2 = 0 MOD 26, 13*4 = (13*2) * 2 = 0 * 2 = 0 MOD 26, 13*6 = (13*2) * 3 = 0 * 3 = 0 MOD 26, etc. Example: D = 3, so $ 3 \times 17 \mod 26 \equiv 25 $ and the letter at rank 25 is Z. For example if we use "abcdefghijklmnopqrstuvwxyz" and a multiplier of 3, gives "adgjmpsvybehknqtwzcfilorux". Why is that? 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. 27=3*3*3, so that only the multiples of the only prime divisor 3 such as a=3, 9 and 27 will not yield a unique encryption, all the other integers will: The good keys a are therefore Z27* = {1,2,4,5,7,8,10,11,13,14,16,17,19,20,22,23,25,26} allowing 18 different unique encryptions, 6 more than before. The three factors in the parentheses already have the same desired format, however, the single 2 destroys it. block cipher - Multiplicative Inverse in AES - Cryptography Stack Exchange It is easy to implement and easy to understand, and it does not require any large amount of computational power. I.e. While you still can simply enter an integer number to calculate its remainder of Euclidean division by a given modulus, this modulo calculator can do much more. Moreover, multiplying any two good keys yields again a good key. 4 Let s be such a reversible function. No, it is not. 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.) 11 Simply: Z26* = {1,3,5,7,9,11,15,17,19,21,23,25}. I accomplish this. Convert each letter in the plain text alphabet to a corresponding integer in the range of 0 to m -1; 2. 3. Each character from the plaintext is always mapped to the same character in the ciphertext as in the Caesar cipher. The best answers are voted up and rise to the top, Not the answer you're looking for? Multiplicative Cipher - Online Decoder, Encoder It is not difficult to find the encoded E in English documents as every 8th letter on average is an E (about 13%), it is therefore by far the most frequent letter. 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. More precisely: Out of the 25 (= p * q - 1) integers that are smaller than 26, we had 12 (=13-1) multiples of 2 {2,4,6,8,10,12,14,16,18,20,22,24} and the 1 (=2-1) multiple of 13 {13} as bad keys, so that 25-12-1=12 good keys are remaining: a = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 Notice that u(26) = 12 = 25-12-1 = (p*q - 1) (p-1) - (q-1) Example2: For M=10=5*2, we obtain u(10)=4 good keys which are obtained by crossing out the 4 (=5-1) multiples of 2 and the 1 (=2-1) multiples of 5 as bad keys: a = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Notice that again u = 4 = 9 4 1 = (p*q - 1) (p-1) (q-1) Example3: For M=15=5*3, we obtain u(15)=8 good keys which are obtained by crossing out the 4 (=5-1) multiples of 2 and the 2 (=3-1) multiples of 5: a = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 Notice that again u = 8 = 14 4 2 = (p*q - 1) (p-1) (q-1) The number of good keys can always be computed by u(p*q) = (p*q - 1) - (p-1) -(q-1). Example: If we use the encoding key a=3, we find that the decoding key a-1 is 9 as the 1 occurs in the J- or 9-column telling us additionally that the plain letter J (=9) encrypts to the cipher letter b (=1). In order to decrypt the message we need a combination of a Caesar and a multiplication cipher decryption. For the M, 12*3=36 would result. 2.5: Application of Matrices in Cryptography that 3 and 9 are inverse to each other because of the commutative property of the MOD-multiplication (exhibited by the diagonal as a line of reflection). 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. 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. =CODE("a") yields 97). This encoding and decoding is working based on alphabet shifting & transforming the letters into numbers . No, 13 is missing. 21 is an inverse to 5 MOD 26, therefore 5 is inverse to 21 and the two 1s are mirrored over the diagonal line. (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. 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. Now, lets come to the highlight of this section: I will show you in a few steps how to compute ((M) for any M from one equation instead of combining the four properties? This is important because if the key is known by an unauthorized party, they will be able to decrypt the message. Solution of Multipilicative Inverse of 7. That is, . Our implementation of Vigenre, Beaufort, etc. . ((25)=____________ as all integers from 1 to 24 except for 5,10,15,20 are relatively prime to 25. 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). The modular multiplicative inverse of a modulo m can be found with the Extended Euclidean algorithm. 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). The monoalphabetic cipher family has one very important feature, namely one letter of the open alphabet corresponds to exactly one letter of the secret alphabet. Why does Acts not mention the deaths of Peter and Paul? 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. Lets consider two options: Option 1: Cracking the cipher code using letter frequencies If plain letters are replaced by cipher letters the underlying letter frequencies remain unchanged. If a single character is encrypted by E(C) = (c * k) % 36 then possible keys k are numbers that are coprime to 36, ie. Agree Example the letter M (12th letter in this zero indexed alphabet) and key 3 would be 12 * 3 = 36. 0x95 = 1001 0101 = x 7 + x 4 + x 2 + 1 0x8A = 1000 1010 = x 7 + x 3 + x And the product of the two polynomial reduced modulo the irreductible polynomial is 1, as expected. 23 8 dCode retains ownership of the "Multiplicative Cipher" source code. In conclusion, we can say that multiplicative cipher is a simple encryption technique that can be easily implemented. Multiplicative Cipher - TutorialsPoint 3) u(p*q) = (p-1)*(q-1), if M is a product of two primes M=p*q. We denote 5-1 the inverse of 5. Multiplicative encryption uses a key $ k $ (an integer) and an alphabet. 17 This, however, limits readability. Before we conclude this section with the highlight of creating a sole formula for ((M) from these four properties, we will consider 2 examples for each of the 4 properties of Eulers (-function. 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. One of the major goals of current Mathematics research is to design faster factoring algorithms as todays are fairly slow. 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. Credit goes to the Swiss Mathematician Leonard Euler (pronounced Oiler, 1707-1783). The following C++ program firstly determines the factors for an entered alphabet length M and secondly their multiples, the bad keys. Reminder : dCode is free to use. 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. the commonly used RSA Cipher is based on the relative slowness of such factoring programs. The MOD 26 calculation leaves the 10 unchanged. How to encrypt using Multiplicative cipher? Affine Cipher is the combination of Multiplicative Cipher and Caesar Cipher algorithm. So there is an infinite number of possible keys, but many will give identical messages, because for a $ k $ key, then the $ k + 26 $ key gives an identical cipher.

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