First of all, you need to know which one of the 12 good keys was used. Try it for yourself! In order to decrypt the message we need a combination of a Caesar and a multiplication cipher decryption. Cite as source (bibliography): This means that the cipher E does not equal 7. For example, take the list L = "ABCD", whose length is 4. For example, if the key is 7 and the plaintext letter is s, the numerical value of s is 18, and so the ciphertext letter is (18*7) mod 26 = 22. 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. Affine Cipher Calculator - Math Celebrity 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. Cipher textanromrjukahhouha=1ANROMRJUKAHHOUHa=3ANXWEXDYMALLWYLa=5ANTISTHECARRIERa=7ANVCYVFOUABBCOBa=9ANZQKZBIEAVVQIVa=11ANLGULPQIADDGQDa=15ANPUGPLKSAXXUKXa=17ANBKQBZSWAFFKSFa=19ANFYCFVMGAZZYMZa=21ANHSIHTWYAJJSWJa=23ANDEWDXCOAPPECPa=25ANJMOJRGQATTMGT MS Excel as a simple encryption and decryption tool: I created the following table in MS Excel with the CHAR and the MOD function: Cipher textanromrjukahhouhaa-101317141217920100771420739ANXWEXDYMALLWYL521ANTISTHECARRIER715ANVCYVFOUABBCOB93ANZQKZBIEAVVQIV1119ANLGULPQIADDGQD157ANPUGPLKSAXXUKX1723ANBKQBZSWAFFKSF1911ANFYCFVMGAZZYMZ215ANHSIHTWYAJJSWJ2317ANDEWDXCOAPPECP2525ANJMOJRGQATTMGT For example, I created the T in the row a=5 using the Excel-formula =CHAR(65+MOD(E$2*$B4,26)) where the cell E$2 contains 17 and the cell $B4 contains 21 as the decoding key a-1. Longer messages reveal the most the letter e equivalent, however, this is not necessarily so for our message. a=4 is inverse to itself modulo 5 since a * a-1 = 4 * 4 = 16 = 1 MOD 5. Here is a non-calculator way to understand why 25 is inverse to itself: Since 25 = -1 MOD 26, it follows 25 * 25 = (-1) * (-1) = 1 MOD 26. 36 modulo 26 = 10 so the letter K would be chosen. 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*. Affine Cipher - Crypto Corner Then the if-condition if (ans=='e') is fulfilled so that we enter the encoding part of the program. It is not difficult to understand that the length of such numbers requires the usage of computers. 1) Learn how to decode the Multiplication Cipher. Ok, lets continue with the encoding part. Method 1: Separated: In each sub-alphabet, mod 16 is calculated (hex addition), since each sub-alphabet contains 16 elements, and it remains in the same partial alphabet from which the plaintext letter originates. 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. Multiplication Cipher Lets simply test all possible keys of the multiplication ciphers MOD 26: PLAIN LETTER 0000000000000000000000000 a ABCDEFGHIJKLMNOPQRSTUVWXYZ00000000000000000000000000010123456789101112131415161718192021222324252024681012141618202224024681012141618202224303691215182124147101316192225258111417202340481216202426101418220481216202426101418225051015202549141924381318232712172216111621606121824410162228142006121824410162228142070714212916234111825613201815223101724512198081624614224122021018081624614224122021018909181101921120312214132251423615247162581710010204142481821222616010204142481821222616110112271831425102161721324920516112238194151201224102282061841621401224102282061841621413013013013013013013013013013013013013013140142164186208221024120142164186208221024121501541982312116520924132176211025143187221116016622122188241442010016622122188241442010170178251672415623145221342112320112191011891801810220124221462416801810220124221462416819019125241710322158120136251811423169221147200201482221610424181260201482221610424181262102116116122171272231813832419149425201510522022181410622420161284022181410622420161284230232017141185225221916131074124211815129632402422201816141210864202422201816141210864225025242322212019181716151413121110987654321 We learned already that the key a=2 (as can be seen in the 3rd row) does not produce a unique encryption. Our ultimate goal is not to develop a formula for the number of bad keys but rather for the number of good keys. Does the increase of our alphabet length by 1 increase the number of unique encryptions obtained? A longer alphabet produces less unique encryptions. Although the function is well-defined when a letter occurs more than once, this makes little sense in encryption algorithms, since the reversibility suffers. This means that the key should be a large, random number that is difficult to guess or factor. The Caesar cipher is a special case of the Affine cipher where A is 1 and B is the shift/offest. Convert each group into a string of numbers by assigning a number to each letter of the message. Note the difference in 'D' and 'd': The index value is the same, but the 'd' is. 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. The number fetched through output is mapped in the table mentioned above and the corresponding letter is taken as the encrypted letter. CRITERION FOR GOOD KEYS A key a produces a unique encryption, if the greatest common divisor of 26 and a equals 1, which we write as: gcd(26, a)=1 Convince yourself that 26 has a greatest common divisor equal to 1 with each of these good keys a = 1,3,5,7,9,11,15,17,19,21,23,25. First we need to calculate the modular multiplicative inverse of keyA. Are these quarters notes or just eighth notes? The statements while(cl!='~') and cout << cl; cin >> pl; are in charge of it. 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). In some secret manner, the sender and the recipient had to agree on the encoding key a. Back to the virus carrier message. 28 equals 2*2*7 so that all the keys that are multiples of 2 or 7 do not and all non-multiples of 2 or 7 do produce a unique encryption: Z28* = {1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27} allowing only 12 different unique encryptions. Decrypt, 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. 1) This program both encodes and decodes. (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 are players required to record the moves in World Championship Classical games? Multiply It! We can combine these two criteria into one easy criterion. In affine cipher each letter in an alphabet is mapped to its numeric equivalent, encrypted using a simple mathematical function, and converted back to a letter. Additional restrictions to the key are imposed by the need to decrypt encrypted text :). 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? Zero has no modular multiplicative inverse. 2.4 Varying the Alphabet Length varies the Number of Good Keys Using an alphabet length of M=27: Say for legibility reasons we add a blank symbol as our 27th plain letter. }. Modulo Arithmetic & Ciphers. This process repeats until M is reduced to 1 and therefore less than the smallest factor possible, 2. It is suitable for small-scale applications but not recommended for practical purposes. 15 or . 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. 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. In this chapter we will study the Multiplicative Cipher. For the English alphabet, where m = 26, this means a cannot be 2, 4, 6, 8 (any even number) or 13. . Multiplicative Cipher : Encryption Decryption Method - YouTube Notice, that property 3) became useless for the calculation process since factors that are relative prime are separated via property 4). In fact, the security of i.e. So, we are left with determining the decoding key a-1 knowing the original encoding key a. The x values are the ones that we can choose independently, here the length of the alphabet M. Each y-value is dependent on the choice of x, i.e. Online calculator: Hill cipher - PLANETCALC 19 Copyright 1998 - 2023 CrypTool Contributors. How does the j decode to the H, and the u decode to the E? Vigenre cipher - online encoder / decoder - Calcoolator.eu 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. Learn how PLANETCALC and our partners collect and use data. Example4: For M= 34 =81, we get u(81) = 34 - 33 = 81 27 = 54. 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. It is a-1=4 since 3*4 = 12 = 1 MOD 11. The formula MOD(E$2*$B4,26) computes the number of the plain letter T, namely 19. Which ones are those? background-color: #620E01; the number of unique encryptions u are dependent on the chosen alphabet length M. Since u can be expressed as a formula that involves M, namely u=M-1, we say that u is a function of M and write u(M)=M-1. Reminder : dCode is free to use. 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. ((3)=3-1=2 as 1 and 2 are relative prime to 3. This encoding and decoding is working based on alphabet shifting & transforming the letters into numbers . gcd(k,36)=1. 6 For the M, 12*3=36 would result. Now when a=25, we have: 25*25 = 625. The algorithm memorizes the alphabet with which it has determined the number of the plaintext. What is the difference between "cipher" and "encryption"? 16 An alphabet[1] is an ordered set of all characters which can occur in a plaintext, a secret text, or the key. 0
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