multiplicative cipher calculator

Which number would that be? an idea ? color: #ffffff; Take a moment now to verify the Rule for finding the decoding key a-1: 1) For a given good key a, find the unique 1 in the a-row, 2) From that 1 go all the way up that column, 3) The letters numerical equivalent that you hit on the very top is the inverse of a. Therefore, since there are no other prime divisors and thus no multiples, all integers less than M serve as good keys. This is important because if the key is known by an unauthorized party, they will be able to decrypt the message. background-color: #620E01; Our good-key-criterion declares those integers to be good keys that are relative prime to 27. Exporting results as a .csv or .txt file is free by clicking on the export icon Decoding aam can either yield NAT or ANT as the plain text. However, it can be simplified further using the fact that we are considering here alphabets of length M that are powers of a prime p: M=pn for some positive integer n. Thus, our formula simplifies to: u(M) = pn pn/p which simplifies further to = pn - pn-1. In order to decrypt the message we need a combination of a Caesar and a multiplication cipher decryption. This is also the case when the letter is in the key. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. 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. } Step 3: Now, apply the formula which is mentioned above. 21 is an inverse to 5 MOD 26, therefore 5 is inverse to 21 and the two 1s are mirrored over the diagonal line. Therefore, each integer less than 29 is a good key MOD 29: Z29* = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28}. Our ultimate goal is not to develop a formula for the number of bad keys but rather for the number of good keys. Each letter is associated with its rank $ c $ in the alphabet (starting from 0). 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. 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. 15 By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. 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 . PLAIN LETTERNATANTSecret key a=2130190131900120012Cipher letteraamaam You can see the dilemma of this message. 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, . This table shows the occurances of the letters in the text (ignoring the case of the letters): This table shows how the text matches a normal probability to text (where 'E' has the highest level of occurance and 'Z' has the least). 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). Each character from the plaintext is always mapped to the same character in the ciphertext as in the Caesar cipher. Modified 8 years, 6 months ago. 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*. 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. 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. So which ones do? color: #ffffff; The command const is used as a safety feature in C++: both variables are constant and can never be modified in any program. 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. Which language's style guidelines should be used when writing code that is supposed to be called from another language? 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. The encryption process is done by multiplying the numerical value of each letter in the plaintext by the key and then taking the result modulo the key. Reciprocal (or) Multiplicative Inverse is: We have to understand why multiplying by a bad key a MOD 26 yields some integers more than once and others not at all. Therefore, all the keys that are multiples of 5 such as a=10,15,20,25,30 will also translate the H into 0(=a). Since a=10 is a bad key he checks the good key a=23. 3) u(p*q) = (p-1)*(q-1), if M is a product of two primes M=p*q. 26, 52, 78, ) have its equivalent key in a=0, a very bad key, since 26=52=78=0 MOD 26. Read about it on wiki, I will provide only example. Affine cipher - encoder / decoder. What is the inverse of 7 MOD 11? 2.5 Counting the Number of Good Keys for various Alphabet Lengths M An Introduction to the Euler Function. These ads use cookies, but not for personalization. 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. 2) u(pn)= pn - pn-1, if M is a power of a prime M= pn. 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). Lets check this for an alphabet length of M=29. Characters not belonging to the alphabet are not encrypted or allowed as keys. Which keys now yield a unique encryption? Code You could also define a to be a different good key. or . For illustration purposes we use the message "GEHEIMNIS" and the key 3. Thus, safer encryptions are necessary. E (x) = (ax + b) mod m D (x) = a -1 (x - b) mod m For more math formulas, check out our Formula Dossier What 4 concepts are covered in the Affine Cipher Calculator? Please, check our dCode Discord community for help requests!NB: for encrypted messages, test our automatic cipher identifier! Parts of Long Multiplication 2 5 6 Multiplicand 3 2 Multiplier + 5 1 2 Partial Product + 7 6 8 Partial Product = 8 1 9 We just had to multiply each cipher letter by a-1. 36 modulo 26 = 10 so the letter K would be chosen. It describes the multiplicative property of (. Equivalently stated: what product of a-1 and 5 equals 1 more than a multiple of 26 such as 27, 53, 79, 105, etc? 1. 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). A corresponding warning is displayed. Notice that we found the good keys indirectly. Is there a generic term for these trajectories? The following steps take place: In the example, an overflow has occurred in the third letter, so that modulo |L| = 4 is calculated. A summary of our explorations for the number of good keys shows: 1) u(p) = p - 1, if M is prime M=p. 4) ((n*m) = ((n) * ((m) when n and m are relatively prime. Notice in the last row that all we need to know are the prime factors p of M without knowing how often they occur. PLAIN LETTER:ABCDEFGHIJKLMNOPQRSTUVWXYZ Secret key: a=2012345678910111213141516171819202122232425 024681012141618202224024681012141618202224 Cipher letter:acegikmoqsuwyacegikmoqsuwy Notice, that only every other cipher letter appears, and that exactly twice. You may see ads that are less relevant to you. This is not a useful encryption system since it may yield ambiguous messages. div#home a { Thank you! The basic modulation function of a multiplicative cipher in Python is as follows . 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. which we used in our virus carrier example. It has to be placed after the cout command as in: cout << setw(2) << j*factor. It surely acquires this simple form for any number of primes or prime powers. 18 This brute force approach will work fast enough for integers M that have 10 digits or less. N (=13) translates into a for any even key a aswell because even keys N 4*13 = 2*(2*13) = 2*0 = 0 MOD 26, 6*13 = 3*(2*13) = 3*0 = 0 MOD 26, 8*13 = 4*(2*13) = 4*0 = 0 MOD 26, etc. This eventually enables us to calculate the number of integers that are relative prime to these primes and prime powers. padding: 12px; Subsequently, that difference is multiplied by the good key a=5 which I defined as such in int a=5. The first character G corresponds to the six. 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. 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. No, 13 is missing. =CODE("a") yields 97). Let us consider the above cipher text i.e. 23 A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. Instead of adding a number as we did in the Caesar Cipher, we will now multiply each plain letter by an integer a, our secret encoding key. 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. To ensure that no two letters are mapped to the same letter, a and m must be coprime. While using Caesar cipher technique, encrypting and decrypting symbols involves converting the values into numbers with a simple basic procedure of addition or subtraction. h2 { We then write them in the form (1-1/p), multiply them and that product by M yielding ((M). So the cipher text symbol will be w for the letter a in this case. 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. To have the solution, the right part of the linear diophantine equation should be a multiple of the . Hey, this shows a great way to produce more unique encryptions which of course makes life harder for an eavesdropper: Recommendation for more security: Choose the alphabet length M to be a prime number to make cracking the cipher text more difficult. Ok, lets continue with the encoding part. . This corresponds to the K. If "GEHEIMNIS" would be completely encoded by this procedure, the ciphertext would be: "SMVMYKNYC". does not work internally with letters, but with numbers. Well, I leave all the entered non-letters such as ! The only disadvantage is that the minus sign itself has to be written as "---", so as not to be confused as a range operator. Note the difference in 'D' and 'd': The index value is the same, but the 'd' is. 0 However, subtracting the number of bad keys from the number of all possible keys (=M-1) yields the number of good keys. Example3: Doing arithmetic MOD 7, the inverse of a=3 is a-1 = 5 because a * a-1 = 3*5 = 15 = 1 MOD 7. The plain letter c is stored as 103, however, I want the c to equal 2 in compliance with our translation a=0, b=1, c=2, etc. Just as 5*1/5 yields 1, 5 * 5-1 shall equal 1 MOD 26. and all data download, script, or API access for "Multiplicative Cipher" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app! affine cipher If a = 1, the Affine cipher is equivalent of a Caesar cipher. Since 625=24*26+1 which means that 625 leaves a remainder of 1 when divided by 26, we have 625 = 1 MOD 26 and altogether 25 * 25 = 625 = 1 MOD 26. Calculates a modular multiplicative inverse of an integer a, which is an integer x such that the product ax is congruent to 1 with respect to the modulus m. ax = 1 (mod m) ax aa1 1 (mod m) a x a a 1 1 ( mod m) Integer a.

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