For now, lets focus on the lower case letters. Therefore, a translation must take place, which can on the one hand transform letters in numbers and, conversely, re-generate letters again. 1) Learn how to decode the Multiplication Cipher. It is actually less secure than the Caesar cipher because the number of possible keys is smaller. Thus, property 4) yields nothing new if our alphabet length is the product of two primes. Notice, that property 3) became useless for the calculation process since factors that are relative prime are separated via property 4). Before Conversion: ABCDEFGHIJKLMNOPQRSTUVWXYZ After Conversion: XYZABCDEFGHIJKLMNOPQRSTUVW Age Calculators Generally: The good keys are those as that are relative prime to M and are denoted as ZM*. How could it be broken? The basic modulation function of a multiplicative cipher in Python is as follows . color: #ffffff; I first subtract 65 =A and then multiply that difference by the good key a=5 yielding 10 again. Example4: If M=39=3*13=p*q, then the formula yields u(39) = (3-1)*(13-1) = 2*12 = 24. By using this website, you agree with our Cookies Policy. 36 modulo 26 = 10 so the letter K would be chosen. This inverse modulo calculator calculates the modular multiplicative inverse of a given integer a modulo m. Multiplicative inverse vs. Modular multiplicative inverse warning First of all, there is a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x, and it is not the same as modular multiplicative inverse. Since the number of unique encryptions u is a function of the alphabet length M, we may write in function notation: u(M) to denote the number of unique encryptions (which equals the number of good keys) as a function of M. I.e. For classical methods, the alphabet often consists only of the uppercase letters (A-Z). And, for this to happen, we need to have a modular inverse of the key matrix in - ring of integers modulo m. If source vector B is multiplied by matrix A to get vector C, then to restore vector B from vector C (decrypt text), one needs to multiply it by the modular inverse of the matrix. Thus, safer encryptions are necessary. Now when a=25, we have: 25*25 = 625. An alphabet[1] is an ordered set of all characters which can occur in a plaintext, a secret text, or the key. For instance, to find the inverse of the good key a=5 we have to look at the fifth row which shows that a-1 equals 21 since the only 1 in this row is in the V- or 21-column (5 * 21 = 1 MOD 26). To have the solution, the right part of the linear diophantine equation should be a multiple of the . Moreover, multiplying any two good keys yields again a good key. See the image attached below for a better understanding. 6*3=18. Sphero Up to 1 Hour Grades: 5 to 8. In some secret manner, the sender and the recipient had to agree on the encoding key a. 9 ~=.., $=.. etc. Please enable JavaScript to use all functions of this website. This formula can be simplified into the product of two factors. 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. The Affine Cipher uses modulo arithmetic to perform a calculation on the numerical value of a letter to create the ciphertext. 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. v l X X X Are these quarters notes or just eighth notes? We can also calculate all the possible keys for the Affine Cipher. How to encrypt using Multiplicative cipher? 2. Key is the matrix; however, it is convenient to use the key phrase, which is transformed into the digit representation and matrix. ((5)=_____ as 1,2,3,4 are relative prime to 5. We can see in the table that an A will always translate into 0 (=a) since the product of any such key a with 0 (=A) yields 0. Are they the odd numbers between 1 and 25? Similarly, the multiples of a=7 will translate an F (=5) into an 0 (=a) because 7 does so. Can we increase the number of unique encryptions by further extending our alphabet? Therefore, a simple prime check program would be sufficient to find the divisors p of M. We then set up the factors of the form (1- 1/p), multiply them and eventually multiply that answer by M. Example1: Say M=180, then a prime check program yields the prime factors 2,3 and 5, so that ((180) = 180 * (1-1/2) * (1-1/3) * (1-1/5) = 180 * (1/2) * (2/3) * (4/5) = 90 * (2/3) * (4/5) = 60 * (4/5) = 48 Example2: Say M=360, since 360=2*180 the prime factors are again 2,3 and 5, so that ((360) = 360 * (1-1/2) * (1-1/3) * (1-1/5) = 360 * (1/2) * (2/3) * (4/5) = 180 * (2/3) * (4/5) = 120 * (4/5) = 96 Example3 is for you: Say M=90, since 90=____ the prime factors are _______, so that ((90) = 90 * (1-1/__) * (1-1/__) * (1-1/__) = 90 * ____________________ = _______________ = _______________ = ___ Of course, I could have computed the answers in the above examples right away but I wanted to give you the chance of brushing up on your skills to multiply fractions. As 36=2*2*3*3, the possible keys are basically all numbers not multiples of 2 and/or 3. 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. I accomplish this. where the operation of multiplication substitutes the operation of division by the modular multiplicative inverse. ((25)=____________ as all integers from 1 to 24 except for 5,10,15,20 are relatively prime to 25. 5 7 11 13 17 19 23 25 29 31 35 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. The procedure to use the multiplicative inverse calculator is as follows: Step 1: Enter the values in the numerator and denominator input field Step 2: Now click the button "Solve" to get the output Step 3: The multiplicative inverse value will be displayed in the "Answer" field What is Multiplicative Inverse? Reminder : dCode is free to use. How a top-ranked engineering school reimagined CS curriculum (Ep. Since, for the standard alphabet, there are 12 numbers less than 26 which are . The trick is now that if we enter more than one letter all but the first entered letter are buffered (which means temporarily stored in the computers RAM) until read in in cin >> pl; inside the while loop. Except for 2 and 13, all prime numbers less than 26 are among the keys (why do they have to?). Thus they have the following restrictions: Modular inverse of a matrix. It converts to the plain letter number 26 so that we now have to encrypt MOD 27. 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. 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. That would additionally require that the good keys form a commutative group with respect to addition. Therefore, an eavesdropper simply has to count letter frequencies to identify the most frequent cipher letter. ((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. In case you wonder why the discussion of cracking codes is made public; why is it not kept secret to maintain the security of ciphers? Mathematically: a-1 * a = a * a-1 = 1. 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. In order to increase the probability of this, the alphabet is expanded, so its length becomes the prime integer. The reason for that is that a prime number has per definition no prime divisor except for 1 and itself. Multiplicative encryption uses a key $ k $ (an integer) and an alphabet. The basic modulation function of a multiplicative cipher in Python is as follows . I want to show you an example where we used it already. 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). The encrypted text is the smallest digit of an addition of plaintext and key when both are hexadecimal digits. Wonderful, that is all we need to solve our encryption function C= a*P MOD 26 for the plain letter P in order to then decode the encrypted message: Multiplying both sides of our encryption equation the equation yields a-1*C = a-1*(a*P) (1) = (a-1*a)*P (2) = 1*P (3) = P MOD 26 (4) Remark: Solving this equation required the 4 group properties: the existence of an inverse and the closure in (1), the associative property in (2), the inverse property in (3) and the unit element property in (4). the commonly used RSA Cipher is based on the relative slowness of such factoring programs. Thus, we now go ahead and practice a bit more computer programming. Which ones are those? It is not difficult to understand that the length of such numbers requires the usage of computers. So are 2 and 3, 2 and 5, 3 and 10, 26 and 27, 45 and 16. Which cracking method should a code cracker use. padding: 12px; For an alphabet length of 26 this corresponds to 12 keys: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25. These ads use cookies, but not for personalization. To do so, I distinguish between upper and lower case letters since they are encoded slightly different. Therefore, the set of all encoding keys must equal the set of all decoding keys. The Caesar cipher is a special case of the Affine cipher where A is 1 and B is the shift/offest. an idea ? If you are able to invent a fast factoring algorithm, you will not have to worry about a future job. For the encryption to be reversible (so that the message can be decrypted), the key must be a coprime number with 26 (where 26 is the number of letters of the alphabet). Example4: For M= 34 =81, we get u(81) = 34 - 33 = 81 27 = 54. 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. 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. The following table shows the numbers relative prime to M for the first 21 integers. A function that performs this is called an alphabet function. 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. 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: This means that the key should be a large, random number that is difficult to guess or factor.
multiplicative cipher calculator