multiplicative cipher calculator
For a check: the same eight integers 1,5,7,11,13,17,19,23 are relative prime to 30 and are thus the good keys for M=30. We have to understand why multiplying by a bad key a MOD 26 yields some integers more than once and others not at all. Variant Beaufort cipher Base32 Hash function Morse code to text Z-Base-32 View Affine cipher Slope / a Step Down Step Up How do you find the key domain of the multiplication cipher efficiently? Example5: Try it yourself! a bug ? I will answer it at the end of this chapter in the Abstract Algebra section. In some secret manner, the sender and the recipient had to agree on the encoding key a. I.e. color: #ffffff; In fact, I always have to subtract 101 from each entered lower case plain letter to get its corresponding number. Since a=10 is a bad key he checks the good key a=23. Two MacBook Pro with same model number (A1286) but different year. As 29 is prime, it has no divisors except for 1 and 29 and thus there are no multiples as bad keys. You are asked to enter your plain letter in cin >> pl; As long as you dont enter ~ the while-condition while(pl!='~') is fulfilled and the entered plain letter (=pl) is being encoded. PLAIN LETTERNATANTSecret key a=2130190131900120012Cipher letteraamaam You can see the dilemma of this message. Thus our decoding function P = a-1*C MOD 26 tells us to simply multiply each cipher letter by the inverse of the encoding key a=5, namely by the decoding key a-1=21 MOD 26 and we can eventually decode: Cipher textanromrjukahhouh013171412179201007714207 0131981819742017178417PLAIN TEXTANTISTHECARRIER For example, multiplying the cipher letter r=17 by a-1 = 21 decodes the r to T=19 since 21*17 = 357 = 19 MOD 26. In our example, after subtracting 101 from the plain letter c we get the desired 2 that is now multiplied by a=5 yielding 10. h2 { Example: Encrypt DCODE with the key k= 17 k = 17 and the 26-letter alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ This weirdness is not really weird. or ? We saw that an alphabet length of M=26 produces 12 unique encryptions, since the even numbers as multiples of 2 and the 13 are the 13 bad keys. 6 How does the j decode to the H, and the u decode to the E? Which ones are those? Again, we just have to find the cipher numbers in the 5th row and then go up that column to the very top to find the corresponding plain letter. This program is an extension of the previous simple factoring program. This inverse modulo calculator calculates the modular multiplicative inverse of a given integer a modulo m. 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. . While using Caesar cipher technique, encrypting and decrypting symbols involves converting the values into numbers with a simple basic procedure of addition or subtraction. A summary of our explorations for the number of good keys shows: 1) u(p) = p - 1, if M is prime M=p. When a letter occurs in several alphabets, the first of these alphabets is used. The modular multiplicative inverse of a modulo m can be found with the Extended Euclidean algorithm. In such case, divide M by that factor: M/=factor; and start checking M/factor for factors less than M/factoretc. 21 is an inverse to 5 MOD 26, therefore 5 is inverse to 21 and the two 1s are mirrored over the diagonal line. Sphero Up to 1 Hour Grades: 5 to 8. Method 2: Merged: In the alphabet, mod 22 is calculated because the alphabet contains 22 elements. Example the letter M (12th letter in this zero indexed alphabet) and key 3 would be 12 * 3 = 36. In conclusion, we can say that multiplicative cipher is a simple encryption technique that can be easily implemented. Verify this now! Try it for yourself! 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). The encrypted text is the smallest digit of an addition of plaintext and key when both are hexadecimal digits. Credit goes to the Swiss Mathematician Leonard Euler (pronounced Oiler, 1707-1783). The first character G corresponds to the six. 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. Do they? Example: Encrypt DCODE with the key $ k = 17 $ and the 26-letter alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ. You can observe this order-doesnt-matter rule in the original 26x26 multiplication table: The diagonal line from the top left to the bottom right forms a reflection line. To show this, let's look at this equation: This is a linear diophantine equation with two unknowns; refer to Linear Diophantine Equations Solver. How many multiples of 3 will not produce a unique encryption? The multiplicative cipher is a special case of the Affine cipher where B is 0. 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. Are the used 12 unique encryptions a set number? "Ordered" means that sorting is possible and we can speak of the n-th character of an alphabet. The key should be kept secret and only shared with authorized parties. 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. 4) ((n*m) = ((n) * ((m) when n and m are relatively prime. All we need to know are the prime divisors of M, but we dont even need to know how often a prime number divides M. Multiply It! where c is the modular multiplicative inverse of a. In an additive cipher, the cipher alphabet is a shift of the plaintext alphabet. 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. 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. "Signpost" puzzle from Tatham's collection, xcolor: How to get the complementary color. The key should not be easily guessable or should not be easily cracked. We then perform matrix multiplication modulo the length of the . Note The advantage with a multiplicative cipher is that it can work with very large keys like 8,953,851. How could it be broken? 25, Encrypt Ubuntu won't accept my choice of password. Our ultimate goal is not to develop a formula for the number of bad keys but rather for the number of good keys. For larger integers, however, dividing by every integer less than M slows the program down enormously. Can I use an 11 watt LED bulb in a lamp rated for 8.6 watts maximum. This shows that when using an encoding key that is one less than the alphabet length M, namely a = M-1, then the decoding key must also equal M-1, a-1 = M-1. } It is actually less secure than the Caesar cipher because the number of possible keys is smaller. What is the difference between "cipher" and "encryption"? 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. No, 9,15, 21 and 25 are not prime and the prime 13 is missing. No, it is not. For the purpose of setting up an explicit formula for ((M), we now try to give the three factors (in parentheses) the same format. For classical methods, the alphabet often consists only of the uppercase letters (A-Z). In order to be able to use the command setw() we have to include the iomanip.h library in #include . It converts to the plain letter number 26 so that we now have to encrypt MOD 27. div#home { Option 2: Cracking the cipher code using trial and error (brute force) Knowing that there are just 12 possible unique encryptions MOD 26, the journalist produces the corresponding 12 rows in the 26 x 26 multiplication table and cracks the code easily. Parabolic, suborbital and ballistic trajectories all follow elliptic paths. Thank you! This means that the key should be a large, random number that is difficult to guess or factor. for M=29 we have u(29)=28. The key should be relatively prime to the length of the alphabet. Introduction to Monotonic Stack - Data Structure and Algorithm Tutorials. Then the Vigenre encryption for an input character in and a key key can be described as: The letters of in and key are converted into numbers, these numbers are added, and the sum is re-converted to a letter. Boolean algebra of the lattice of subspaces of a vector space? 7 11 Reciprocal (or) Multiplicative Inverse is: 3 After finding each factor of M, I just print them out in for (j=1;j #include #include #include void main() { int M, m, j, factor, factor2; bool prime; clrscr(); cout << "This program finds the 'bad' keys for an entered alphabet length M." << endl; cout << "===========================================================================" << endl; do { cout << "Enter the alphabet length or 0 to exit: M="; cin >> M; m=M; factor=2; prime=0; //initialization while(factor <= m) { if (m%factor==0) { if (factor!=M) { cout << "Divisor of "<< M << " =" << setw(3) <. A=65, B=66, C=67, .., Z=100, a=101, b=102, c=103, z=125. If you are able to invent a fast factoring algorithm, you will not have to worry about a future job. Extracting arguments from a list of function calls. ((3)=3-1=2 as 1 and 2 are relative prime to 3. 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. The three factors in the parentheses already have the same desired format, however, the single 2 destroys it. 23 The command const is used as a safety feature in C++: both variables are constant and can never be modified in any program. The only two keys that are inverse to themselves are 1 and 25, which means that the encoding key equals its decoding key. That would additionally require that the good keys form a commutative group with respect to addition. 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. 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. where the operation of multiplication substitutes the operation of division by the modular multiplicative inverse. The following steps take place: In the example, an overflow has occurred in the third letter, so that modulo |L| = 4 is calculated. Learn more about Stack Overflow the company, and our products. 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*. 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.) Which language's style guidelines should be used when writing code that is supposed to be called from another language? Modulo Arithmetic & Ciphers. We can also calculate all the possible keys for the Affine Cipher. 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. This encoding and decoding is working based on alphabet shifting & transforming the letters into numbers . 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. RSA Express Encryption/Decryption Calculator This worksheet is provided for message encryption/decryption with the RSA Public Key scheme. Each character in the message is multiplied with this key. 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. ((25)=____________ as all integers from 1 to 24 except for 5,10,15,20 are relatively prime to 25. Multiplicative encryption uses a key $ k $ (an integer) and an alphabet. 14 I accomplish this. The solution shows the work for the Standard Algorithm. Fraction calculator - subtracting fractions step by step with explanation With the Fractions Calculator, you can subtract any two mixed numbers or proper and improper fractions. I leave the translation from an upper case plain letter to a lower case cipher letter as an easy exercise for you. So it will look like this after calculation. Finally I understand how to calculate the modular multiplicative inverse :) $\endgroup$ - np00. Alphabets (yes, there may be several: more below) can be described by a list L of letters. Let us consider the above cipher text i.e. 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. Example1: If M=24=3*8=3*23, then ((24) = ((3*23) using property 4) yields = ((3)*((23). background-image: none; If a=4,6,8,,24, we encounter the same dilemma as for a=2. Thus, the number of bad keys can simply be found by dividing the alphabet length M by the only prime divisor p and subtracting 1 from that fraction: M/p - 1. Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. To verify this: 262 = 676 =1 MOD 27. Try to explain this, than continue reading! A corresponding warning is displayed. The Multiplicative Cipher is an Affine cipher (ax+b) with the value b null (equal to 0), so a multiplication by $ a $. An alphabet[1] is an ordered set of all characters which can occur in a plaintext, a secret text, or the key. What is the inverse of 5 MOD 11? Each row that contains each integer from 0 to 25 exactly once and therefore yields a unique cipher letter will serve. We have explored the first three properties already, however, the 4th property is new - but not totally new. That is, they mustn't have any common divisors. Example3: Doing arithmetic MOD 7, the inverse of a=3 is a-1 = 5 because a * a-1 = 3*5 = 15 = 1 MOD 7. 1. Back to the virus carrier message. 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 . Just as the regular multiplication of two integers is commutative (i.e. 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. (Definition). We make use of First and third party cookies to improve our user experience. Each character is multiplied with this key and the corresponding letter is substituted. However, there is no 7 the numerical equivalent of letter h - in the E column. 25 The reason is (M-1) * (M-1) = (-1) * (-1) = 1 MOD M. For example: when using an alphabet length of M = 27 and an encoding key a=26 then its decoding key is a-1 =26. Which was the first Sci-Fi story to predict obnoxious "robo calls"? Is there such a thing as "right to be heard" by the authorities? Coincidence? 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. 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. 10 2) u(pn)= pn - pn-1, if M is a power of a prime M= pn. A little computer program turns out to be again very valuable as the number of good keys can be easily determined by first finding all prime factors of M to then use the above explicit formula. A multiplicative cipher is a type of cipher that comes under a monoalphabetic cipher, in which each letter that is present in the plaintext is replaced by a corresponding letter of the ciphertext, according to a fixed multiplication key. Can you? 1 Are there any canonical examples of the Prime Directive being broken that aren't shown on screen? Thus they have the following restrictions: The bad key a=2 yields an ambiguous message as we saw in the introductory example: each A turns into 0 (=a) since 2*0 = 0 MOD 26 just as each N turns into 0 since 2*13 = 26 = 0 MOD 26. For illustration purposes we use the message "GEHEIMNIS" and the key 3. 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 number fetched through output is mapped in the table mentioned above and the corresponding letter is taken as the encrypted letter. We can combine these two criteria into one easy criterion. Then we perform the reverse operations performed by the encryption algorithm. The copy-paste of the page "Multiplicative Cipher" or any of its results, is allowed as long as you cite dCode! According to the definition in wikipedia, in classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. 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! ~=.., $=.. etc. Certainly none of the cryptosystems we have considered thus far. In order to decrypt the message we need a combination of a Caesar and a multiplication cipher decryption. The encryption function looks like this: f (x)= ax+b mod . It thus gives a great example that we are only guaranteed to solve this equation for numbers that form a group with respect to multiplication MOD 26. But the modular multiplicative inverse is a different thing, that's why you can see our inverse modulo calculator below. Characters not belonging to the alphabet are not encrypted or allowed as keys. So, we are left with determining the decoding key a-1 knowing the original encoding key a. 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. 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. 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). 3) u(p*q) = (p-1)*(q-1), if M is a product of two primes M=p*q. Affine Cipher is the combination of Multiplicative Cipher and Caesar Cipher algorithm. Example4: For M= 34 =81, we get u(81) = 34 - 33 = 81 27 = 54. Builds the Affine Cipher Translation Algorithm from a string given an a and b value This calculator has 3 inputs. Except for 2 and 13, all prime numbers less than 26 are among the keys (why do they have to?). This formula can be simplified into the product of two factors. Example4: What is the inverse of 3 MOD 11? 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. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. 9 Firstly I have no idea how they derived this formula, but I think I have a general idea. This is it. You could also define a to be a different good key. I found a-1 = 2 by simply testing the integers in Z5*={1,2,3,4}. Encrypted text: The quick brown fox jumps over the lazy dog. For letters that do not occur in L, the alphabet function sL is undefined. An easier way to determine the decoding key a-1 Decoding a message turns out to be really easy once we know the decoding key a-1. color: #ffffff; Now the cipher letter cl equals k and we can end the lower case encoding. color: #ffffff; Here is the C++ Code for the encryption and decryption of the multiplication cipher: //Multiplication Cipher using the good key a=5 //Author: Nils Hahnfeld, 9/22/99 #include #include void main() { char cl,pl,ans; int a=5, ainverse=21; //as a-1*a=21*5=105=1 MOD 26 clrscr(); do { cout << "Multiplication Cipher: (e)ncode or (d)ecode or (~) to exit:" ; cin >> ans; if (ans=='e') { cout<< "Enter plain text: "<< endl; cin >> pl; while(pl!='~') { if ((pl>='a') && (pl<='z')) cl='a' + (a*(pl -'a'))%26; else if ((pl>='A') && (pl<='Z')) cl='A' + (a*(pl -'A'))%26; else cl=pl; cout << cl; cin >> pl; } } else if (ans=='d') { cout << "Enter cipher text: " << endl; cin >> cl; while(cl!='~') { if ((cl>='a') && (cl<='z')) pl='a' + (ainverse*(cl -'a'))%26; else if ((cl>='A') && (cl<='Z')) pl='A' + (ainverse*(cl -'A'))%26; else pl=cl; cout << pl; cin >> cl; } } } while(ans!='~'); } Programmers Remarks: Can you understand the code yourself?
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