If 2 would be used as key, then C=2 -> 2*2 = 4 -> so the character C is encoded as an E. Analogous, P=15 -> 15*2 = 30 -> 30 modulo 26 = 4 -> the character P would also be encoded as an E.Therefore, it is not possible to determine if an E in a ciphertext corresponds to a C or a P. In order to achieve a unique matching only keys that are co-prime to the length of the alphabet can be used. The first character G corresponds to the six. Information Security Stack Exchange is a question and answer site for information security professionals. a feedback ? It is a-1=4 since 3*4 = 12 = 1 MOD 11. However, converting 19 to its character does not yield the desired T. The T is stored as 84 which you could see by entering the Excel formula =CODE("T"). I accomplish this. Step 1: So here we are going to cipher text a simple plain text, let us assume the plain text is GEEKSFORGEEKS and let us consider the key as 7. It describes the multiplicative property of (. You can change your choice at any time on our, Modular Multiplicative Inverse Calculator. Lets investigate this in the following section. We also turn the plaintext into digraphs (or trigraphs) and each of these into a column vector. The command const is used as a safety feature in C++: both variables are constant and can never be modified in any program. PLAIN LETTER:ABCDEFGHIJKLMNOPQRSTUVWXYZ Secret key: a=2012345678910111213141516171819202122232425 024681012141618202224024681012141618202224 Cipher letter:acegikmoqsuwyacegikmoqsuwy Notice, that only every other cipher letter appears, and that exactly twice. Each character is multiplied with this key and the corresponding letter is substituted. Code 20 Also, there is no general match on how to handle digits or special characters. Combining our three formulas for the number of good keys, we will then be able to develop a general formula for the number of good keys for any given alphabet length M. Lets start with Example1: M=26=p*q=2*13. Try it for yourself. All symbols to be encrypted must belong to alphabet, Everyone who receives the link will be able to view this calculation, Copyright PlanetCalc Version: The Multiplicative Cipher is an Affine cipher (ax+b) with the value b null (equal to 0), so a multiplication by $ a $. Sphero Up to 1 Hour Grades: 5 to 8. //Author: Nils Hahnfeld 10-16-99 //Program to determine ((M)using M*(1-1/p1)*(1-1/p2)* #include #include void main() { int factor, M, m; float phi; clrscr(); cout << "This programs uses M*(1-1/p1)*(1-1/p2)* to calculate phi(M). We can therefore always find a-1 for a given good key a. 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. It would take quite a long time for a computer to brute-force through a majority of nine million keys. Examples for property 4): 24 and 28 are products of primes and prime powers. 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. Are there any canonical examples of the Prime Directive being broken that aren't shown on screen? Definition of an inverse number: A number a-1 that yields 1 when multiplied by a is called the inverse of a. 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. 24 The Affine Cipher uses modulo arithmetic to perform a calculation on the numerical value of a letter to create the ciphertext. Reminder : dCode is free to use. Encrypted text: The quick brown fox jumps over the lazy dog. Credit goes to the Swiss Mathematician Leonard Euler (pronounced Oiler, 1707-1783). The message "ACDC" should be encrypted with the key "ABBA" according to the Vigenre method. Finding the decoding keys for each good key a in the same manner, we obtain the following key pairs: Good Encoding key aIts decoding key a-111395217159311191571723191121523172525 Three important observations: All decoding keys a-1 in the right column are among the set of all encoding keys a. Notice, that property 3) became useless for the calculation process since factors that are relative prime are separated via property 4). Now, how do you decrypt the above message? We can combine these two criteria into one easy criterion. Lets add a dot to our alphabet to denote the end of a sentence in the original message. The solution can be found with the Extended Euclidean algorithm. 1 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. 36 modulo 26 = 10 so the letter K would be chosen. How do you find the key domain of the multiplication cipher efficiently? We just had to multiply each cipher letter by a-1. 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. 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. Those are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75 and 78 as the multiples of 3 that are less than 81. 2) If M is a prime power, M=pn: Now lets look back at M=27 as an example where we only have the one prime factor p=3, such that M=33. 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. Example3: Doing arithmetic MOD 7, the inverse of a=3 is a-1 = 5 because a * a-1 = 3*5 = 15 = 1 MOD 7. //Author: Nils Hahnfeld 10/15/99 //Factoring program #include #include #include void main() { int M, factor ; clrscr(); do { cout << "Enter the integer that you want to factor or 0 to exit: M="; cin >> M; factor=2; while(factor <= M) { if (M%factor==0) //check all integers less than M as factors { cout << factor << endl; M/=factor; factor=1; } factor++; } }while(M!=0); } Programmers remarks: Starting with 2, this program checks the integers from 2 to M-1 as potential factors of M in if (M%factor==0). If a = 1, the Affine cipher is equivalent of a Caesar cipher. 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.) 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). 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. Moreover, since a=13 is a bad key its multiples 26, 39, must also be bad keys. Well, I leave all the entered non-letters such as ! I will answer it at the end of this chapter in the Abstract Algebra section. 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. 3 * 9 = 9 * 3 =27) the MOD- multiplication is commutative (3 * 9 = 9 * 3 = 1 MOD 26). However, there are some additional integers that are not prime (i.e. Mathematically: a-1 * a = a * a-1 = 1. It uses genetic algorithm over text fitness function to break the encoded text. The modular multiplicative inverse of an integer a modulo m is an integer b such that 2) u(pn)= pn - pn-1, if M is a power of a prime M= pn. 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. There are several way to implement the inversion and the affine transformation described in the AES to get the final SBox. Multiplicative Cipher : Encryption Decryption Method | Mono-alphabetic Substitution Cryptography Quick Trixx 5.13K subscribers Subscribe 38K views 5 years ago Cryptography and Network Security. Say you first want to encode the letter c then you have to enter e when asked. As an attentive reader, we realize that the MOD multiplication of the keys is closed (recall the group properties in the previous chapter). background-color: #620E01; In some secret manner, the sender and the recipient had to agree on the encoding key a. This is it. Since, for the standard alphabet, there are 12 numbers less than 26 which are . Additional restrictions to the key are imposed by the need to decrypt encrypted text :). Characters not belonging to the alphabet are not encrypted or allowed as keys. Simply by looking at the table, we find that the following keys (whose rows are bold) produce a unique encryption and therefore call them the good keys: a = 1,3,5,7,9,11,15,17,19,21,23,25 Why those and what do they have in common? 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. Contributed by: Shawna Martell (March 2011) Open content licensed under CC BY-NC-SA Snapshots 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. div#home a:active { Therefore, no matter how he decides to crack the cipher text, it wont take long. unchanged so that you can detect the format of the original message easier. Example1: When using fractions, 5-1=1/5 is the inverse number to 5, 3-1=1/3 is the inverse number to 3, 3/2 is the inverse number to 2/3. Divide the letters of the message into groups of two or three. 3) u(p*q) = (p-1)*(q-1), if M is a product of two primes M=p*q. Learn more about Stack Overflow the company, and our products. What is the symbol (which looks similar to an equals sign) called? How do we deal with non-letters? for M=29 we have u(29)=28. 17 Step 2: The basic formula that can be used to implement Multiplicative Cipher is: Decryption= (C * Multiplication inverse of the key) Mod 26. 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. 11 To find a multiplicative inverse We need to find a number x such that: If we find the number x such that the equation is true, then x is the inverse of a, and we call it a^-1. Our alphabet length of 28 now yields how many unique encryptions? In the next chapter, I will show you one principle of increasing the safety of a cipher code. Since a=10 is a bad key he checks the good key a=23. WAP to implement Additive cipher(key=20), Multiplicative cipher(key=15)and affine cipher(key=15,20). 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. Determining the bad keys for a given alphabet length M is a perfect task for a computer. 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. You can find the reciprocal quite easily. If a=4,6,8,,24, we encounter the same dilemma as for a=2. 3 Right, we have to add 101 to the 10 which we do by adding a=101 in cl='a' + (a*(pl -'a'))%26. Test it yourself. Modular arithmetic is used; that is, all operations (addition, subtraction, and multiplication) are done in the ring of integers, where the modulus is m - the length of the alphabet. The same alphabet is used to generate the encrypted text. Example: Encrypt DCODE with the key k= 17 k = 17 and the 26-letter alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ Options regulate the case when a letter does not appear in any alphabet: it is not encrypted, but transferred directly to the output. Moreover, you can see that the plain letter V encrypts to the cipher text letter b (=1) when using a=5 as the encoding key. Not every key phrase is qualified to be the key; however, there are still more than enough. If a single character is encrypted by E(C) = (c * k) % 36 then possible keys k are numbers that are coprime to 36, ie. Modified 8 years, 6 months ago. "Ordered" means that sorting is possible and we can speak of the n-th character of an alphabet. ((28) = _____________________________ as 1,3,5,9,11,13,15,17,19,23,25,27 are relative prime to 28. In this lab, you'll learn about the multiplication cipher, a monoalphabetic cipher. Let us understand this by implementing a simple example using the Multiplicative Cipher. The number obtained indicates the rank in the alphabet of the corresponding numbered letter. For a given alphabet, there are only a few possible keys. In fact, the sets of the encoding and decoding keys are identical. That is weird! The grey rows show what would be expected for the order, and the red one shows what your text gives for the order: If we use a value which is not co-prime, such as 2, we will not get unique characters for the mapping: Bib: @misc{asecuritysite_99257, title = {Multiplication Cipher}, year={2023}, organization = {Asecuritysite.com}, author = {Buchanan, William J}, url = {https://asecuritysite.com/coding/mult}, note={Accessed: May 02, 2023}, howpublished={\url{https://asecuritysite.com/coding/mult}} }. It only takes a minute to sign up. 2.1 Encryption using the Multiplication Cipher Instead of encoding by adding a constant number, we multiply each plain letter by our secret key a. b ^ ^ ^ 8 ^ a G n n n n n R R R f h h h h h h $ u R ` R R R n n n n 7 R j n n n f R f k \ ^ % n n `d P ^ v$ .$ r % T 0 G $ r 2 % n n n n Chapter 2 Multiplicative Cipher In this chapter we will study the Multiplicative Cipher. The statements while(cl!='~') and cout << cl; cin >> pl; are in charge of it. Zero has no modular multiplicative inverse. 3) If the alphabet length M is a product of two prime numbers p and q The last case we have to study is when M is a product of two primes. 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. The next two lines then show us that the variable false is defined as 0 and true as 1. First of all, you need to know which one of the 12 good keys was used. ((25)=____________ as all integers from 1 to 24 except for 5,10,15,20 are relatively prime to 25. For example if we use "abcdefghijklmnopqrstuvwxyz" and a multiplier of 3, gives "adgjmpsvybehknqtwzcfilorux". Introduction to Monotonic Stack - Data Structure and Algorithm Tutorials. I first subtract 65 =A and then multiply that difference by the good key a=5 yielding 10 again. background-color: #620E01; As a small example we consider Vigenre with the following two alphabets: In both cases, both the plaintext and the key should both consist of the text "0123456789abcdABCD". Thus, property 4) yields nothing new if our alphabet length is the product of two primes. While using Caesar cipher technique, encrypting and decrypting symbols involves converting the values into numbers with a simple basic procedure of addition or subtraction. 1) Learn how to decode the Multiplication Cipher. 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. 15 The handling of non-alphabet characters (convert, skip, ) can be set in the options - but this is not a function of the actual encryption process itself. For each character of the plain message, apply the following calculation: ($ 26 $ being the number of letters in the alphabet). 1. Parabolic, suborbital and ballistic trajectories all follow elliptic paths. Multiplying such answers yields the number of good keys for any given alphabet length. Example5: If M=65=5*13=p*q, then the formula yields u(65) = (5-1)*(13-1) = 4*12 = 48. Modulus m. padding: 12px; 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. Thus, dividing is performed slightly different: instead of dividing by 5 or multiplying by 1/5, we first write 5-1 (instead of 1/5) where 5-1 now equals an integer and multiply both sides by that integer 5-1. Note: This cipher is closely related to the. Try to understand as much as possible first, then continue reading.
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