It only takes a minute to sign up. How to recognize a Multiplicative ciphertext? color: #ffffff; That is weird! 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. The number fetched through output is mapped in the table mentioned above and the corresponding letter is taken as the encrypted letter. 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). Example3: Doing arithmetic MOD 7, the inverse of a=3 is a-1 = 5 because a * a-1 = 3*5 = 15 = 1 MOD 7. I will couple the Multiplication Cipher with the Caesar Cipher (which produces 26 unique encryptions) to obtain a super encryption that will allow 12*26=312 possible unique encryptions. Example the letter M (12th letter in this zero indexed alphabet) and key 3 would be 12 * 3 = 36. Notice that we found the good keys indirectly. A=65, B=66, C=67, .., Z=100, a=101, b=102, c=103, z=125. It was encoded MOD 26. Try to explain this, than continue reading! We can also calculate all the possible keys for the Affine Cipher. cryptography - Affine cipher - Modular multiplicative inverse 24 DOC Chapter 2 : Multiplicative Cipher - TI89 Implementation of Affine Cipher - GeeksforGeeks rev2023.5.1.43405. color: #ffffff; 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. Solution of Multipilicative Inverse of 7. an idea ? In conclusion, we can say that multiplicative cipher is a simple encryption technique that can be easily implemented. If a single character is encrypted by E(C) = (c * k) % 36 then possible keys k are numbers that are coprime to 36, ie.gcd(k,36)=1.Furthermore it makes not much sense to consider numbers not between 1 and 36, because of the modulo. Convert each group of numbers into column matrices. It is possible to distinguish between 2 types of actions in the plain text: uppercase letters [A-Z] and digits [0-9]. ((15)=((3*5)=(3-1)*(5-1)=2*4=8 as 1,2,4,7,8,11,13,14 are relative prime to 15. Encrypted text: The quick brown fox jumps over the lazy dog. Code No, it is not. I found a-1 = 2 by simply testing the integers in Z5*={1,2,3,4}. The calculator logic is explained below the calculator. Let us understand this by implementing a simple example using the Multiplicative Cipher. From now on we will use a handy Notation for the set of possible and good keys: 1) All the possible keys for an alphabet length of 26 are clearly all the numbers between 1 and 26, denoted as Z26. To show this, let's look at this equation: This is a linear diophantine equation with two unknowns; refer to Linear Diophantine Equations Solver. This is important because even if a key is secure when it is first chosen, it can become less secure over time as technology and methods for breaking encryption increase. 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. 39, 65, 91, ) have its equivalent key in a=13, another bad key, since 39=65=91=13 MOD 26. or . , 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 solution can be found with the Extended Euclidean algorithm. Alternatively, the non-alphabet letters in the key and the plain text can also be filtered out to increase the security. This is important because if the key shares a factor with the plaintext, it can be easily broken by factoring in the key. Our alphabet length of 28 now yields how many unique encryptions? 3. In fact, I always have to subtract 101 from each entered lower case plain letter to get its corresponding number. Text is divided into blocks of size n, and each block forms a vector of size n. Each vector is multiplied by the key matrix of n x n. The result, vector of size n, is a block of encrypted text. To do so, I distinguish between upper and lower case letters since they are encoded slightly different. Thirdly, listing the good keys would be best done using C++ vectors or even C-style arrays which you might know. . 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. How to encrypt using Multiplicative cipher? This process repeats until M is reduced to 1 and therefore less than the smallest factor possible, 2. Options: Multiplier: filter whitespace characters group 5 characters filter non-alphabet characters convert to first alphabet The modular multiplicative inverse of an integer a modulo m is an integer b such that 11 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. How would anyone ever break even this basic, amateurish cipher/encryption scheme? This modulo calculator performs arithmetic operations modulo p over a given math expression. 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. margin-bottom: 16px; Cryptography Tutorial - Multiplication Cipher : Decode It - TI89 The inverse function returns the n-th character for a number n in L. To n, the length of the list L is added or subtracted as often as necessary until the index lies in the list. Consider the letters and the associated numbers to be used as shown below , The numbers will be used for multiplication procedure and the associated key is 7. Try it! Remember that a function, per definition, assigns to each x-value one particular y-value. The theory can be found after the calculator. using properties 1) and 2) yields = (3-1)*(23-22) = 2*4 = 8. Each letter is associated with its rank $ c $ in the alphabet (starting from 0). 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. While using Caesar cipher technique, encrypting and decrypting symbols involves converting the values into numbers with a simple basic procedure of addition or subtraction. The key should be relatively prime to the length of the alphabet. 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"). will translate the H (=7) into a (=0), because 5*7 = 35 = 0 MOD 35. This, however, limits readability. We have to understand why multiplying by a bad key a MOD 26 yields some integers more than once and others not at all. Now when a=25, we have: 25*25 = 625. The algorithm memorizes the alphabet with which it has determined the number of the plaintext. I want to show you an example where we used it already. 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: "<. Since a=10 is a bad key he checks the good key a=23. As an attentive reader, we realize that the MOD multiplication of the keys is closed (recall the group properties in the previous chapter). M23456789101112131415161718192021( (M)12242648121041268816618812 Similar to our notation, the properties of Eulers (-function that computes the number of integers that are relatively prime to M and wrote similarly to our notation: Eulers (-function: 1) ((p) = p-1 for a prime p. 2) ((pn) = pn - pn-1 for a prime power pn. } Example1: If M=24=3*8=3*23, then ((24) = ((3*23) using property 4) yields = ((3)*((23). To find the multiplicative inverse of a real number, simply divide 1 by that number. We then write them in the form (1-1/p), multiply them and that product by M yielding ((M). 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. WAP to find the solutions of equations: a.14x=12mod 18 b.3x+4=6 mod 132. Now, lets come to the highlight of this section: I will show you in a few steps how to compute ((M) for any M from one equation instead of combining the four properties? Example4: If M=39=3*13=p*q, then the formula yields u(39) = (3-1)*(13-1) = 2*12 = 24. Therefore, an eavesdropper simply has to count letter frequencies to identify the most frequent cipher letter. 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". Multiplicative Cipher - Tutorial - scanftree Now, lets look at examples for MOD arithmetic: Example2: The inverse of a=3 is a-1 = 2 MOD 5 because a * a-1 = 3*2 = 6 = 1 MOD 5. ((25)=____________ as all integers from 1 to 24 except for 5,10,15,20 are relatively prime to 25. 1 Additionally, you will learn that the RSA Cipher uses prime numbers as well. 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. Not every key phrase is qualified to be the key; however, there are still more than enough. 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 when multiplied by 5: a-1 * 5 = 1 MOD 26. 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. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Mathematically: a-1 * a = a * a-1 = 1. 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. ((5)=_____ as 1,2,3,4 are relative prime to 5. This yields the correct plain text: Cipher textanromrjukahhouh013171412179201007714207 0131981819742017178417PLAIN TEXTANTISTHECARRIER As you can see, detecting the most frequent cipher letter is of enormous help in cryptography. C = (a * P) mod 26 In order to create unique cipher characters, we must use a multiplier which is co-prime (the values do not share any factors when dividing - see Try GCD of 5) in relation to the size of the alphabet (26), so you should use either 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 or 25. No provisions are made for high precision arithmetic, nor have the algorithms been encoded for efficiency when dealing with large numbers. Decoding aam can either yield NAT or ANT as the plain text. CacheSleuth - Multiplicative Cipher 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. In formula: u(M) = (M-1) b(M) using the above formula for the number of bad keys yields = M-1 - (M/p -1) distributing the minus sign to the terms in the parenthesis yields = M-1 - M/p + 1 canceling out the 1s yields = M - M/p This turns out to be a handy formula for the number of good keys. For the English alphabet, where m = 26, this means a cannot be 2, 4, 6, 8 (any even number) or 13. 3) ((p*q) = (p-1)*(q-1) for two distinct primes p and q. What Should Be the Length of the Symmetric Key in Cryptography? } 3.0.4224.0. Finally, I have to add the usual 65 = A (why?) Thus, among those numbers that occur twice in the cipher code, 14, 17 and 20, we can eliminate the odd 17. Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. The 26-letter Latin alphabet allows only 11 keys: 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25 (these are coprime numbers with 26). He obtains: Cipher textanromrjukahhouh013171412179201007714207 013116711232140151519215PLAIN TEXTANLGHLXCOAPPTCP That message does not reveal a virus carrier. Calculator Use Multiplication of positive or negative whole numbers or decimal numbers as the multiplicand and multiplier to calculate the product using long multiplication. 15 However, when using MOD arithmetic and solving 23=5*P MOD 26, we dont deal with fractions but only integers. Which number would that be? the commonly used RSA Cipher is based on the relative slowness of such factoring programs. 8 You should have realized that decoding means to undo the original multiplication. 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. 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. 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? They seem to not follow any apparent pattern. Subsequently, that difference is multiplied by the good key a=5 which I defined as such in int a=5. In this lab, you'll learn about the multiplication cipher, a monoalphabetic cipher. 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. (Definition). Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. For separate partial alphabets the following results: For a merged alphabets, the encrypted text is "02468ACEacACEae024". 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. What 1 formula is used for the Affine Cipher Calculator? } If a=1 is used as a key, each cipher letter equals its plain letter which shows that it does produce a unique encryption. By subtracting a (=101) from the entered plain letter in (pl -'a');. It describes the multiplicative property of (. This requires additional meta-information of the letters that must be recorded before encryption. Are the used 12 unique encryptions a set number? An alphabet[1] is an ordered set of all characters which can occur in a plaintext, a secret text, or the key. Contributed by: Shawna Martell (March 2011) Open content licensed under CC BY-NC-SA Snapshots In an additive cipher, the cipher alphabet is a shift of the plaintext alphabet. 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. Since there are 9 threes (or 9 multiples of 3) in 27 and therefore 8 threes when counting only up to 26 yielding the 8 listed bad keys. Therefore, we first have to add 65 to the 19 in order to translate the 84 eventually into the desired T using =CHAR(65+MOD(E$2*$B4,26)). What is the difference between "cipher" and "encryption"? Affordable solution to train a team and make them project ready. 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. 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. 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. So it will look like this after calculation. 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. If we had a video livestream of a clock being sent to Mars, what would we see? 1) This program both encodes and decodes. It has to be placed after the cout command as in: cout << setw(2) << j*factor. Its numerical equivalent reveals the row and therefore the key a as follows: PLAIN LETTER 0000000000000000000000000 ABCDEFGHIJKLMNOPQRSTUVWXYZ101234202468303691240481216505101520254914192438131823271217221611162160612182470714212808162469091811010010204141101122718120122410221301301301401421641501541981601662212170178251618018102201901912524200201482210211611622022181410230232017141185225221916131074124211815129632402422201825025242322 After intercepting the cipher text, an eavesdropper simply finds the most frequent letter of this rather brief message.
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