Try it for yourself! 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}. Furthermore it makes not much sense to consider numbers not between 1 and 36, because of the modulo. color: #ffffff; 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. 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? I found a-1 = 2 by simply testing the integers in Z5*={1,2,3,4}. 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. Step 2: The basic formula that can be used to implement Multiplicative Cipher is: Decryption= (C * Multiplication inverse of the key) Mod 26 Here, c = ciphertext Mod = Modulo Step 3: Let's see how decryption can be done using the above formula: Ciphertext = QCCSWJUPQCCSW and multiplication inverse key = 15 unchanged so that you can detect the format of the original message easier. 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. For a given alphabet, there are only a few possible keys. a bug ? div#home a:link { 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. Say a=5 was chosen. This formula can be simplified into the product of two factors. Examples are: 4 and 5 are relatively prime because gcd(4,5)=1. Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. 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. This is also the case when the letter is in the key. . Say you first want to encode the letter c then you have to enter e when asked. 9 Vice versa, the cost of detecting the most frequent cipher letter in the first approach is at the gain of producing only one plain text provided that the most frequent cipher letter turns out to be unique. Note The advantage with a multiplicative cipher is that it can work with very large keys like 8,953,851. We can combine these two criteria into one easy criterion. This website would like to use cookies for Google Analytics. Generally: The good keys are those as that are relative prime to M and are denoted as ZM*. That would additionally require that the good keys form a commutative group with respect to addition. Even products of 3 primes or prime powers like 30 or 60 can now be dealt with due to the 4th property: Example2: If M=30=2*3*5, then ((30) = ((2*3*5) using property 4) yields = ((2)*((3*5) again property 4) yields = ((2)*((3)*((5) now using property 1) yields = 1*2*4 = 8. You noticed, that the multiplicative property of Eulers (-function, expressed in property 4), is used to decompose any integer M into its prime factors or prime power factors to then apply the first two properties to each prime or prime power. 23 v l X X X 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. Our alphabet length of 28 now yields how many unique encryptions? Since a=10 is a bad key he checks the good key a=23. =CODE("a") yields 97). For a check: the eight integers 1,5,7,11,13,17,19,23 are relative prime to 24 and thus the good keys for M=24. 25 The reason for that is that a prime number has per definition no prime divisor except for 1 and itself. //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). Therefore, we just have to add a number in order to get k=111. 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. How could it be broken? This is important because if the key shares a factor with the plaintext, it can be easily broken by factoring in the key. If a = 1, the Affine cipher is equivalent of a Caesar cipher. 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. 3 Why did US v. Assange skip the court of appeal? Simply: Z26 = {0,1,2,3,, 24,25}. Before Conversion: ABCDEFGHIJKLMNOPQRSTUVWXYZ After Conversion: XYZABCDEFGHIJKLMNOPQRSTUVW Age Calculators Example1: M=9=32 has the only prime divisor 3 and thus b=9/3 1 = 2 bad keys which are 3 and 6 as the multiples of 3 that are less than 9. } You could also define a to be a different good key. Thus, the encryption process is a Caesar cipher merged with a multiplication cipher. Right, we have to add 101 to the 10 which we do by adding a=101 in cl='a' + (a*(pl -'a'))%26. It uses genetic algorithm over text fitness function to break the encoded text. First, symbols of the used alphabet (alphabet as a set of symbols, for example, the alphabet in the above calculator includes space, comma, and dot symbols) are encoded with digits, for example, symbol's order number in the set. does not work internally with letters, but with numbers. I'm learning and will appreciate any help. Affine cipher - Modular multiplicative inverse. 2 20 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. Now the cipher letter cl equals k and we can end the lower case encoding. So, we are left with determining the decoding key a-1 knowing the original encoding key a. 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. The modular multiplicative inverse of an integer a modulo m is an integer b such that "Signpost" puzzle from Tatham's collection, xcolor: How to get the complementary color. So there is an infinite number of possible keys, but many will give identical messages, because for a $ k $ key, then the $ k + 26 $ key gives an identical cipher. We can therefore always find a-1 for a given good key a. 17 3.0.4224.0, The greatest common divisor of two integers, The greatest common divisor and the least common multiple of two integers, Solution of nonhomogeneous system of linear equations using matrix inverse. We have explored the first three properties already, however, the 4th property is new - but not totally new. Say, the lower case plain letter c is entered, then the condition if ((pl>='a') && (pl<='z')) is fulfilled and the encryption is being executed by this one seemingly weird command cl='a' + (a*(pl -'a'))%26; Let me explain that to you in detail: First you need to know that each letter is stored as a number: i.e. or . Feedback and suggestions are welcome so that dCode offers the best 'Multiplicative Cipher' tool for free! Consequently, the longer a cipher text, the easier the cipher E can be detected. For the English alphabet, where m = 26, this means a cannot be 2, 4, 6, 8 (any even number) or 13. The alphabet function sL returns the smallest index at which it occurs to a letter that is present in L. The index of the first character can be configured. How would anyone ever break even this basic, amateurish cipher/encryption scheme? This weirdness is not really weird. In general we have the: Formula for the number of good keys if M is a prime If the alphabet length M=p is prime, the number of good keys is u(p) = p-1. We know already that: ((60) = ((22*3*5) = (22-21)*(3-1)*(5-1)((M) = ((p12* p2* p3) = (p12- p11)*( p2-1)*( p3-1). These are valuable information for an eavesdropper that help cracking the message. It would take quite a long time for a computer to brute-force through a majority of nine million keys. If we dont want to give an eavesdropper any additional information about our secret message, we would firstly either not use such characters at all or we would expand our alphabet length and encode them just like the other plain letters. In some secret manner, the sender and the recipient had to agree on the encoding key a. 4 Affordable solution to train a team and make them project ready. Verify this now! 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. I will answer it at the end of this chapter in the Abstract Algebra section. The letter A remains unchanged ans id always encoded A. ((8)= ((23)=23 -22 =4 as 1,3,5,7 are relative prime to 8. This is very likely in English texts and virtually certain in the German language where on average every 5th letter is an E. Even if an eavesdropper decides to produce all 12 possible plain texts, they can be generated with the help of a computer within a few seconds. 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? How to pick a symmetric cipher for a given cipher text size? dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!A suggestion ? 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. The next two lines then show us that the variable false is defined as 0 and true as 1. Try to understand as much as possible first, then continue reading. 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) <. Among the 12 good keys we pick a=5 to encode the virus carrier message as follows: PLAIN TEXTANTISTHECARRIER0131981819742017178417 013171412179201007714207Cipher textanromrjukahhouh Exercise1: Encrypt the same plain text using the key a=7. This program is an extension of the previous simple factoring program. The only disadvantage is that the minus sign itself has to be written as "---", so as not to be confused as a range operator. 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. 8 In the next chapter, I will show you one principle of increasing the safety of a cipher code. 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. That is, . color: #ffffff; These calculations were correct but almost required a calculator. Code Or are they possibly the primes between 1 and 25? Sphero Up to 1 Hour Grades: 5 to 8. 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). The final formula uses determinant and the transpose of . 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. Cipher textanromrjukahhouha=1ANROMRJUKAHHOUHa=3ANXWEXDYMALLWYLa=5ANTISTHECARRIERa=7ANVCYVFOUABBCOBa=9ANZQKZBIEAVVQIVa=11ANLGULPQIADDGQDa=15ANPUGPLKSAXXUKXa=17ANBKQBZSWAFFKSFa=19ANFYCFVMGAZZYMZa=21ANHSIHTWYAJJSWJa=23ANDEWDXCOAPPECPa=25ANJMOJRGQATTMGT MS Excel as a simple encryption and decryption tool: I created the following table in MS Excel with the CHAR and the MOD function: Cipher textanromrjukahhouhaa-101317141217920100771420739ANXWEXDYMALLWYL521ANTISTHECARRIER715ANVCYVFOUABBCOB93ANZQKZBIEAVVQIV1119ANLGULPQIADDGQD157ANPUGPLKSAXXUKX1723ANBKQBZSWAFFKSF1911ANFYCFVMGAZZYMZ215ANHSIHTWYAJJSWJ2317ANDEWDXCOAPPECP2525ANJMOJRGQATTMGT For example, I created the T in the row a=5 using the Excel-formula =CHAR(65+MOD(E$2*$B4,26)) where the cell E$2 contains 17 and the cell $B4 contains 21 as the decoding key a-1. If multiplication is used to convert to cipher text, it is called a wrap-around situation. Write to dCode! 6 The 14 as the possible cipher E then tells him to test the keys a=10 and a=23. So, lets understand why the bad keys a = 2,4,6,8,10,12,13,14,16,18,20,22,24 dont produce a unique encryption. This corresponds to the K. If "GEHEIMNIS" would be completely encoded by this procedure, the ciphertext would be: "SMVMYKNYC". A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. 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. Mathematically, calculate the modular inverse $ k^{-1} $ of the key modulo 26 and apply the calculation for each letter: Example: The key $ 17 $ has the inverse modulo 26 of the value $ 23 $ so Z (index 25) becomes $ 25 \times 23 \mod 26 \equiv 3 $ and 3 corresponds to D in the alphabet. So in our case, it was GEEKSFORGEEKS, so it will become: Multiplicative Cipher text = QCCSWJUPQCCSW. Moreover, since a=13 is a bad key its multiples 26, 39, must also be bad keys. It is actually less secure than the Caesar cipher because the number of possible keys is smaller. Additionally, you will learn that the RSA Cipher uses prime numbers as well. This requires additional meta-information of the letters that must be recorded before encryption. Does the increase of our alphabet length by 1 increase the number of unique encryptions obtained? background-color: #620E01; By subtracting a (=101) from the entered plain letter in (pl -'a');. What are the variants of the Multiplicative cipher. 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. 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. Are they the odd numbers between 1 and 25? color: #ffffff; Each character from the plaintext is always mapped to the same character in the ciphertext as in the Caesar cipher. 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). Read about it on wiki, I will provide only example. 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. The Multiplicative Cipher is an Affine cipher (ax+b) with the value b null (equal to 0), so a multiplication by a a. for the RSA encryption. 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? So which ones do? To show this, let's look at this equation: This is a linear diophantine equation with two unknowns; refer to Linear Diophantine Equations Solver. The modular multiplicative inverse of a modulo m can be found with the Extended Euclidean algorithm. Parts of Long Multiplication 2 5 6 Multiplicand 3 2 Multiplier + 5 1 2 Partial Product + 7 6 8 Partial Product = 8 1 9 One of the major goals of current Mathematics research is to design faster factoring algorithms as todays are fairly slow.