multiplicative cipher calculator
((28) = _____________________________ as 1,3,5,9,11,13,15,17,19,23,25,27 are relative prime to 28. Read about it on wiki, I will provide only example. div#home { 565), Improving the copy in the close modal and post notices - 2023 edition, New blog post from our CEO Prashanth: Community is the future of AI, Link between Cipher suites and certificate key. By using this website, you agree with our Cookies Policy. 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. The only disadvantage is that the minus sign itself has to be written as "---", so as not to be confused as a range operator. A function that performs this is called an alphabet function. Similarly, the multiples of a=7 will translate an F (=5) into an 0 (=a) because 7 does so. As an attentive reader, we realize that the MOD multiplication of the keys is closed (recall the group properties in the previous chapter). 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. Which number would that be? Affine cipher - Modular multiplicative inverse. 3. Solution of Multipilicative Inverse of 7. The answer is a simple No: Only those encryption systems that withstand all possible attacks are secure and thus useful. So the cipher text symbol will be w for the letter a in this case. As an example, lets encode and decode NAT and ANT. PLAIN LETTERNATANTSecret key a=2130190131900120012Cipher letteraamaam You can see the dilemma of this message. 14 26, 52, 78, ) have its equivalent key in a=0, a very bad key, since 26=52=78=0 MOD 26. Here is how: u = (p*q - 1) - (p-1) (q 1) getting rid of the first two parentheses yields = p*q -1 - p + 1 (q 1) the two 1s cancel each other out yielding = p*q p (q 1) factoring the p yields = p*(q-1) (q 1) (q-1) in both terms can be factored yielding = (q-1) * (p 1) which can also be written as = (p-1) * (q 1) Formula for the number of good keys if M is the product of two primes: The number of good keys is u(M) = u(p*q) = (p-1)*(q-1). Since, for the standard alphabet, there are 12 numbers less than 26 which are . We just had to multiply each cipher letter by a-1. Just as 5*1/5 yields 1, 5 * 5-1 shall equal 1 MOD 26. This is not a useful encryption system since it may yield ambiguous messages. 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. This weirdness is not really weird. 5 7 11 13 17 19 23 25 29 31 35 WAP to implement Additive cipher(key=20), Multiplicative cipher(key=15)and affine cipher(key=15,20). 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. We can therefore always find a-1 for a given good key a. ((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. How to pick a symmetric cipher for a given cipher text size? Try it for yourself! 11 For the fraction a/b, the multiplicative inverse is b/a. It was encoded MOD 26. Since the bool.h library is very short I want to show you its contents: typedef int bool; const int false = 0; const int true = 1; In the first line the new data type bool is defined of type int so that the (two) bool-variables are just regular integers. h2 { They are very special primes as they must consist of 100 digits or more. By subtracting a (=101) from the entered plain letter in (pl -'a');. 19 18 Network Security: Multiplicative InverseTopics discussed:1) Explanation on the basics of Multiplicative Inverse for a given number.2) Explanation on the basi. These are valuable information for an eavesdropper that help cracking the message. This principle of finding the number of bad keys holds true for any alphabet length that is a prime power: There are M/p multiples of p less or equal to M, and therefore M/p - 1 many less than M. And we are only interested in those integers less than M since we are calculating MOD M which involves the integers 0 to M-1. Lets check why: 1*1=1 MOD 26 which explains a = a-1 = 1 (Big deal!). How would anyone ever break even this basic, amateurish cipher/encryption scheme? Multiply It! (I.e. Example: Encrypt DCODE with the key k= 17 k = 17 and the 26-letter alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ 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. The Affine Cipher uses modulo arithmetic to perform a calculation on the numerical value of a letter to create the ciphertext. Exporting results as a .csv or .txt file is free by clicking on the export icon 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. They seem to not follow any apparent pattern. The formula MOD(E$2*$B4,26) computes the number of the plain letter T, namely 19. It only takes a minute to sign up. PLAIN LETTER:ABCDEFGHIJKLMNOPQRSTUVWXYZ Secret key: a=2012345678910111213141516171819202122232425 024681012141618202224024681012141618202224 Cipher letter:acegikmoqsuwyacegikmoqsuwy Notice, that only every other cipher letter appears, and that exactly twice. 2) Learn how to compute and use the modular inverse to decode. if the letter e (the most frequent letter in the English language) occurs 20 times in the plain text its replacement letter will appear 20 times in the cipher text. In fact, any character is stored as a number: i.e. Code 5 Agree 3) ((p*q) = (p-1)*(q-1) for two distinct primes p and q. Our ultimate goal is not to develop a formula for the number of bad keys but rather for the number of good keys. RSA Express Encryption/Decryption Calculator This worksheet is provided for message encryption/decryption with the RSA Public Key scheme. On the right we ended up with the explicit formula for ((M) when M consists of one prime power and two primes. 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? for the RSA encryption. 9,15,21 and 25). } a bug ? You have 36 possible "characters" here. Parabolic, suborbital and ballistic trajectories all follow elliptic paths. Is there a generic term for these trajectories? ((25)=____________ as all integers from 1 to 24 except for 5,10,15,20 are relatively prime to 25. (Identification), How to decipher Multiplicative cipher without key? When doing so we will discover very important mathematical encryption tools such as Eulers (-function, Eulers and Lagranges Theorem and study further examples of groups, rings and fields. 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. 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 get the following encoding and decoding table. Before considering such encoding techniques, we go ahead and check if the other frequent number, 20, is the cipher E. Checking the E column, we can see that the possible two keys are the bad one a=18 and the good one a=5. For the same reason, an alphabet length of M=31 produces u=30 unique encryptions. Determining the bad keys for a given alphabet length M is a perfect task for a computer. We then write them in the form (1-1/p), multiply them and that product by M yielding ((M). 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. If you choose to do so, dont forget to also redefine the corresponding decoding key in int a=5, ainverse=21; . In this lab, you'll learn about the multiplication cipher, a monoalphabetic cipher. Examples are: 4 and 5 are relatively prime because gcd(4,5)=1. Moreover, we build the mathematical foundation to understand secure encryption systems such as the RSA encryption. 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? 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. For example, if the key is 7 and the plaintext letter is s, the numerical value of s is 18, and so the ciphertext letter is (18*7) mod 26 = 22. We also turn the plaintext into digraphs (or trigraphs) and each of these into a column vector. 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. To use this worksheet, you must supply: a modulus N, and either: 3.0.4224.0. Thank you! an idea ? Why is that? If we had a video livestream of a clock being sent to Mars, what would we see? does not work internally with letters, but with numbers. For a given alphabet, there are only a few possible keys. color: #ffffff; 2.5 Counting the Number of Good Keys for various Alphabet Lengths M An Introduction to the Euler Function. I.e. } Therefore, we just have to add a number in order to get k=111. The statements while(cl!='~') and cout << cl; cin >> pl; are in charge of it. The final formula uses determinant and the transpose of . The message "ACDC" should be encrypted with the key "ABBA" according to the Vigenre method. } 12 For illustration purposes we use the message "GEHEIMNIS" and the key 3. 21 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. Multiplicative encryption uses a key k k (an integer) and an alphabet. 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. 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. So are 2 and 3, 2 and 5, 3 and 10, 26 and 27, 45 and 16. This website would like to use cookies for Google Analytics. Our alphabet length of 28 now yields how many unique encryptions? Multiplying such answers yields the number of good keys for any given alphabet length. Thus, safer encryptions are necessary. Say a=5 was chosen. 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. What is the symbol (which looks similar to an equals sign) called? 0 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). 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. Example3: Now, it is your turn. We then perform matrix multiplication modulo the length of the . https://de.wikipedia.org/wiki/Alphabet_(Kryptologie). a feedback ? By using our site, you For example, take the list L = "ABCD", whose length is 4. //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). The three factors in the parentheses already have the same desired format, however, the single 2 destroys it. 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? That would additionally require that the good keys form a commutative group with respect to addition. "Ordered" means that sorting is possible and we can speak of the n-th character of an alphabet. 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. The conversion to letters takes place modulo to the alphabet length: If a 1 is added to the last character, the result of the sum is the first character of the alphabet. 6*3=18. In conclusion, we can say that multiplicative cipher is a simple encryption technique that can be easily implemented. 9 The number fetched through output is mapped in the table mentioned above and the corresponding letter is taken as the encrypted letter. 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. Example 1: For M=27=33: Inserting 3 for p and 3 for n in pn - pn-1 yields u(27) = 33 - 32 = 27-9 = 18 which is just what we wanted. Playfair cipher online encoder and decoder. Try to answer it for yourself. However, when using MOD arithmetic and solving 23=5*P MOD 26, we dont deal with fractions but only integers. Can we do even better with M=28 ? As you can see on the wiki, decryption function for affine cipher for the following encrytption function: E (input) = a*input + b mod m is defined as: D (enc) = a^-1 * (enc - b) mod m The only possible problem here can be computation of a^-1, which is modular multiplicative inverse. The following C++ program firstly determines the factors for an entered alphabet length M and secondly their multiples, the bad keys. If M=60=22*3*5, then ((60) = ((22*3*5) using property __ yields = ((22)*((3*5) using property __ yields = ((22)*((3)*((5) using properties __ and __ yields = (22 21)*2*4 = 2*2*4 = 16. color: #ffffff; padding-right: 20px; 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. The MOD 26 calculation leaves the 10 unchanged. Our good-key-criterion declares those integers to be good keys that are relative prime to 27. 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. Connect and share knowledge within a single location that is structured and easy to search. 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. 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. So which ones do? You could also define a to be a different good key. ((5)=_____ as 1,2,3,4 are relative prime to 5. 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. According to the definition in wikipedia, in classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. Information Security Stack Exchange is a question and answer site for information security professionals. Lab 2. 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. 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. Alphabets (yes, there may be several: more below) can be described by a list L of letters. Learn more. It has to be placed after the cout command as in: cout << setw(2) << j*factor. Since a=10 is a bad key he checks the good key a=23. Zero has no modular multiplicative inverse. . Equivalently stated: what product of a-1 and 5 equals 1 more than a multiple of 26 such as 27, 53, 79, 105, etc? The basic task behind the multiplicative cipher is to use a large prime number as a multiplication key, and then use the modular arithmetic of the integers modulo, the key to encode and decode the plaintext. Other frequent letters such as T, A, O and N occurring with about (8%) might be of further help to crack the cipher text. where the operation of multiplication substitutes the operation of division by the modular multiplicative inverse. Since each plain letter turns into 0 for a=0 and remains unchanged for a=1, we start with a=2. 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. Which ones are those? padding: 12px; For example if we use "abcdefghijklmnopqrstuvwxyz" and a multiplier of 3, gives "adgjmpsvybehknqtwzcfilorux". The command const is used as a safety feature in C++: both variables are constant and can never be modified in any program. Therefore, an eavesdropper simply has to count letter frequencies to identify the most frequent cipher letter. Cite as source (bibliography): The basic implementation of affine cipher is as shown in the image below In this chapter, we will implement affine cipher by creating its corresponding class that includes two basic functions for encryption and decryption. You can try the sample button which uses a multiplication of 3, and a message of "knowledgeispower" gives enqohmjsmyctqomz. Example3: Doing arithmetic MOD 7, the inverse of a=3 is a-1 = 5 because a * a-1 = 3*5 = 15 = 1 MOD 7. A longer alphabet produces less unique encryptions. 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. Test it yourself. 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. For larger integers, however, dividing by every integer less than M slows the program down enormously. Notice, that all we need to find are the different primes, say p1, p2,, pn, as our explicit formula for the number of unique encryptions appears to be: Formula for the number of good keys for any alphabet length M: For an alphabet length M, there are ((M) = M * (1- 1/p1) * (1- 1/p2) ** (1- 1/pn) good keys where each pi is a prime divisor of M. It is really enjoyable to use this simple formula as we just need to find all prime divisors of M and dont have to worry about how often they occur. Decoding aam can either yield NAT or ANT as the plain text. How a top-ranked engineering school reimagined CS curriculum (Ep. 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.)
