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Hill Cipher menggunakan perhitungan matematika yang disebut Aljabar linier, dan khususnya mengharuskan pengguna untuk memiliki pemahaman dasar tentang matriks. Ini juga memanfaatkan Modulo Arithmetic seperti the Affine Cipher. Karena itu, hill cipher memiliki sifat matematika yang jauh lebih penting daripada beberapa yang lain.

Namun, sifat inilah yang memungkinkannya bertindak relatif dengan mudah pada blok huruf yang lebih besar. Dalam contoh yang diberikan, kita akan berjalan melalui semua langkah untuk menggunakan sandi ini untuk bertindak pada digraf dan trigraf. Hal ini dapat diperpanjang lebih jauh, tapi ini kemudian membutuhkan pengetahuan yang jauh lebih dalam tentang latar belakang matematika.

Tentukan Plaintext pesan selanjutnya, susun plaintext dalam bentuk blok matriks 2x1 jika ordo kunci 2x2, 3x1 jika ordo kunci 3x3. Tentukan matriks kunci dengan persyaratan nilai determinasi matriks kunci harus nilai bilangan ganjil postif atau negatif.

Tentukan Nilai Determinan Matriks Kunci 2. Lakukan perhitungan menggunakan rumus sampai hasil perhitungannya mendapatkan nilai bilangan bulat postif atau negatif, maka hasil perhitungan rumus tersebut adalah hasil untuk nilai Invers Modulo, hindari bilangan pecahan. Tentukan Kunci dekripsi Hill Cipher Mk-1 5.

Modulus yang di gunakan untuk mengambil nilai index huruf untuk plaintext dan ciphertext menggunakan susunan abjad huruf A — Z, yaitu Mod Invers Matriks Kunci Mk 4. Related Papers. Algoritma hill chiper. By jepri ENDO. By Taronisokhi Zebua. By Anita Sindar. By Muhammad Rhifky Wayahdi.

### Hill Cipher

Need an account? Click here to sign up.In the second chapter, we discussed the fundamentals of modern cryptography. We equated cryptography with a toolkit where various cryptographic techniques are considered as the basic tools. One of these tools is the Symmetric Key Encryption where the key used for encryption and decryption is the same. In this chapter, we discuss this technique further and its applications to develop various cryptosystems.

Unlike modern systems which are digital and treat data as binary numbers, the earlier systems worked on alphabets as basic element. These earlier cryptographic systems are also referred to as Ciphers. In general, a cipher is simply just a set of steps an algorithm for performing both an encryption, and the corresponding decryption.

It is a mono-alphabetic cipher wherein each letter of the plaintext is substituted by another letter to form the ciphertext. It is a simplest form of substitution cipher scheme. This cryptosystem is generally referred to as the Shift Cipher. This number which is between 0 and 25 becomes the key of encryption.

In order to encrypt a plaintext letter, the sender positions the sliding ruler underneath the first set of plaintext letters and slides it to LEFT by the number of positions of the secret shift. The plaintext letter is then encrypted to the ciphertext letter on the sliding ruler underneath. The result of this process is depicted in the following illustration for an agreed shift of three positions. On receiving the ciphertext, the receiver who also knows the secret shift, positions his sliding ruler underneath the ciphertext alphabet and slides it to RIGHT by the agreed shift number, 3 in this case.

He then replaces the ciphertext letter by the plaintext letter on the sliding ruler underneath. Caesar Cipher is not a secure cryptosystem because there are only 26 possible keys to try out. An attacker can carry out an exhaustive key search with available limited computing resources. It is an improvement to the Caesar Cipher. Instead of shifting the alphabets by some number, this scheme uses some permutation of the letters in alphabet.

For example, A. Z and Z. A are two obvious permutation of all the letters in alphabet. Permutation is nothing but a jumbled up set of alphabets.In classical cryptographythe Hill cipher is a polygraphic substitution cipher based on linear algebra. Invented by Lester S. Hill init was the first polygraphic cipher in which it was practical though barely to operate on more than three symbols at once. The following discussion assumes an elementary knowledge of matrices. Each letter is first encoded as a number.

If one uses a larger number than 26 for the modular base, then a different number scheme can be used to encode the letters, and spaces or punctuation can also be used. A Hill cipher is another way of working out the equation of a matrix. Now, suppose that our message is instead 'CAT', or:.

Every letter has changed. The Hill cipher has achieved Shannon 's diffusionand an n-dimensional Hill cipher can diffuse fully across n symbols at once. There are standard methods to calculate the inverse matrix; see matrix inversion for details. We have not yet discussed one complication that exists in picking the encrypting matrix. Not all matrices have an inverse see invertible matrix.

The matrix will have an inverse if and only if its determinant is not zero, and does not have any common factors with the modular base. Thus, if we work modulo 26 as above, the determinant must be nonzero, and must not be divisible by 2 or If the determinant is 0, or has common factors with the modular base, then the matrix cannot be used in the Hill cipher, and another matrix must be chosen otherwise it will not be possible to decrypt.

Fortunately, matrices which satisfy the conditions to be used in the Hill cipher are fairly common. So, modulo 26, the determinant is Since this has no common factors with 26, this matrix can be used for the Hill cipher. The risk of the determinant having common factors with the modulus can be eliminated by making the modulus prime. Consequently a useful variant of the Hill cipher adds 3 extra symbols such as a space, a period and a question mark to increase the modulus to Unfortunately, the basic Hill cipher is vulnerable to a known-plaintext attack because it is completely linear.

Calculating this solution by standard linear algebra algorithms then takes very little time. While matrix multiplication alone does not result in a secure cipher it is still a useful step when combined with other non-linear operations, because matrix multiplication can provide diffusion. For example, an appropriately chosen matrix can guarantee that small differences before the matrix multiplication will result in large differences after the matrix multiplication.

Some modern ciphers use indeed a matrix multiplication step to provide diffusion. The function g in Twofish is a combination of non-linear S-boxes with a carefully chosen matrix multiplication MDS.

Recently, some publications tried to make the Hill cipher secure.

Hill Cipher Explained (with Example)

The key size is the binary logarithm of the number of possible keys. This is only an upper bound because not every matrix is invertible and thus usable as a key. The number of invertible matrices can be computed via the Chinese Remainder Theorem. It is. Equally, the number of invertible matrices modulo 13 i. Additionally it seems to be prudent to avoid too many zeroes in the key matrix, since they reduce diffusion.To start with the program directly is not a good idea here.

Basically Hill cipher is a cryptography algorithm to encrypt and decrypt data to ensure data security. In this example we are going to take up a 2X2 matrix for better understanding and simplification.

The same method can be applied to 3X3 matrix to get the desired results. So the first thing we have to do in encrypting the data using hill cipher is to take up a string of characters as key matrix to encrypt data and convert this key matrix to number matrix. For this purpose we will need to convert this plain text into diagraphs. For this purpose we will write this from the starting of our first column vector having first letter at the top and second letter at the bottom and after this jumping on to the second column vector having third letter at the top and fourth letter at the bottom and so on.

Now we need to multiple each column vector from the key matrix and obtain the result. The result after multiplication is shown down here:. After this, as all the numbers are greater than 26 so we need to divide these column vectors with 26 and note the remainder i.

To do this first find the determinant of our key matrix. In our case determinant evaluates to 37, which is again greater than 26 so we will find mod26 of out determinant i.

The next step is to find a number which gives the answer 1 when mod26 is found after multiplying that number by the modulo of our determinant. This can be done with the help of hit and trial method. Next step is to multiply this adjoint with the number we found above 19 and find mod26 to keep the range under On doing this, we get.

## Hill cipher

So first we will write this string in column vectors, and next convert this column vectors into corresponding number and multiply it with the inverse of key matrix we found above and find mod26 then. After that we need to transfer these numbers back to letters to get our actual string. One is done here. Menu: 1: Encryption 2: Decryption 1 Enter the line: helloworld Enter the key: mble Result: kpnjiidofd. Menu: 1: Encryption 2: Decryption 2 Enter the line: kpnjiidofd Enter the key: mble Result: helloworld.

IOException ; import java.You can directly visit one of the CrypTool topic pages by clicking one of the five buttons here below. Plaintext: The quick brown fox jumps over the lazy dog. Encrypted text: Xfg cwuwa vhkyp rne ytxqw mgfp hkj ivyf ehz. The Hill cipher was the first cipher purely based on mathematics linear algebra.

Upper case and lower case characters are treated equally. Then the encryption is done by multiplying the numbers with an n x n key matrix modulo 26 if we have A-Z as our alphabet.

### Hill Cipher Encryption and Decryption

The result is converted back to text producing the ciphertext. This result 15, 14, 7 can be decoded by "POH" which would be the output of the Hill cipher for the chosen message and the used key. To decode the message, one would have to multiply the ciphertext with the inverse matrix of the key and apply modulo 26 to the result.

The key has to be chosen in such a way that there exists an inverse matrix for the key matrix because it would be impossible to decode the message otherwise. Therefore the determinant of the key matrix modulo 26 has to be co-prime to Numbers co-prime to 26 are: 1,3,5,7,9,11,15,17,19,21,23, The determinant of the key matrix shown above is therefore calculated as such:.

The Hill cipher was invented in by Lester S. The cipher is based on linear algebra only. When parts of the plaintext are known, an attacker could try to find out the key by using a system of linear equations.

So unfortunately, the basic Hill cipher is vulnerable to known-plaintext attacks. This Webpage requires JavaScript. Add alphabet. Compressed alphabet notation:. Keyword for alphabet:. Details: The key has to be chosen in such a way that there exists an inverse matrix for the key matrix because it would be impossible to decode the message otherwise. This is better for security but no requirement of the original method. If a length like 26 is used, then this website complains e. Implementations without this additional restriction and with the possibility to choose matrix dimensions n other than 2 or 3 are: CrypTool 1, CrypTool 2, and SageMath.

Imprint Contact us Privacy.Invented by Lester S. Hill inthe Hill cipher is a polygraphic substitution cipher based on linear algebra. Hill used matrices and matrix multiplication to mix up the plaintext. To counter charges that his system was too complicated for day to day use, Hill constructed a cipher machine for his system using a series of geared wheels and chains. However, the machine never really sold. Hill's major contribution was the use of mathematics to design and analyse cryptosystems.

It is important to note that the analysis of this algorithm requires a branch of mathematics known as number theory. Many elementary number theory text books deal with the theory behind the Hill cipher, with several talking about the cipher in detail e.

Elementary Number Theory and its applicationsRosen, It is advisable to get access to a book such as this, and to try to learn a bit if you want to understand this algorithm in depth. For a guide on how to break Hill ciphers, see Cryptanalysis of the Hill Cipher. This example will rely on some linear algebra and some number theory.

The key for a hill cipher is a matrix e. To encipher this, we need to break the message into chunks of 3. We now take the first 3 characters from our plaintext, ATT and create a vector that corresponds to the letters replace A with 0B with Z with 25 etc. To get our ciphertext we perform a matrix multiplication you may need to revise matrix multiplication if this doesn't make sense :. This process is performed for all 3 letter blocks in the plaintext.

The plaintext may have to be padded with some extra letters to make sure that there is a whole number of blocks. Now for the tricky part, the decryption.

We need to find an inverse matrix modulo 26 to use as our 'decryption key'. If our 3 by 3 key matrix is called Kour decryption key will be the 3 by 3 matrix K -1which is the inverse of K. To find K -1 we have to use a bit of maths. It turns out that K -1 above can be calculated from our key. A lengthy discussion will not be included here, but we will give a short example.

The important things to know are inverses mod mdeterminants of matricesand matrix adjugates. Let K be the key matrix. Let d be the determinant of K. The following formula tells us how to find K -1 given K :.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. In the english language, the most common digraph is TH which is then followed by HE. In this particular example let's say the digraphs with the most frequencies are RH and NI. Unfortunately my use of matrix notation is limited and I fear that I would clog up the screen with my poor attempt so I'll just put the result of my work.

I basically combined the key matrix of a, b, c, and d with the pairs TH and HE to get:. Assuming this work is correct, I believe that I can just set these values equal to the values of RH and NI and solve for a, b, c, or d. However, I'm not entirely sure if this is correct. I am also not entirely sure how I would proceed after creating this equation.