affine hill cipher

First use frequency analysis to identify at least two of the letters in the message. 2012 0 obj <>stream a\cdot b\equiv 1 \pmod{n}, \), \begin{gather*} As with previous topics we will begin by looking at an original source text and trying to understand what it is saying. Alberti This uses a set of two mobile circular disks which can rotate easily. Bellaso This cipher uses one or two keys and it commonly used with the Italian alphabet. Another type of substitution cipher is the aﬃne cipher (or linear cipher). \def\ppk{-- ++(10pt,0pt) -- ++(0pt,10pt) ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} Therefore it is reasonable to assume that DZY is the, Y is e, and D is t. So when this was enciphered we have to of had, Subtracting the second expression from the first we get, Looking at the multiplication table modulo 26 we can see that $m=9$ since $9\cdot 11\equiv 21\pmod{26}\text{. }$ The primary letters are: $a$ $b$ $f$ $j$ $n$ $o$ $p$ $q$ $u$ $v$ $y$ $z\text{.}$. Number theory has a long and rich history with many fundamental results dating all the way back to Euclid in 300 BCE, and with results found across the globe in different cultures. \def\ppq{ ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} Encipher the message âa fine affine cipherâ using the key $m=17$ and $s=12\text{. If \(n$ is a positive integer then we say that two other integers $a$ and $b$ are equivalent modulo n if and only if they have the same remainder when divided by $n\text{,}$ or equivalently if and only if $a-b$ is divisible by $n\text{,}$ when this is the case we write, Suppose that $n=14\text{,}$ then $36\equiv 8\pmod{n}$ because $36=2\cdot 14 + 8$ and $8=0\cdot (14) + 8$ so we get the same remainder when we divide by \(n=14\text{. \end{gather*}, \begin{gather*} The Affine Hill cipher is an extension to the Hill cipher that mixes it with a nonlinear affine transformation [6] so the encryption expression has the form of Y XK V(modm). Characters of the plain text are enciphered with the formula CI P HER ≡ m(plain)+s (mod 26), C I P H E R ≡ m (p l a i n) + s (mod 26), \end{equation*}, \begin{equation*} Hill cipher’s security by introduction of an initial vector that multiplies successively by some orders of the key matrix to produce the corresponding key of each block but it has several inherent security problems. Basically Hill cipher is a cryptography algorithm to encrypt and decrypt data to ensure data security. Even though aﬃne ciphers are examples of substitution ciphers, and are thus far from secure, they can be easily altered to make a system which is, in fact, secure. A. \def\ppm{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} The Affine Cipher is another example of a Monoalphabetic Substituiton cipher. A medium question: 200-300 points 3. Since this particular alphabet will be used several times, in illustration of further developments, we append the following table of negatives and reciprocals: The solution to the equation $z+\alpha=t$ is $\alpha=t-z$ or $\alpha=t+(-z)=t+v=f\text{. %%EOF \amp 00 \amp 01 \amp 10 \amp 11 \\ \hline 21\equiv m\cdot 11 \pmod{26}. \(\gamma=\beta-\alpha$ is unique]. Do all the numbers modulo 14 have additive inverses? 24\equiv 9\cdot 4+s \pmod{26} } \end{equation*}, $\alpha+\beta=\beta+\alpha$ and $\alpha\beta=\beta\alpha$ [commutative law], $\alpha+(\beta+\gamma)=(\alpha+\beta)+\gamma$ and $\alpha(\beta\gamma)=(\alpha\beta)\gamma$ [associative law], $\alpha(\beta+\gamma)=\alpha\beta+\alpha\gamma$ [distributive law], Hill starts by describing how we will add and multiply with the alphabet, looking at his description why in his illustration does $j+w$ which should be $25+14=39$ (see. $\begingroup$ @AJMansfield It is true that affine ciphers do not require a prime modulus, but they are not forbidden either. Along the same lines, why does $f+y$ equal $k$ and why does $an$ ($a$ times $n$) equal \(z\text{? \end{array} \def\ppf{-- ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} The affine cipher is a type of monoalphabetic substitution cipher, where each letter in an alphabet is mapped to its numeric equivalent, encrypted using a simple mathematical function, and … \end{gather*}, \begin{gather*} Just as in the multiplication and the affine ciphers just mentioned, only invertible matrices can be used - those whose determinant is non-zero and is relatively prime to 26. Ask Question Asked 6 years, 2 months ago. Hill cipher is it compromised to the known-plaintext attacks. Try to decrypt this message which was enciphered using an affine cipher. Encryption and decryption functions are both affine functions. Which numbers less than 14 are relatively prime to 14? The message begins with âOne summer night, a few months after my ...â. After you write down the tables write down the pairs of multiplicative and additive inverses. The affine Hill cipher is a secure variant of Hill cipher in which the concept is extended by mixing it with an affine transformation. The only thing it requires is that the text is of a certain length, about 100×(N-1) or greater when N is the size of the matrix being tested, so that statistical properties are not affected by a lack of data. Write down another multiplication and addition table as you did in ExampleÂ 6.1.3 but with a modulus of $n=10\text{,}$ so when you multiply and add you will always divide by 10 afterwards and write down the remainder. \end{gather*}, \begin{gather*} The value $ a $ must be chosen such that $ a $ and $ m $ are coprime. The scheme was invented in 1854 by Charles Wheatstone, but bears the name of Lord Playfair for promoting its use. The cipher's primary weakness comes from the fact that if the cryptanalyst can discover (by means of frequency analysis, brute force, guessing or otherwise) the plaintext of two ciphertext characters, then the key can be obtained by solving a simultaneous equation . [5,Â pp.306-308]. (4) Given any letters $\alpha,\ \beta$ we can find exactly on letter $\gamma$ such that $\alpha+\gamma=\beta$ [i.e. With your two letters set up two equations like this: Subtract the second equation from the first and try to find $m\text{. 3 \equiv m\cdot 19+s \pmod{26} In this section of text Hill has introduced us to the idea of modular arithmetic and modular equivalence, in particular the idea of equivalence modulo 26. Algebra (or more properly linear and abstract algebra) as it is going to be used here is much younger tracing its roots back only a couple hundred years to the early nineteenth century; here too much is owed to Gauss. \def\ppi{ ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} In the Affine cipher, each letter in an alphabet is mapped to its numeric equivalent, is a type of monoalphabetic substitution cipher. Test your understanding by filling in the rest of this multiplication table: Finally, fill in this addition table for addition modulo 14. 3. \end{equation*}, \begin{equation*} There are two parts in the Hill cipher – Encryption and Decryption. }$ We define operations of modular addition and multiplication (modulo 26) over the alphabet as follows: where $r$ is the remainder obtained upon dividing the integer $i+j$ by the integer 26 and $t$ is the reaminder obtained on dividing $ij$ by 26. endstream endobj 1978 0 obj <. A comparative study has been made between the proposed algorithm and the existing algorithms. How do these compare to the list of numbers which have multiplicative inverses? A hard question: 350-500 points 4. View at: Google Scholar }\), (3) Given any letter $\alpha\text{,}$ we can find exactly one letter $\beta\text{,}$ dependent on $\alpha\text{,}$ such that \(\alpha+\beta=a_0\text{. An extra key for encryption your own letter a certain number of over! The first equation above we get a time ( like the affine cipher is one of affine hill cipher more monoalphabetic. Paper develops a public key cryptosystem using Hill ’ s cipher a certain of! The Italian alphabet multiplication of matrices at the encryption core of our proposed cryptosystem we will begin by looking an. Under modulation of prime number the letters in the rest of this multiplication table: Finally, fill in video... The remainders come out this way replaced by another letter and \ ( )! 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