\end{equation*}, \begin{equation*} In summary, aﬃne encryption on the English alphabet using encryption key (α,β) is accomplished via the formula y ≡ αx + β (mod 26). \def\ppv{-- ++(5pt,10pt) -- ++(5pt,-10pt) ++(5pt,0pt)} Look back at ExampleÂ 6.1.3 and write down the pairs of additive and multiplicative inverses. an=z,\ hm=k,\ cr=s,\ etc. 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. \end{array} }\) Characters of the plain text are enciphered with the formula, and characters of the cipher text are deciphered with the formula. \(\gamma=\beta-\alpha\) is unique]. }\) Note that \(m^{-1}\equiv 19\pmod{26}\) and \(-s\equiv 22\pmod{26}\text{. \def\ppk{-- ++(10pt,0pt) -- ++(0pt,10pt) ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} \def\ppj{-- ++(10pt,0pt) -- ++(0pt,10pt) ++(-5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} In this paper, a modified version of Hill cipher is proposed to overcome all the drawbacks mentioned above. However, we can also take advantage of the fact that it is an affine cipher. Now let's decipher the message AJINF CVCSI JCAKU which was enciphered using an affine cipher and a key of \(m=11\) and \(s=4\text{. No matter which modulus you use, do all the numbers have additive inverses, i.e. \def\ppp{ ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} \end{gather*}, \begin{gather*} Hill cipher is it compromised to the known-plaintext attacks. %%EOF
The cipher we will focus on here, Hill's Cipher, is an early example of a cipher based purely in the mathematics of number theory and algebra; the areas of mathematics which now dominate all of modern cryptography. }\) Alternately, we can observe that \(36-8=28\) and \(28=2\cdot(14)\) is divisible by \(n=14\text{.}\). A hard question: 350-500 points 4. 5\cdot 4+16\equiv 10\pmod{26} A random matrix key, RMK is introduced as an extra key for encryption. \end{equation*}, \begin{equation*} \begin{array}{|c|c|c|c|c|}\hline In this paper, we extend this concept in the encryption core of our proposed cryptosystem. }\), Decipher the message RXGTM CHUHJ CFWM which was enciphered using the key \(m=3\) and \(s=7\text{.}\). } How do these compare to the list of numbers which have multiplicative inverses? The amount of points each question is worth will be distributed by the following: 1. Next e is replaced by 4 and we get, and 10 is K, so plain e becomes cipher K. The plain l corresponds to 11 and. Invented by Lester S. Hill in 1929, it was the first polygraphic cipher in which it was practical (though barely) to operate on more than three symbols at once. The de… With your two letters set up two equations like this: Subtract the second equation from the first and try to find \(m\text{. Try to decrypt this message which was enciphered using an affine cipher. \end{equation*}, \begin{equation*} A. \def\ppo{-- ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} \amp 00 \amp 01 \amp 10 \amp 11 \\ \hline Similar to the Hill cip her the affine Hill cipher is polygraphic cipher, encrypting/decrypting letters at a time. 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. According to the definition in wikipedia, in classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. \def\ppi{ ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} \newcommand{\lt}{<} How do these compare to the list of numbers which have multiplicative inverses? Let \(a_0,\ a_1,\ \ldots,\ a_{25}\) denote any permutation of the letters of the English alphabet; and let us associate the letter \(a_i\) with the integer \(i\text{. Now that you have the key you should be able to decipher the message as you had previously. 24-10\equiv s \pmod{26} \def\ppt{ ++(0pt,10pt) -- ++(10pt,-5pt) -- ++(-10pt,-5pt) ++(15pt,0pt)} Along the same lines, why does \(f+y\) equal \(k\) and why does \(an\) (\(a\) times \(n\)) equal \(z\text{? It also make use of Modulo Arithmetic (like the Affine Cipher). which is p. Try to decipher the remaining characters in the message on your own. a_i\, a_j=a_t, \amp 00 \amp 01 \amp 10 \amp 11 \\ \hline In his illustration he also says \(hm\) which should be 4 times 13, or 52, is \(k\) which is 0, why is this the case? }\) Take the A and replace it by 0 and then using the formula above we get, so we replace cipher A with plain text c. The J is replaced by 9 and, therefore cipher J becomes plain r. To use the other formula for deciphering we need \(m^{-1}s\equiv 2\pmod{26}\text{. \def\ppd{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(15pt,-10pt)} }\), Substitute your value for \(m\) into the first equation and use it to find \(s\text{.}\). \def\ppn{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} The Additive (or shift) Cipher System The first type of monoalphabetic substitution cipher we wish to examine is called the additive cipher. Test your understanding by filling in the rest of this multiplication table: Finally, fill in this addition table for addition modulo 14. \end{equation*}, \begin{equation*} 21\equiv m\cdot 11 \pmod{26}. Hi guys, in this video we look at the encryption process behind the affine cipher. For example the greatest common divisor of 7 and 36 is 1 so they are relatively prime, however the greatest common divisor of 30 and 36 is 6 so they are not relatively prime. 's Scheme We call 0 the additive identity because for all \(a\) and all possible moduli \(n\) we get, We say that \(a\) and \(b\) are additive inverses modulo \(n\) if, We call 1 the multiplicative identity because for all \(a\) and all possible moduli \(n\) we get, We say that \(a\) and \(b\) are multiplicative inverses modulo \(n\) if. In the Affine cipher, each letter in an alphabet is mapped to its numeric equivalent, is a type of monoalphabetic substitution cipher. (Now we can see why a shift cipher is just a special case of an aﬃne cipher: A shift cipher with encryption key ‘ is the same as an aﬃne cipher with encryption key (1,‘).) Gronsfeld This is also very similar to vigenere cipher. with subscripts prime to 26, as âprimaryâ letters, we make the assertion, easily proved: If \(\alpha\) is any primary letter and \(\beta\) is any letter, there is exactly one letter \(\gamma\) for which \(\alpha\gamma=\beta\text{.}\). [5,Â pp.306-308]. 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. Encryption – Plain text to Cipher text. 10 \amp 11 \amp 00 \amp 01 \amp 00 \\ \hline \end{gather*}, \begin{equation*} 1999 0 obj
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Alberti This uses a set of two mobile circular disks which can rotate easily. \end{equation*}, \begin{equation*} In this cipher method, each plaintext letter is replaced by another character whose position in the alphabet is a certain number of units away. Encryption and decryption functions are both affine functions. Encryption is done using a simple mathematical function and converted back to a letter. Here, we have a prime modulus, period. a\cdot 1\equiv a\pmod{n}\text{.} If you look at the numbers which do have multiplicative inverses how do they relate to those which Hill described as prime to 26? 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{. Encipher the message âa fine affine cipherâ using the key \(m=17\) and \(s=12\text{. 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. 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), An affine cipher is a cipher with a two part key, a multiplier m m and a shift s s and calculations are carried out using modular arithmetic; typically the modulus is n= 26. n = 26. The integers \(i\) and \(j\) may be the same or different. $ An Affine-Hill Cipher is the following modification of a Hill Cipher: Let m be a positive integer, and define P = C = (Z26)". This is a cipher based on the multiplication of matrices. 3. 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