Using the same value for \(n\) we get that \(3\cdot 5\equiv 1\pmod{n}\) because \(15=1\cdot (14) +1\text{,}\) so the remainder when \(3\cdot 5\) is divided by \(n\) is 1. \def\ppy{ ++(10pt,10pt) -- ++(-10pt,-5pt) -- ++(10pt,-5pt) ++(-5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} h�b```���l�B ��ea�� ��0_Ќ�+��r�b���s^��BA��e���⇒,.���vB=/���M��[Z�ԳeɎ�p;�)
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\end{equation*}, \begin{equation*} Similar to the Hill cip her the affine Hill cipher is polygraphic cipher, encrypting/decrypting letters at a time. \def\ppx{ ++(0pt,10pt) -- ++(10pt,-5pt) -- ++(-10pt,-5pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} \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. 24\equiv 9\cdot 4+s \pmod{26} According to the definition in wikipedia, in classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. The proposed method increases the security of the system because it involves two or more digital signatures under modulation of prime number. 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 \(n=14\text{. }\), Decipher the message RXGTM CHUHJ CFWM which was enciphered using the key \(m=3\) and \(s=7\text{.}\). Which numbers less than 14 are relatively prime to 14? \def\ppf{-- ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} 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. \def\ppd{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(15pt,-10pt)} The key used to encrypt and decrypt and it also needs to be a number. \def\ppj{-- ++(10pt,0pt) -- ++(0pt,10pt) ++(-5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} 11 \amp 11 \amp 01 \amp 11 \amp 10 \\ \hline Number theory as we understand and use it today is due in large part to Carl Friedrich Gauss and his text Disquisitiones Arithmeticae published in 1801 (when Gauss was 24). Since we assume that A does not have repeated elements, the mapping f: A ⟶ Z / nZ is bijective. \end{equation*}, \begin{equation*} \mbox{ 24\equiv m\cdot 4+s \pmod{26}\\ Encryption is converting plain text into ciphertext. What is the difference between the even and odd rows (excluding row 7)? \end{gather*}, \begin{equation*} The Affine Cipher is another example of a Monoalphabetic Substituiton cipher. a\equiv b \pmod{n}. The amount of points each question is worth will be distributed by the following: 1. Also, be sure you understand how to encipher and decipher by hand. so that \(s=14\text{. \def\ppw{ ++(0pt,10pt) -- ++(5pt,-10pt) -- ++(5pt,10pt) ++(-5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} Encryption and decryption functions are both affine functions. Now let's decipher the message AJINF CVCSI JCAKU which was enciphered using an affine cipher and a key of \(m=11\) and \(s=4\text{. 5\cdot 4+16\equiv 10\pmod{26} The affine cipher is a type of monoalphabetic substitution cipher, where each letter in an alphabet is mapped to its numeric equivalent, encrypted using a simple mathematical function, and … \def\ppv{-- ++(5pt,10pt) -- ++(5pt,-10pt) ++(5pt,0pt)} $ }\) 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. a+0\equiv a\pmod{n}\text{.} It also make use of Modulo Arithmetic (like the Affine Cipher). \end{equation*}, \begin{equation*} \end{equation*}, \begin{equation*} }\) Characters of the plain text are enciphered with the formula, and characters of the cipher text are deciphered with the formula. \def\ppq{ ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} (4) Given any letters \(\alpha,\ \beta\) we can find exactly on letter \(\gamma\) such that \(\alpha+\gamma=\beta\) [i.e. 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. 11–23, 2018. 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 \(\alpha\gamma=\beta\text{.}\). As with previous topics we will begin by looking at an original source text and trying to understand what it is saying. The letters of an alphabet of size m are first mapped to the integers in the range 0 … m-1, in the Affine cipher, \newcommand \sboxTwo{ Which numbers less than 10 are relatively prime to 10? \end{array} \def\ppp{ ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} First, modern explanations of Hill's cipher focus on the simplest case when the matrix has dimension \(2\times 2\) and there is no shift. ciphers.) \begin{array}{|c|c|c|c|c|}\hline Ask Question Asked 6 years, 2 months ago. plain\,\equiv\, m^{-1}CIPHER-m^{-1}s\pmod{26}. With your two letters set up two equations like this: Subtract the second equation from the first and try to find \(m\text{. %%EOF
} \mbox{ An Affine-Hill Cipher is the following modification of a Hill Cipher: Let m be a positive integer, and define P = C = (Z26)". \def\ppr{ ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} 00 \amp 01 \amp 11 \amp 10 \amp 11 \\ \hline \def\ppu{ ++(10pt,10pt) -- ++(-10pt,-5pt) -- ++(10pt,-5pt) ++(5pt,0pt)} \def\ppg{ ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(15pt,-10pt)} $\begingroup$ @AJMansfield It is true that affine ciphers do not require a prime modulus, but they are not forbidden either. \end{gather*}, \begin{gather*} Analyzing this we get that the most common characters are Y, D, I, O and U; the most common bigrams are DZ, ZY, YG, and OB; the most common trigrams are DZY, OBO, LDZ, and DZO. a\cdot 1\equiv a\pmod{n}\text{.} An affine cipher is a cipher with a two part key, a multiplier m m and a shift s s and calculations are carried out using modular arithmetic; typically the modulus is n= 26. n = 26. 's Scheme \end{equation*}, \begin{equation*} \def\ppb{-- ++(10pt,0pt) -- ++(0pt,10pt) ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} Hi guys, in this video we look at the encryption process behind the affine cipher. The value $ a $ must be chosen such that $ a $ and $ m $ are coprime. }\) We can then get the inverse keys \(m^{-1}\equiv 3\pmod{26}\) and \(-m^{-1}s\equiv 10\pmod{26}\text{. Do all of them have multiplicative inverses? 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. Gronsfeld This is also very similar to vigenere cipher. CIPHER\,\equiv\, m(plain)+s\pmod{26}, c+x=t,\ j+w=m,\ f+y=k,\ -f=y,\ -y=f,\ etc.\\ $ An easy question: 100-150 points 2. Note that the multiplier \(m\) must be relatively prime to the modulus so that it has a multiplicative inverse. (You will want to use FigureÂ C.0.13. View at: Google Scholar Because of this, the cipher has a significantly more mathematical nature than some of … 19(0+22)\equiv 2\pmod{26} \def\ppo{-- ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} plain\,\equiv\, m^{-1}(CIPHER-s)\pmod{26}, Also Read: Java Vigenere Cipher 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. Look back at ExampleÂ 6.1.3 and write down the pairs of additive and multiplicative inverses. 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), We actually shift each letter a certain number of places over. If you look at the numbers which do have multiplicative inverses how do they relate to those which Hill described as prime to 26? The scheme was invented in 1854 by Charles Wheatstone, but bears the name of Lord Playfair for promoting its use. Viewed 2k times 0 $\begingroup$ Prove that the affine cipher over Z26 has perfect secrecy if every key is used with equal probability of 1/312. The Affine cipher is a type of monoalphabetic substitution cipher, wherein each letter in an alphabet is mapped to its numeric equivalent, encrypted using a simple mathematical function, and converted back to a letter. However, given the importance of this material to the rest of what we will be discussing in subsequent chapters, we will look at the material from a more modern perspective. 01 \amp 10 \amp 00 \amp 01 \amp 11 \\ \hline 24-10\equiv s \pmod{26} This is a concept which will be central to most everything else we do so we need to spend a little more time trying to precisely understand modular equivalence. }\), (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{. \end{gather*}, \begin{equation*} \(\gamma=\beta-\alpha\) is unique]. Alberti This uses a set of two mobile circular disks which can rotate easily. 19(8)+2\equiv 24\pmod{26} }\) We call \(\beta\) the ânegativeâ of \(\alpha\text{,}\) and we write: \(\beta=-\alpha\text{.}\). }\) Substituting \(m=9\) into the first equation above we get. \end{gather*}, \begin{gather*} This means the message encrypted can be broken if the attacker gains enough pairs of plaintexts and ciphertexts. Also Read: Caesar Cipher in Java. a_1,\ a_3,\ a_5,\ a_7,\ a_9,\ a_{11},\ a_{15},\ a_{17},\ a_{19},\ a_{21},\ a_{23},\ a_{25}, The Affine cipher is a special case of the more general monoalphabetic substitutioncipher. }\) Alternately, we can observe that \(36-8=28\) and \(28=2\cdot(14)\) is divisible by \(n=14\text{.}\). (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,‘).) 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. After you write down the tables write down the pairs of multiplicative and additive inverses. %PDF-1.5
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Do all the numbers modulo 10 have additive inverses? \( Now that you have the key you should be able to decipher the message as you had previously. }\), Substitute your value for \(m\) into the first equation and use it to find \(s\text{.}\). \def\ppl{-- ++(10pt,0pt) ++(-10pt,0pt) -- ++(0pt,10pt) ++(5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} In this cryptosystem, a key K consists of a pair (L, b), where L is an m x m invertible matrix over Z26, and be (Z26)". A hard question: 350-500 points 4. 3. Cryptanalysis of Lin et al. 21\equiv m\cdot 11 \pmod{26}. Basically Hill cipher is a cryptography algorithm to encrypt and decrypt data to ensure data security. The method described above can solve a 4 by 4 Hill cipher in about 10 seconds, with no known cribs. 1999 0 obj
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A ciphertext is a formatted text which is not understood by anyone. \newcommand \sboxOne{ 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? \end{gather*}, \begin{gather*} However, we can also take advantage of the fact that it is an affine cipher. 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. In this cipher method, each plaintext letter is replaced by another character whose position in the alphabet is a certain number of units away. In summary, aﬃne encryption on the English alphabet using encryption key (α,β) is accomplished via the formula y ≡ αx + β (mod 26). 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. Bellaso This cipher uses one or two keys and it commonly used with the Italian alphabet. \end{equation*}, \begin{equation*} In mathematics, an affine function is defined by addition and multiplication of the variable (often $ x $) and written $ f (x) = ax + b $. To decrypt hill ciphertext, compute the matrix inverse modulo 26 (where 26 is the alphabet length), requiring the matrix to … Often the simple scheme A = 0, B = 1, …, Z = 25 is used, but this is not an essential feature of the cipher. $ \def\ppa{-- ++(10pt,0pt) -- ++(0pt,10pt) ++(5pt,-10pt)} In the Affine cipher, each letter in an alphabet is mapped to its numeric equivalent, is a type of monoalphabetic substitution cipher. A medium question: 200-300 points 3. Test your understanding by filling in the rest of this multiplication table: Finally, fill in this addition table for addition modulo 14. M. G. V. Prasad and P. Sundarayya, “Generalized self-invertiblekey generation algorithm by using reflection matrix in hill cipher and affine hill cipher,” in Proceedings of the IEEE Symposium Series on Computational Intelligence, vol. \def\ppe{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} No matter which modulus you use, do all the numbers have multiplicative inverses, i.e. 00 \amp 00 \amp 01 \amp 10 \amp 11 \\ \hline In this paper, a modified version of Hill cipher is proposed to overcome all the drawbacks mentioned above. h�bbd```b``v��A$��d�f[�Hƹ`5�`����� L� �����+`6X=�[�.0�"s*�$c�{F.���������v#E���_ ?�X
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. 10 \amp 00 \amp 10 \amp 01 \amp 11 \\ \hline Bazeries This system combines two grids commonly called (Polybius) and a single key for encryption. The message begins with âOne summer night, a few months after my ...â. 10 \amp 11 \amp 00 \amp 01 \amp 00 \\ \hline 3 \equiv m\cdot 19+s \pmod{26} 11 \amp 10 \amp 01 \amp 10 \amp 11 \\ \hline }\) The primary letters are: \(a\) \(b\) \(f\) \(j\) \(n\) \(o\) \(p\) \(q\) \(u\) \(v\) \(y\) \(z\text{.}\). \def\ppc{-- ++(10pt,0pt) ++(-10pt,0pt) -- ++(0pt,10pt) ++(15pt,-10pt)} The affine Hill cipher is a secure variant of Hill cipher in which the concept is extended by mixing it with an affine transformation. Do all the numbers modulo 14 have additive inverses? for involutory key matrix generation is also implemented in the proposed algorithm. \def\ppm{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} What is strange or different about the row for 7? Try to decrypt this message which was enciphered using an affine cipher. } Here, we have a prime modulus, period. There are two parts in the Hill cipher – Encryption and Decryption. \def\ppt{ ++(0pt,10pt) -- ++(10pt,-5pt) -- ++(-10pt,-5pt) ++(15pt,0pt)} \end{array} Let's encipher the message âhello worldâ with an affine cipher and a key of \(m=5\) and \(s=16\text{;}\) assume that we match up the alphabet with the integers from 0 to 25 in the usual way so that a is 0, b is 1, c is 2, etc.. Last Updated : 14 Oct, 2019 Hill cipher is a polygraphic substitution cipher based on linear algebra.Each letter is represented by a number modulo 26. As per Wikipedia, Hill cipher is a polygraphic substitution cipher based on linear algebra, invented by Lester S. Hill in 1929. The de… a_i+a_j=a_r,\\ \begin{array}{|c|c|c|c|c|}\hline endstream
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The plaintext is divided into vectors of length n, and the key is a nxn matrix. 5\cdot 11+16\equiv 19\pmod{26}\text{,} The integers \(i\) and \(j\) may be the same or different. \end{equation*}, \begin{equation*} \newcommand{\gt}{>} \def\ppi{ ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} Lin et al. Prove that the affine cipher over Z26 has perfect secrecy if every key is used with equal probability of 1/312. How do these compare to the list of numbers which have multiplicative inverses? No matter which modulus you use, do all the numbers have additive inverses, i.e. 1977 0 obj
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An algorithm proposed by Bibhudendra et al. Along the same lines, why does \(f+y\) equal \(k\) and why does \(an\) (\(a\) times \(n\)) equal \(z\text{? Active 4 years, 9 months ago. A random matrix key, RMK is introduced as an extra key for encryption. \end{gather*}, \begin{gather*} \), \begin{gather*} \end{equation*}, \begin{equation*} In this way the letter h is replaced by the number 7 and when we encipher it we get, and 25 is Z, so plain h becomes cipher Z. \end{equation*}, \begin{equation*} } How do these compare to the list of numbers which have multiplicative inverses? Which numbers less than 26 are relatively prime to 26? In this paper, we extend this concept in the encryption core of our proposed cryptosystem. The remaining ciphers – Atbash, Caesar, Affine, Vigenère, Baconian, Hill, Running-Key, and RSA – fall under the non-monoalphabetic category. \newcommand{\lt}{<}
}\), The system of linear equations: \(o\, \alpha+u\, \beta = x\text{,}\) \(n\, \alpha+i\, \beta = q\) has solution \(\alpha = u\text{,}\) \(\beta=o\text{,}\) which may be obtained by the familiar method of elimination or by formula. a+ b\equiv 0 \pmod{n}, To decipher you will need to use the second formula listed in DefinitionÂ 6.1.17. It is slightly different to the other examples encountered here, since the encryption process is substantially mathematical. Hi guys, in this video we look at the encryption process behind the affine cipher. \amp 00 \amp 01 \amp 10 \amp 11 \\ \hline endstream
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\end{equation*}, \begin{gather*} \end{equation*}, \begin{equation*} \end{gather*}, \begin{gather*} Encryption is done using a simple mathematical function and converted back to a letter. 2012 0 obj
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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. How do these compare to the list of numbers which have multiplicative inverses? Another type of substitution cipher is the aﬃne cipher (or linear cipher). A. The proposed algorithm is an extension from Affine Hill cipher. The affine cipher is similar to the $ f $ function as it uses the values $ a $ and $ b $ as a coefficient and the variable $ x $ is the letter to be encrypted. \newcommand{\amp}{&} M.K. Which numbers, other than 7, that are less than 36 are relatively prime to 36? A very hard question: 550-700 points In the case of a tie, select questions predetermined by the event supervisor wil… 01 \amp 11 \amp 10 \amp 01 \amp 00 \\ \hline ), An affine cipher is a cipher with a two part key, a multiplier \(m\) and a shift \(s\) and calculations are carried out using modular arithmetic; typically the modulus is \(n=26\text{. Decryption involves matrix computations such as matrix inversion, and arithmetic calculations such as modular inverse. Hill cipher is it compromised to the known-plaintext attacks. 21\equiv m\cdot -15 \pmod{26} The whole process relies on working modulo m (the length of the alphabet used). \amp 00 \amp 01 \amp 10 \amp 11 \\ \hline To decrypt, as opposed to just decipher, an affine cipher you can use the techniques we learned in ChapterÂ 2 since they are a type of monoalphabetic substitution cipher. A comparative study has been made between the proposed algorithm and the existing algorithms. 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 \(i\text{. 19(13)+2\equiv 15\pmod{26} Do all of them have multiplicative inverses? Hill cipher decryption needs the matrix and the alphabet used. The cipher's primary weakness comes from the fact that if the cryptanalyst can discover (by means of frequency analysis, brute force, guessing or otherwise) the plaintext of two ciphertext characters, then the key can be obtained by solving a simultaneous equation . Viswanath in [1] proposed the concepts a public key cryptosystem using Hill’s Cipher. numbers you can multiply them by in order to get 1? an=z,\ hm=k,\ cr=s,\ etc. }\) 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{. The cipher is less secure than a substitution cipher as it is vulnerable to all of the attacks that work against substitution ciphers, in addition to other attacks. To set up an aﬃne cipher, you pick two values a and b, and then set ϵ(m) = am + b mod 26. This is a cipher based on the multiplication of matrices. (6) In any algebraic sum of terms, we may clearly omit terms of which the letter \(a_0\) is a factor; and we need not write the letter \(a_1\) explicitly as a factor in any product. \end{equation*}, \begin{equation*} 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. That used an affine cipher have multiplicative inverses two mobile circular disks which rotate... Listed in DefinitionÂ 6.1.17 is done using a simple mathematical function and converted back to a letter Java vigenere a... 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