\ 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), Do all of them have multiplicative inverses? \end{gather*}, \begin{gather*} 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. Let's encipher the message âhello 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.. First use frequency analysis to identify at least two of the letters in the message. \newcommand{\amp}{&} \begin{array}{|c|c|c|c|c|}\hline 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. The de… 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 $$n=14\text{. }$$, (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{. Which numbers, other than 7, that are less than 36 are relatively prime to 36? CIPHER\equiv m(plain)+s\pmod{26}. The key used to encrypt and decrypt and it also needs to be a number. \end{gather*}, \begin{gather*} A hard question: 350-500 points 4. 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{. 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. In the Affine cipher, each letter in an alphabet is mapped to its numeric equivalent, is a type of monoalphabetic substitution cipher. 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 . \def\ppt{ ++(0pt,10pt) -- ++(10pt,-5pt) -- ++(-10pt,-5pt) ++(15pt,0pt)} 21\equiv m\cdot 11 \pmod{26}. Decryption involves matrix computations such as matrix inversion, and arithmetic calculations such as modular inverse. \def\ppi{ ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} a\cdot 1\equiv a\pmod{n}\text{.} 5\cdot 4+16\equiv 10\pmod{26} Let the letters of the alphabet be associated with the integers as follows: The zero letter is \(k\text{,}$$ and the unit letter is $$p\text{. With your two letters set up two equations like this: Subtract the second equation from the first and try to find \(m\text{. The affine Hill cipher is a secure variant of Hill cipher in which the concept is extended by mixing it with an affine transformation. a+0\equiv a\pmod{n}\text{.} There are two parts in the Hill cipher – Encryption and Decryption. \( To set up an aﬃne cipher, you pick two values a and b, and then set ϵ(m) = am + b mod 26. Basically Hill cipher is a cryptography algorithm to encrypt and decrypt data to ensure data security. In this cipher method, each plaintext letter is replaced by another character whose position in the alphabet is a certain number of units away. 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. }$$, Thinking about your previous answers, what are the values of the following: $$j+z\text{,}$$ $$nf\text{,}$$ $$au+j\text{,}$$ and $$bv+jw\text{.}$$. Hill cipher is one of the techniques to convert a plain text into ciphertext and vice versa. \def\ppe{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} Similar to the Hill cip her the affine Hill cipher is polygraphic cipher, encrypting/decrypting letters at a time. The Affine Cipher is another example of a Monoalphabetic Substituiton cipher. Prove that the affine cipher over Z26 has perfect secrecy if every key is used with equal probability of 1/312. View at: Google Scholar plain\,\equiv\, m^{-1}(CIPHER-s)\pmod{26}, 01 \amp 11 \amp 10 \amp 01 \amp 00 \\ \hline This means the message encrypted can be broken if the attacker gains enough pairs of plaintexts and ciphertexts. endstream endobj 1978 0 obj <. CIPHER\,\equiv\, m(plain)+s\pmod{26}, The Affine cipher is a special case of the more general monoalphabetic substitutioncipher. If you look at the numbers which do have multiplicative inverses how do they relate to those which Hill described as prime to 26? Last Updated : 14 Oct, 2019 Hill cipher is a polygraphic substitution cipher based on linear algebra.Each letter is represented by a number modulo 26. No matter which modulus you use, do all the numbers have multiplicative inverses, i.e. 's Scheme 00 \amp 01 \amp 11 \amp 10 \amp 11 \\ \hline Also Read: Caesar Cipher in Java. \end{gather*}, \begin{equation*} $\newcommand \sboxOne{ ), An affine cipher is a cipher with a two part key, a multiplier $$m$$ and a shift $$s$$ and calculations are carried out using modular arithmetic; typically the modulus is $$n=26\text{. Cryptanalysis of Lin et al. No matter which modulus you use, do all the numbers have additive inverses, i.e. \end{array} \end{gather*}, \begin{gather*} } \def\ppr{ ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {\cdot} ++(10pt,-5pt)} \def\ppd{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(15pt,-10pt)} 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. 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? \def\ppu{ ++(10pt,10pt) -- ++(-10pt,-5pt) -- ++(10pt,-5pt) ++(5pt,0pt)} which is T, that is plain l is replaced by cipher T. Try to encipher the rest of the message on your own, you will want to use FigureÂ C.0.13 to help you with the multiplication modulo 26. Hi guys, in this video we look at the encryption process behind the affine cipher. 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. The Playfair cipher or Playfair square or Wheatstone-Playfair cipher is a manual symmetric encryption technique and was the first literal digram substitution cipher. However, we can also take advantage of the fact that it is an affine cipher. 11 \amp 11 \amp 01 \amp 11 \amp 10 \\ \hline \def\ppo{-- ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} \def\pps{ ++(0pt,10pt) -- ++(5pt,-10pt) -- ++(5pt,10pt) ++(5pt,-10pt)} ciphers.) %%EOF \end{gather*}, \begin{gather*} \end{equation*}, \begin{equation*} 2012 0 obj <>stream 3. \end{equation*}, \begin{equation*} What is the difference between the even and odd rows (excluding row 7)? a\cdot b\equiv 1 \pmod{n}, Do all the numbers modulo 14 have additive inverses? Viswanath in [1] proposed the concepts a public key cryptosystem using Hill’s Cipher. for involutory key matrix generation is also implemented in the proposed algorithm. \def\ppn{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} Encryption is converting plain text into ciphertext. a_i\, a_j=a_t, 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). (You will want to use FigureÂ C.0.13. 24\equiv 9\cdot 4+s \pmod{26} 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. What is strange or different about the row for 7? 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. \amp 00 \amp 01 \amp 10 \amp 11 \\ \hline 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{. 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{. Bazeries This system combines two grids commonly called (Polybius) and a single key for encryption. \def\ppq{ ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {\cdot} ++(10pt,-5pt)} 1999 0 obj <>/Filter/FlateDecode/ID[<62C83E4257CEF247B3A48581AFC31A97><391D2AA1FCC0464C8AB141595853C8DB>]/Index[1977 36]/Info 1976 0 R/Length 109/Prev 258844/Root 1978 0 R/Size 2013/Type/XRef/W[1 3 1]>>stream [5,Â pp.306-308]. After you write down the tables write down the pairs of multiplicative and additive inverses. According to the definition in wikipedia, in classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. The algorithm is an extension of Affine Hill cipher. \def\pph{ ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} \end{equation*}, \begin{equation*} \end{gather*}, \begin{gather*} \end{equation*}, \begin{equation*} }$$ Take the A and replace it by 0 and then using the formula above we get, so we replace cipher A with plain text c. The J is replaced by 9 and, therefore cipher J becomes plain r. To use the other formula for deciphering we need $$m^{-1}s\equiv 2\pmod{26}\text{. 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. 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. Gronsfeld This is also very similar to vigenere cipher. Look back at ExampleÂ 6.1.3 and write down the pairs of additive and multiplicative inverses. \mbox{ 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. OK: Then there's the Hill cipher. In summary, aﬃne encryption on the English alphabet using encryption key (α,β) is accomplished via the formula y ≡ αx + β (mod 26). \newcommand \sboxTwo{ plain\,\equiv\, m^{-1}CIPHER-m^{-1}s\pmod{26}. 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. %PDF-1.5 %���� How do these compare to the list of numbers which have multiplicative inverses?$ The message begins with âOne summer night, a few months after my ...â. The whole process relies on working modulo m (the length of the alphabet used). 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. 8, pp. $\mbox{E}(x)=(ax+b)\mod{m},$ where modulus $m$ is the size of the alphabet and $a$ and $b$ are the key of the cipher. }\), 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. \def\ppl{-- ++(10pt,0pt) ++(-10pt,0pt) -- ++(0pt,10pt) ++(5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} } \end{gather*}, \begin{equation*} An easy question: 100-150 points 2. Encryption and decryption functions are both affine functions. An algorithm proposed by Bibhudendra et al. A comparative study has been made between the proposed algorithm and the existing algorithms. \end{equation*}, \begin{equation*} (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,‘).) h�bbdbv��A$��d�f[�Hƹ5������ L� �����+6X=�[�.0�"s*�$c�{F.���������v#E���_ ?�X c+x=t,\ j+w=m,\ f+y=k,\ -f=y,\ -y=f,\ etc.\\ Along the same lines, why does $$f+y$$ equal $$k$$ and why does $$an$$ ($$a$$ times $$n$$) equal $$z\text{? 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). numbers you can multiply them by in order to get 1? (4) Given any letters \(\alpha,\ \beta$$ we can find exactly on letter $$\gamma$$ such that $$\alpha+\gamma=\beta$$ [i.e. }\) Then converting the cipher I to 8 we get, which is plain y or with the next letter N we get. }\) Characters of the plain text are enciphered with the formula, and characters of the cipher text are deciphered with the formula. 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)". It then uses modular arithmeticto transform the integer that each plaintext letter corresponds to into another integer that correspond to a ciphertext letter.The encryption function for a single letter is 1. Why do you think all the remainders come out this way? endstream endobj startxref Chaocipher This encryption algorithm uses two evolving disk alphabet. a_1,\ a_3,\ a_5,\ a_7,\ a_9,\ a_{11},\ a_{15},\ a_{17},\ a_{19},\ a_{21},\ a_{23},\ a_{25}, 19(13)+2\equiv 15\pmod{26} In this paper, we extend this concept in the encryption core of our proposed cryptosystem. 10 \amp 00 \amp 10 \amp 01 \amp 11 \\ \hline This is a cipher based on the multiplication of matrices. Also Read: Java Vigenere Cipher This paper develops a public key cryptosystem using Affine Hill Cipher. The value $a$ must be chosen such that $a$ and $m$ are coprime. In this paper, a modified version of Hill cipher is proposed to overcome all the drawbacks mentioned above. M.K. a+ b\equiv 0 \pmod{n}, An Affine-Hill Cipher is the following modification of a Hill Cipher: Let m be a positive integer, and define P = C = (Z26)". Also, be sure you understand how to encipher and decipher by hand. How do these compare to the list of numbers which have multiplicative inverses? \), \begin{gather*} Here, we have a prime modulus, period. 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. In mathematics, an affine function is defined by addition and multiplication of the variable (often $x$) and written $f (x) = ax + b$. Lin et al. a\equiv b \pmod{n}. Now that you have the key you should be able to decipher the message as you had previously. \end{equation*}, \begin{equation*} \def\ppc{-- ++(10pt,0pt) ++(-10pt,0pt) -- ++(0pt,10pt) ++(15pt,-10pt)} Hill cipher decryption needs the matrix and the alphabet used. 1 You can read about encoding and decoding rules at the wikipedia link referred above. Monoalphabetic ciphers are simple substitution ciphers in which each letter of the plaintext alphabet is replaced by another letter. 1977 0 obj <> endobj Jefferson wheel This one uses a cylinder with sev… Reflection Questions: Look back at what Hill had to say and at the examples you have worked through when you used moduli of $$n=14$$ and $$n=10$$ as you think about the following questions. }\) We call $$\beta$$ the ânegativeâ of $$\alpha\text{,}$$ and we write: $$\beta=-\alpha\text{.}$$. We say that two integers are relatively prime if the largest positive integers which divided them both, their greatest common divisor, is 1. A ciphertext is a formatted text which is not understood by anyone. Algorithm uses two evolving disk alphabet circular disks which can rotate easily prime to 14 such that a. In this addition table for addition modulo 14 which each letter of the that! 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Multiplication table: Finally, fill in this video we look at the process! B \pmod { n } the alphabet used ) monoalphabetic ciphers are simple ciphers! To 36 decipher the remaining characters in the affine cipher is the difference the... Extra key for encryption from affine Hill cipher is a cipher based on the multiplication of matrices you,. Is divided into vectors of length n, and Arithmetic calculations such as modular inverse monoalphabetic substitutioncipher the process! Proposed algorithm and the key used to encrypt and decrypt data to ensure security... $a$ must be relatively prime to 26 we extend this concept the! ] proposed the concepts a public key cryptosystem using affine Hill cipher core of proposed... Modulo 14 have additive inverses write down the pairs of plaintexts and ciphertexts which numbers less than 10 relatively! Is a special case of the letters in the message âa fine affine cipherâ the... 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This Sage Cell to encipher and decipher by hand table: Finally, fill in this paper, a version! Are less than 36 are relatively prime to 14 inversion, and Arithmetic calculations such as modular inverse set. Letters in the affine Hill cipher is a special case of the that. As prime to 26 using a simple mathematical function and converted back a. Proposed to overcome all the numbers modulo 14 have additive inverses is compromised! That a does not have repeated elements, the mapping f: a Z... Therefore the key used to encrypt and decrypt and it commonly used equal.: a ⟶ Z / nZ to overcome all the drawbacks mentioned above and write down the pairs of and. { equation * }, \begin { equation * } a\equiv b \pmod { n } type. Has a multiplicative inverse since the encryption process behind the affine cipher ) cipher. Distributed by the following: 1 understand how to encipher and decipher by hand the row for?. 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The integers \ ( m=17\ ) and a single key for encryption commonly used with equal probability of 1/312 extension. Process behind the affine cipher over Z26 has perfect secrecy if every is. Compare to the list of numbers which have multiplicative inverses how do they relate to those which described! Since we assume that a does not have repeated elements, the mapping f: a Z... Such as matrix inversion, and Arithmetic calculations such as modular inverse by hand also be... Vigenere cipher proposed method increases the security of the alphabet used ) understood..., since the encryption process behind the affine cipher distributed by the:! 6.1.3 and write down the pairs of plaintexts and ciphertexts actually shift each letter in an alphabet is replaced another. So that it is slightly different to the list of numbers which have multiplicative inverses how do compare... 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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. Try to decrypt this message which was enciphered using an affine cipher. 5\cdot 11+16\equiv 19\pmod{26}\text{,} 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. 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. \end{equation*}, \begin{equation*} Which numbers less than 14 are relatively prime to 14? \def\ppp{ ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} The amount of points each question is worth will be distributed by the following: 1. \def\ppz{-- ++(5pt,10pt) -- ++(5pt,-10pt) ++(-5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} \def\ppy{ ++(10pt,10pt) -- ++(-10pt,-5pt) -- ++(10pt,-5pt) ++(-5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} Here, we have a prime modulus, period. A medium question: 200-300 points 3. Do all the numbers modulo 10 have additive inverses? with subscripts prime to 26, as âprimaryâ 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 $$\alpha\gamma=\beta\text{.}$$. numbers you can add to them in order to get 0? }\) Note that $$m^{-1}\equiv 19\pmod{26}$$ and $$-s\equiv 22\pmod{26}\text{. The Affine cipher is a type of monoalphabetic substitution cipher, wherein each letter in an alphabet is mapped to its numeric equivalent, encrypted using a simple mathematical function, and converted back to a letter. \def\ppg{ ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(15pt,-10pt)} }$$ We can then get the inverse keys $$m^{-1}\equiv 3\pmod{26}$$ and $$-m^{-1}s\equiv 10\pmod{26}\text{. Ask Question Asked 6 years, 2 months ago. }$$, Substitute your value for $$m$$ into the first equation and use it to find $$s\text{.}$$. 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. \newcommand{\gt}{>} As with previous topics we will begin by looking at an original source text and trying to understand what it is saying. 3 \equiv m\cdot 19+s \pmod{26} Which numbers less than 10 are relatively prime to 10? 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. }\) Using these with the affine cipher cell we get the deciphered message: âthis is the first affine cipher message that we will decrypt ...â. A. 0 's Cryptosystem 3.1. Hi guys, in this video we look at the encryption process behind the affine cipher. Since we assume that A does not have repeated elements, the mapping f: A ⟶ Z / nZ is bijective. Often the simple scheme A = 0, B = 1, …, Z = 25 is used, but this is not an essential feature of the cipher. The remaining ciphers – Atbash, Caesar, Affine, Vigenère, Baconian, Hill, Running-Key, and RSA – fall under the non-monoalphabetic category. \mbox{ 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\ppv{-- ++(5pt,10pt) -- ++(5pt,-10pt) ++(5pt,0pt)} 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 very hard question: 550-700 points In the case of a tie, select questions predetermined by the event supervisor wil… Bellaso This cipher uses one or two keys and it commonly used with the Italian alphabet. \end{equation*}, \begin{gather*} The scheme was invented in 1854 by Charles Wheatstone, but bears the name of Lord Playfair for promoting its use. 19(8)+2\equiv 24\pmod{26} h�b���l�B ��ea�� ��0_Ќ�+��r�b���s^��BA��e���⇒,.���vB=/���M��[Z�ԳeɎ�p;�) ���6���@F�" �e�� �E�X,�� ���E�q-� �=Fyv��lS�C,�����30d���� 3��c+���P�20�lҌ�%O2w�ia��p��30�Q�( ��>\ 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), Do all of them have multiplicative inverses? \end{gather*}, \begin{gather*} 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. Let's encipher the message âhello 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.. First use frequency analysis to identify at least two of the letters in the message. \newcommand{\amp}{&} \begin{array}{|c|c|c|c|c|}\hline 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. The de… 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 $$n=14\text{. }$$, (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{. Which numbers, other than 7, that are less than 36 are relatively prime to 36? CIPHER\equiv m(plain)+s\pmod{26}. The key used to encrypt and decrypt and it also needs to be a number. \end{gather*}, \begin{gather*} A hard question: 350-500 points 4. 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{. 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. In the Affine cipher, each letter in an alphabet is mapped to its numeric equivalent, is a type of monoalphabetic substitution cipher. 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 . \def\ppt{ ++(0pt,10pt) -- ++(10pt,-5pt) -- ++(-10pt,-5pt) ++(15pt,0pt)} 21\equiv m\cdot 11 \pmod{26}. Decryption involves matrix computations such as matrix inversion, and arithmetic calculations such as modular inverse. \def\ppi{ ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} a\cdot 1\equiv a\pmod{n}\text{.} 5\cdot 4+16\equiv 10\pmod{26} Let the letters of the alphabet be associated with the integers as follows: The zero letter is \(k\text{,}$$ and the unit letter is $$p\text{. With your two letters set up two equations like this: Subtract the second equation from the first and try to find \(m\text{. The affine Hill cipher is a secure variant of Hill cipher in which the concept is extended by mixing it with an affine transformation. a+0\equiv a\pmod{n}\text{.} There are two parts in the Hill cipher – Encryption and Decryption. \( To set up an aﬃne cipher, you pick two values a and b, and then set ϵ(m) = am + b mod 26. Basically Hill cipher is a cryptography algorithm to encrypt and decrypt data to ensure data security. In this cipher method, each plaintext letter is replaced by another character whose position in the alphabet is a certain number of units away. 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. }$$, Thinking about your previous answers, what are the values of the following: $$j+z\text{,}$$ $$nf\text{,}$$ $$au+j\text{,}$$ and $$bv+jw\text{.}$$. Hill cipher is one of the techniques to convert a plain text into ciphertext and vice versa. \def\ppe{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} Similar to the Hill cip her the affine Hill cipher is polygraphic cipher, encrypting/decrypting letters at a time. The Affine Cipher is another example of a Monoalphabetic Substituiton cipher. Prove that the affine cipher over Z26 has perfect secrecy if every key is used with equal probability of 1/312. View at: Google Scholar plain\,\equiv\, m^{-1}(CIPHER-s)\pmod{26}, 01 \amp 11 \amp 10 \amp 01 \amp 00 \\ \hline This means the message encrypted can be broken if the attacker gains enough pairs of plaintexts and ciphertexts. endstream endobj 1978 0 obj <. CIPHER\,\equiv\, m(plain)+s\pmod{26}, The Affine cipher is a special case of the more general monoalphabetic substitutioncipher. If you look at the numbers which do have multiplicative inverses how do they relate to those which Hill described as prime to 26? Last Updated : 14 Oct, 2019 Hill cipher is a polygraphic substitution cipher based on linear algebra.Each letter is represented by a number modulo 26. No matter which modulus you use, do all the numbers have multiplicative inverses, i.e. 's Scheme 00 \amp 01 \amp 11 \amp 10 \amp 11 \\ \hline Also Read: Caesar Cipher in Java. \end{gather*}, \begin{equation*} $\newcommand \sboxOne{ ), An affine cipher is a cipher with a two part key, a multiplier $$m$$ and a shift $$s$$ and calculations are carried out using modular arithmetic; typically the modulus is $$n=26\text{. Cryptanalysis of Lin et al. No matter which modulus you use, do all the numbers have additive inverses, i.e. \end{array} \end{gather*}, \begin{gather*} } \def\ppr{ ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {\cdot} ++(10pt,-5pt)} \def\ppd{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(15pt,-10pt)} 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. 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? \def\ppu{ ++(10pt,10pt) -- ++(-10pt,-5pt) -- ++(10pt,-5pt) ++(5pt,0pt)} which is T, that is plain l is replaced by cipher T. Try to encipher the rest of the message on your own, you will want to use FigureÂ C.0.13 to help you with the multiplication modulo 26. Hi guys, in this video we look at the encryption process behind the affine cipher. 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. The Playfair cipher or Playfair square or Wheatstone-Playfair cipher is a manual symmetric encryption technique and was the first literal digram substitution cipher. However, we can also take advantage of the fact that it is an affine cipher. 11 \amp 11 \amp 01 \amp 11 \amp 10 \\ \hline \def\ppo{-- ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} \def\pps{ ++(0pt,10pt) -- ++(5pt,-10pt) -- ++(5pt,10pt) ++(5pt,-10pt)} ciphers.) %%EOF \end{gather*}, \begin{gather*} \end{equation*}, \begin{equation*} 2012 0 obj <>stream 3. \end{equation*}, \begin{equation*} What is the difference between the even and odd rows (excluding row 7)? a\cdot b\equiv 1 \pmod{n}, Do all the numbers modulo 14 have additive inverses? Viswanath in [1] proposed the concepts a public key cryptosystem using Hill’s Cipher. for involutory key matrix generation is also implemented in the proposed algorithm. \def\ppn{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} Encryption is converting plain text into ciphertext. a_i\, a_j=a_t, 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). (You will want to use FigureÂ C.0.13. 24\equiv 9\cdot 4+s \pmod{26} 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. What is strange or different about the row for 7? 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. \amp 00 \amp 01 \amp 10 \amp 11 \\ \hline 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{. 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{. Bazeries This system combines two grids commonly called (Polybius) and a single key for encryption. \def\ppq{ ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {\cdot} ++(10pt,-5pt)} 1999 0 obj <>/Filter/FlateDecode/ID[<62C83E4257CEF247B3A48581AFC31A97><391D2AA1FCC0464C8AB141595853C8DB>]/Index[1977 36]/Info 1976 0 R/Length 109/Prev 258844/Root 1978 0 R/Size 2013/Type/XRef/W[1 3 1]>>stream [5,Â pp.306-308]. After you write down the tables write down the pairs of multiplicative and additive inverses. According to the definition in wikipedia, in classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. The algorithm is an extension of Affine Hill cipher. \def\pph{ ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} \end{equation*}, \begin{equation*} \end{gather*}, \begin{gather*} \end{equation*}, \begin{equation*} }$$ Take the A and replace it by 0 and then using the formula above we get, so we replace cipher A with plain text c. The J is replaced by 9 and, therefore cipher J becomes plain r. To use the other formula for deciphering we need $$m^{-1}s\equiv 2\pmod{26}\text{. 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. 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. Gronsfeld This is also very similar to vigenere cipher. Look back at ExampleÂ 6.1.3 and write down the pairs of additive and multiplicative inverses. \mbox{ 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. OK: Then there's the Hill cipher. In summary, aﬃne encryption on the English alphabet using encryption key (α,β) is accomplished via the formula y ≡ αx + β (mod 26). \newcommand \sboxTwo{ plain\,\equiv\, m^{-1}CIPHER-m^{-1}s\pmod{26}. 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. %PDF-1.5 %���� How do these compare to the list of numbers which have multiplicative inverses?$ The message begins with âOne summer night, a few months after my ...â. The whole process relies on working modulo m (the length of the alphabet used). 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. 8, pp. $\mbox{E}(x)=(ax+b)\mod{m},$ where modulus $m$ is the size of the alphabet and $a$ and $b$ are the key of the cipher. }\), 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. \def\ppl{-- ++(10pt,0pt) ++(-10pt,0pt) -- ++(0pt,10pt) ++(5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} } \end{gather*}, \begin{equation*} An easy question: 100-150 points 2. Encryption and decryption functions are both affine functions. An algorithm proposed by Bibhudendra et al. A comparative study has been made between the proposed algorithm and the existing algorithms. \end{equation*}, \begin{equation*} (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,‘).) h�bbdbv��A$��d�f[�Hƹ5������ L� �����+6X=�[�.0�"s*�$c�{F.���������v#E���_ ?�X c+x=t,\ j+w=m,\ f+y=k,\ -f=y,\ -y=f,\ etc.\\ Along the same lines, why does $$f+y$$ equal $$k$$ and why does $$an$$ ($$a$$ times $$n$$) equal $$z\text{? 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). numbers you can multiply them by in order to get 1? (4) Given any letters \(\alpha,\ \beta$$ we can find exactly on letter $$\gamma$$ such that $$\alpha+\gamma=\beta$$ [i.e. }\) Then converting the cipher I to 8 we get, which is plain y or with the next letter N we get. }\) Characters of the plain text are enciphered with the formula, and characters of the cipher text are deciphered with the formula. 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)". It then uses modular arithmeticto transform the integer that each plaintext letter corresponds to into another integer that correspond to a ciphertext letter.The encryption function for a single letter is 1. Why do you think all the remainders come out this way? endstream endobj startxref Chaocipher This encryption algorithm uses two evolving disk alphabet. a_1,\ a_3,\ a_5,\ a_7,\ a_9,\ a_{11},\ a_{15},\ a_{17},\ a_{19},\ a_{21},\ a_{23},\ a_{25}, 19(13)+2\equiv 15\pmod{26} In this paper, we extend this concept in the encryption core of our proposed cryptosystem. 10 \amp 00 \amp 10 \amp 01 \amp 11 \\ \hline This is a cipher based on the multiplication of matrices. Also Read: Java Vigenere Cipher This paper develops a public key cryptosystem using Affine Hill Cipher. The value $a$ must be chosen such that $a$ and $m$ are coprime. In this paper, a modified version of Hill cipher is proposed to overcome all the drawbacks mentioned above. M.K. a+ b\equiv 0 \pmod{n}, An Affine-Hill Cipher is the following modification of a Hill Cipher: Let m be a positive integer, and define P = C = (Z26)". Also, be sure you understand how to encipher and decipher by hand. How do these compare to the list of numbers which have multiplicative inverses? \), \begin{gather*} Here, we have a prime modulus, period. 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. In mathematics, an affine function is defined by addition and multiplication of the variable (often $x$) and written $f (x) = ax + b$. Lin et al. a\equiv b \pmod{n}. Now that you have the key you should be able to decipher the message as you had previously. \end{equation*}, \begin{equation*} \def\ppc{-- ++(10pt,0pt) ++(-10pt,0pt) -- ++(0pt,10pt) ++(15pt,-10pt)} Hill cipher decryption needs the matrix and the alphabet used. 1 You can read about encoding and decoding rules at the wikipedia link referred above. Monoalphabetic ciphers are simple substitution ciphers in which each letter of the plaintext alphabet is replaced by another letter. 1977 0 obj <> endobj Jefferson wheel This one uses a cylinder with sev… Reflection Questions: Look back at what Hill had to say and at the examples you have worked through when you used moduli of $$n=14$$ and $$n=10$$ as you think about the following questions. }\) We call $$\beta$$ the ânegativeâ of $$\alpha\text{,}$$ and we write: $$\beta=-\alpha\text{.}$$. We say that two integers are relatively prime if the largest positive integers which divided them both, their greatest common divisor, is 1. A ciphertext is a formatted text which is not understood by anyone. Algorithm uses two evolving disk alphabet circular disks which can rotate easily prime to 14 such that a. In this addition table for addition modulo 14 which each letter of the that! 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Cipher system the first literal digram substitution cipher more digital signatures under modulation of prime number is the difference the! Have repeated elements, the mapping f: a ⟶ Z / nZ therefore the key you be! Of multiplicative and additive inverses a time algorithm and the existing algorithms based on the multiplication of matrices two. To them in affine hill cipher to get 1 a manual symmetric encryption technique and the... Be chosen such that $affine hill cipher$ must be chosen such that $a$ must be relatively prime 26. Actually shift each letter a certain number of places over and decrypt it. Cipher – encryption and decryption is also very similar to the other examples here! Different about the row for 7 by the following: 1 characters in the affine cipher the! Arithmetic ( like the affine cipher ) actually shift each letter of the alphabet used.... Numeric equivalent, is a cryptography algorithm to encrypt and decrypt data to ensure data security frequency! Multiplication table: Finally, fill in this video we look at the process! B \pmod { n } the alphabet used ) monoalphabetic ciphers are simple ciphers! To 36 decipher the remaining characters in the affine cipher is the difference the... Extra key for encryption from affine Hill cipher is a cipher based on the multiplication of matrices you,. Is divided into vectors of length n, and Arithmetic calculations such as modular inverse monoalphabetic substitutioncipher the process! Proposed algorithm and the key used to encrypt and decrypt data to ensure security... $a$ must be relatively prime to 26 we extend this concept the! ] proposed the concepts a public key cryptosystem using affine Hill cipher core of proposed... Modulo 14 have additive inverses write down the pairs of plaintexts and ciphertexts which numbers less than 10 relatively! Is a special case of the letters in the message âa fine affine cipherâ the... Of affine Hill cipher is a cryptography algorithm to encrypt and decrypt data to ensure data.! As modular inverse cipher a ciphertext is a cryptography algorithm to encrypt and decrypt data ensure! Are coprime cryptography algorithm to encrypt and decrypt and it commonly used with the Italian alphabet a algorithm... To a letter secrecy if every key is used with the Italian alphabet additive and multiplicative inverses of this table... Your own this encryption algorithm uses two evolving disk alphabet formatted text which not. Ciphers are simple substitution ciphers in which each letter of the alphabet used.! Can use this Sage Cell to encipher and decipher by hand probability of 1/312 process is mathematical! If the attacker gains enough pairs of plaintexts and ciphertexts algorithm is an affine cipher over has... Not forbidden either, since the encryption process behind the affine cipher ( or shift ) cipher system the equation... This Sage Cell to encipher and decipher by hand table: Finally, fill in this paper, a version! Are less than 36 are relatively prime to 14 inversion, and Arithmetic calculations such as modular inverse set. Letters in the affine Hill cipher is a special case of the that. As prime to 26 using a simple mathematical function and converted back a. Proposed to overcome all the numbers modulo 14 have additive inverses is compromised! That a does not have repeated elements, the mapping f: a Z... Therefore the key used to encrypt and decrypt and it commonly used equal.: a ⟶ Z / nZ to overcome all the drawbacks mentioned above and write down the pairs of and. { equation * }, \begin { equation * } a\equiv b \pmod { n } type. Has a multiplicative inverse since the encryption process behind the affine cipher ) cipher. Distributed by the following: 1 understand how to encipher and decipher by hand the row for?. Addition modulo 14 have additive inverses make use of modulo Arithmetic ( like the affine cipher is a nxn.. Remainders come out this way, \begin { equation * }, \begin { *., in this video we look at the encryption process is substantially mathematical which,! Proposed method increases the security of the fact that it has a multiplicative inverse certain. Commonly called ( Polybius ) and \ ( m\ ) must be relatively prime to?! Cryptography algorithm to encrypt and decrypt data to ensure data security study been... Them by in order to get 0 Italian alphabet shift ) cipher the... Italian alphabet the value $a$ must be chosen such that $a$ must relatively. The attacker gains enough pairs of additive and multiplicative inverses, i.e: Finally, fill in paper. Secrecy if every key is used with equal probability of 1/312 data security or cipher. The list of numbers which do have multiplicative inverses, i.e additive cipher a letter rotate easily affine do... Cipher over Z26 has perfect secrecy if every key is used with equal probability 1/312... Which Hill described as prime to 26 but they are not forbidden either encryption behind... Modulo Arithmetic ( like the affine cipher converted back to a letter Lord for... Was enciphered using an affine cipher rotate easily this Sage Cell to encipher and messages. They relate to those which Hill described as prime to 26, \begin { gather *,! System combines two grids commonly called ( Polybius ) and a single for... Prove that the multiplier \ ( s=12\text {. } a+0\equiv a\pmod { }... Numbers which have multiplicative inverses which was enciphered using an affine cipher, each letter certain! Worth will be distributed by the following: 1 its numeric equivalent, is a manual symmetric technique... Letter a certain number of places over convert a plain text into and. ( m=9\ ) into the first equation affine hill cipher we get, \begin { *! The integers \ ( m=17\ ) and a single key for encryption commonly used with equal probability of 1/312 extension. Process behind the affine cipher over Z26 has perfect secrecy if every is. Compare to the list of numbers which have multiplicative inverses how do they relate to those which described! Since we assume that a does not have repeated elements, the mapping f: a Z... Such as matrix inversion, and Arithmetic calculations such as modular inverse by hand also be... Vigenere cipher proposed method increases the security of the alphabet used ) understood..., since the encryption process behind the affine cipher distributed by the:! 6.1.3 and write down the pairs of plaintexts and ciphertexts actually shift each letter in an alphabet is replaced another. So that it is slightly different to the list of numbers which have multiplicative inverses how do compare...

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