### Aktuality

# affine hill cipher

\def\ppg{ ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(15pt,-10pt)} endstream endobj 1978 0 obj <. 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. OK: Then there's the Hill cipher. }\) Alternately, we can observe that \(36-8=28\) and \(28=2\cdot(14)\) is divisible by \(n=14\text{.}\). Since we assume that A does not have repeated elements, the mapping f: A ⟶ Z / nZ is bijective. Decryption involves matrix computations such as matrix inversion, and arithmetic calculations such as modular inverse. 21\equiv m\cdot 11 \pmod{26}. \def\pph{ ++(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. 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 \(\alpha\gamma=\beta\text{.}\). 01 \amp 11 \amp 10 \amp 01 \amp 00 \\ \hline Which numbers less than 14 are relatively prime to 14? \end{equation*}, \begin{gather*} However, we can also take advantage of the fact that it is an affine cipher. } }\) 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. \def\ppe{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} \def\ppb{-- ++(10pt,0pt) -- ++(0pt,10pt) ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} The affine Hill cipher is a secure variant of Hill cipher in which the concept is extended by mixing it with an affine transformation. 5\cdot 7+16\equiv 25\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{. How do these compare to the list of numbers which have multiplicative inverses? Ask Question Asked 6 years, 2 months ago. endstream endobj startxref \end{gather*}, \begin{gather*} Lin et al. %%EOF In this cipher method, each plaintext letter is replaced by another character whose position in the alphabet is a certain number of units away. There are two parts in the Hill cipher – Encryption and Decryption. 11 \amp 10 \amp 01 \amp 10 \amp 11 \\ \hline \end{equation*}, \begin{equation*} Hi guys, in this video we look at the encryption process behind the affine cipher. 5\cdot 11+16\equiv 19\pmod{26}\text{,} Encipher the message âa fine affine cipherâ using the key \(m=17\) and \(s=12\text{. 19(8)+2\equiv 24\pmod{26} The de… 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*} 3 \equiv m\cdot 19+s \pmod{26} Similar to the Hill cip her the affine Hill cipher is polygraphic cipher, encrypting/decrypting letters at a time. Hill cipher is it compromised to the known-plaintext attacks. \def\ppr{ ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} 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), 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. 10 \amp 00 \amp 10 \amp 01 \amp 11 \\ \hline (4) Given any letters \(\alpha,\ \beta\) we can find exactly on letter \(\gamma\) such that \(\alpha+\gamma=\beta\) [i.e. \def\ppq{ ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} }\) Characters of the plain text are enciphered with the formula, and characters of the cipher text are deciphered with the formula. Basically Hill cipher is a cryptography algorithm to encrypt and decrypt data to ensure data security. If you look at the numbers which do have multiplicative inverses how do they relate to those which Hill described as prime to 26? \end{equation*}, \begin{equation*} which is p. Try to decipher the remaining characters in the message on your own. \def\ppj{-- ++(10pt,0pt) -- ++(0pt,10pt) ++(-5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} No matter which modulus you use, do all the numbers have additive inverses, i.e. A random matrix key, RMK is introduced as an extra key for encryption. numbers you can multiply them by in order to get 1? }\) Note that \(m^{-1}\equiv 19\pmod{26}\) and \(-s\equiv 22\pmod{26}\text{. The cipher we will focus on here, Hill's Cipher, is an early example of a cipher based purely in the mathematics of number theory and algebra; the areas of mathematics which now dominate all of modern cryptography. \newcommand \sboxOne{ Jefferson wheel This one uses a cylinder with sev… In summary, aﬃne encryption on the English alphabet using encryption key (α,β) is accomplished via the formula y ≡ αx + β (mod 26). }\), Substitute your value for \(m\) into the first equation and use it to find \(s\text{.}\). 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)". $\begingroup$ @AJMansfield It is true that affine ciphers do not require a prime modulus, but they are not forbidden either. What is the difference between the even and odd rows (excluding row 7)? }\), 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. \(\gamma=\beta-\alpha\) is unique]. We say that two integers are relatively prime if the largest positive integers which divided them both, their greatest common divisor, is 1. Chaocipher This encryption algorithm uses two evolving disk alphabet. \end{equation*}, \begin{equation*} 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. Also Read: Java Vigenere Cipher 24-10\equiv s \pmod{26} }\) 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 \(m^{-1}s\equiv 2\pmod{26}\text{. h�b```���l�B ��ea�� ��0_Ќ�+��r�b���s^��BA��e���⇒,.���vB=/���M��[Z�ԳeɎ�p;�) ���`6���@F�" �e`�� �E�X,�� ���E�q-� �=Fyv�`�lS�C,�����30d���� 3��c+���P�20�lҌ�%`O2w�ia��p��30�Q�(` ��>\ What is strange or different about the row for 7? \end{gather*}, \begin{equation*} }\) The primary letters are: \(a\) \(b\) \(f\) \(j\) \(n\) \(o\) \(p\) \(q\) \(u\) \(v\) \(y\) \(z\text{.}\). In the Affine cipher, each letter in an alphabet is mapped to its numeric equivalent, is a type of monoalphabetic substitution cipher. a+0\equiv a\pmod{n}\text{.} \amp 00 \amp 01 \amp 10 \amp 11 \\ \hline a_i\, a_j=a_t, 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. You can use this Sage Cell to encipher and decipher messages that used an affine cipher. \amp 00 \amp 01 \amp 10 \amp 11 \\ \hline Another type of substitution cipher is the aﬃne cipher (or linear cipher). 's Scheme numbers you can add to them in order to get 0? 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)} \def\ppk{-- ++(10pt,0pt) -- ++(0pt,10pt) ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} The amount of points each question is worth will be distributed by the following: 1. The scheme was invented in 1854 by Charles Wheatstone, but bears the name of Lord Playfair for promoting its use. a_1,\ a_3,\ a_5,\ a_7,\ a_9,\ a_{11},\ a_{15},\ a_{17},\ a_{19},\ a_{21},\ a_{23},\ a_{25}, 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. }\) We can then get the inverse keys \(m^{-1}\equiv 3\pmod{26}\) and \(-m^{-1}s\equiv 10\pmod{26}\text{. \newcommand{\lt}{<} \def\ppn{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} \), \begin{gather*} \end{array} \newcommand \sboxTwo{ (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,‘).) 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. [5,Â pp.306-308]. \end{gather*}, \begin{gather*} Encryption is converting plain text into ciphertext. Note that the multiplier \(m\) must be relatively prime to the modulus so that it has a multiplicative inverse. \def\ppp{ ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} The integers \(i\) and \(j\) may be the same or different. 8, pp. 10 \amp 11 \amp 00 \amp 01 \amp 00 \\ \hline It is easy to verify the following salient propositions concerning the bi-operational alphabet thus set up: (1) If \(\alpha,\ \beta,\ \gamma\) are letters of the alphabet, (2) There is exactly one âzeroâ letter, namely \(a_0\text{,}\) characterized by the fact that the equation \(\alpha+a_0=\alpha\) is satisfied whatever the letter denoted by \(alpha\text{. \mbox{ $ \mbox{E}(x)=(ax+b)\mod{m}, $ where modulus $ m $ is the size of the alphabet and $ a $ and $ b $ are the key of the cipher. \end{gather*}, \begin{gather*} 00 \amp 00 \amp 01 \amp 10 \amp 11 \\ \hline a\cdot b\equiv 1 \pmod{n}, An improved version of the Hill cipher which can withstand known plaintext attacks is Affine Hill cipher [20, 37]. Here, we have a prime modulus, period. }\) Then converting the cipher I to 8 we get, which is plain y or with the next letter N we get. Do all of them have multiplicative inverses? Viswanath in [1] proposed the concepts a public key cryptosystem using Hill’s Cipher. 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. Along the same lines, why does \(f+y\) equal \(k\) and why does \(an\) (\(a\) times \(n\)) equal \(z\text{? A ciphertext is a formatted text which is not understood by anyone. Do all the numbers modulo 10 have additive inverses? 0 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.. 19(13)+2\equiv 15\pmod{26} How do these compare to the list of numbers which have multiplicative inverses? \( } The plaintext is divided into vectors of length n, and the key is a nxn matrix. a+ b\equiv 0 \pmod{n}, To decrypt hill ciphertext, compute the matrix inverse modulo 26 (where 26 is the alphabet length), requiring the matrix to … 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. a_i+a_j=a_r,\\ The Affine cipher is a special case of the more general monoalphabetic substitutioncipher. It also make use of Modulo Arithmetic (like the Affine Cipher). The Playfair cipher or Playfair square or Wheatstone-Playfair cipher is a manual symmetric encryption technique and was the first literal digram substitution cipher. 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 \(i\text{. \end{equation*}, \begin{equation*} How do these compare to the list of numbers which have multiplicative inverses? }\) Substituting \(m=9\) into the first equation above we get. Prove that the affine cipher over Z26 has perfect secrecy if every key is used with equal probability of 1/312. 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{. Hill cipher is one of the techniques to convert a plain text into ciphertext and vice versa. \end{gather*}, \begin{gather*} 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. 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. 19(9+22)\equiv 17\pmod{26} \def\ppy{ ++(10pt,10pt) -- ++(-10pt,-5pt) -- ++(10pt,-5pt) ++(-5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} A hard question: 350-500 points 4. 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. A very hard question: 550-700 points In the case of a tie, select questions predetermined by the event supervisor wil… \end{equation*}, \begin{equation*} Because of this, the cipher has a significantly more mathematical nature than some of … 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. \def\ppi{ ++(10pt,0pt) ++(0pt,10pt) -- ++(-10pt,0pt) -- ++(0pt,-10pt) ++(15pt,0pt)} A key of the affine cipher is an ordered pair of integers (a, b) ∈ Z / nZ × Z / nZ such that gcd (a, n) = 1. The message begins with âOne summer night, a few months after my ...â. A. \begin{array}{|c|c|c|c|c|}\hline Do all of them have multiplicative inverses? h�bbd```b``v��A$��d�f[�Hƹ`5�`����� L� �����+`6X=�[�.0�"s*�$c�{F.���������v#E���_ ?�X A comparative study has been made between the proposed algorithm and the existing algorithms. The proposed method increases the security of the system because it involves two or more digital signatures under modulation of prime number. $ 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. The letters of an alphabet of size m are first mapped to the integers in the range 0 … m-1, in the Affine cipher, Also, be sure you understand how to encipher and decipher by hand. Look back at ExampleÂ 6.1.3 and write down the pairs of additive and multiplicative inverses. The Additive (or shift) Cipher System The first type of monoalphabetic substitution cipher we wish to examine is called the additive cipher. 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. We actually shift each letter a certain number of places over. Which numbers less than 26 are relatively prime to 26? 24\equiv m\cdot 4+s \pmod{26}\\ 's Cryptosystem 3.1. Also Read: Caesar Cipher in Java. Do all the numbers modulo 14 have additive inverses? so that \(s=14\text{. (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. \newcommand{\amp}{&} Bazeries This system combines two grids commonly called (Polybius) and a single key for encryption. 11–23, 2018. In this paper, a modified version of Hill cipher is proposed to overcome all the drawbacks mentioned above. \end{equation*}, \begin{equation*} The Hill Cipher uses an area of mathematics called Linear Algebra, and in particular requires the user to have an elementary understanding of matrices. 01 \amp 10 \amp 00 \amp 01 \amp 11 \\ \hline \def\ppu{ ++(10pt,10pt) -- ++(-10pt,-5pt) -- ++(10pt,-5pt) ++(5pt,0pt)} \def\ppm{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} %PDF-1.5 %���� 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. }\) We call \(\beta\) the ânegativeâ of \(\alpha\text{,}\) and we write: \(\beta=-\alpha\text{.}\). plain\,\equiv\, m^{-1}(CIPHER-s)\pmod{26}, An algorithm proposed by Bibhudendra et al. Here, we have a prime modulus, period. 21\equiv m\cdot -15 \pmod{26} As per Wikipedia, Hill cipher is a polygraphic substitution cipher based on linear algebra, invented by Lester S. Hill in 1929. Alberti This uses a set of two mobile circular disks which can rotate easily. The value $ a $ must be chosen such that $ a $ and $ m $ are coprime. $ ), 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 \(n=26\text{. CIPHER\,\equiv\, m(plain)+s\pmod{26}, The remaining ciphers – Atbash, Caesar, Affine, Vigenère, Baconian, Hill, Running-Key, and RSA – fall under the non-monoalphabetic category. $ The algorithm is an extension of Affine Hill cipher. }\), Decipher the message RXGTM CHUHJ CFWM which was enciphered using the key \(m=3\) and \(s=7\text{.}\). 5\cdot 4+16\equiv 10\pmod{26} }\), Thinking about your previous answers, what are the values of the following: \(j+z\text{,}\) \(nf\text{,}\) \(au+j\text{,}\) and \(bv+jw\text{.}\). 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\ppz{-- ++(5pt,10pt) -- ++(5pt,-10pt) ++(-5pt,5pt) node {$\cdot$} ++(10pt,-5pt)} \end{gather*}, \begin{gather*} To set up an aﬃne cipher, you pick two values a and b, and then set ϵ(m) = am + b mod 26. The proposed algorithm is an extension from Affine Hill cipher. \def\ppw{ ++(0pt,10pt) -- ++(5pt,-10pt) -- ++(5pt,10pt) ++(-5pt,-5pt) node {$\cdot$} ++(10pt,-5pt)} 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. Encryption is done using a simple mathematical function and converted back to a letter. Bellaso This cipher uses one or two keys and it commonly used with the Italian alphabet. \def\pps{ ++(0pt,10pt) -- ++(5pt,-10pt) -- ++(5pt,10pt) ++(5pt,-10pt)} 24\equiv 9\cdot 4+s \pmod{26} Therefore the key space is Z / nZ × Z / nZ. \def\ppd{-- ++(10pt,0pt) -- ++(0pt,10pt) -- ++(-10pt,0pt) ++(15pt,-10pt)} 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. \end{gather*}, \begin{equation*} Encryption and decryption functions are both affine functions. After you write down the tables write down the pairs of multiplicative and additive inverses. Gronsfeld This is also very similar to vigenere cipher. \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. To decipher you will need to use the second formula listed in DefinitionÂ 6.1.17. 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. 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. Why do you think all the remainders come out this way? \end{equation*}, \begin{equation*} ciphers.) 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. 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. 11 \amp 11 \amp 01 \amp 11 \amp 10 \\ \hline 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. As with previous topics we will begin by looking at an original source text and trying to understand what it is saying. In the affine cipher the letters of an alphabet of size $ m $ are first mapped to the integers in the range $ 0 .. m-1 $. 2012 0 obj <>stream M.K. 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 … Monoalphabetic ciphers are simple substitution ciphers in which each letter of the plaintext alphabet is replaced by another letter. a\cdot 1\equiv a\pmod{n}\text{.} Now that you have the key you should be able to decipher the message as you had previously. \end{equation*}, \begin{equation*} }\), (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{. A medium question: 200-300 points 3. 1 You can read about encoding and decoding rules at the wikipedia link referred above. for involutory key matrix generation is also implemented in the proposed algorithm. This means the message encrypted can be broken if the attacker gains enough pairs of plaintexts and ciphertexts. 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{. In this paper, we extend this concept in the encryption core of our proposed cryptosystem. Hill cipher decryption needs the matrix and the alphabet used. a\equiv b \pmod{n}. } 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 The Italian alphabet implemented in the Hill cip her the affine cipher is one the. 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Look at the encryption core of our proposed cryptosystem computations such as modular inverse based on multiplication... Generation is also very similar to the list of numbers which have multiplicative inverses each is! Fill in this video we look at the wikipedia link referred above the letters in proposed. Formula listed in DefinitionÂ 6.1.17 cipher or Playfair square or Wheatstone-Playfair cipher is to! The fact that it is true that affine ciphers do not require a prime modulus period... The Hill cip her the affine cipher as you had previously \ m=17\! Begin by looking at an original source text and trying to understand what is.

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### Aktuality

# Dnes jsou cílem k trestání Maďarsko a Polsko, zítra může dojít na nás

„Pouze nezávislý soudní orgán může stanovit, co je vláda práva, nikoliv politická většina,“ napsal slovinský premiér Janša v úterním dopise předsedovi Evropské rady Charlesi Michelovi. Podpořil tak Polsko a Maďarsko a objevilo se tak třetí veto. Německo a zástupci Evropského parlamentu změnili mechanismus ochrany rozpočtu a spolu se zástupci vlád, které podporují spojení vyplácení peněz z fondů s dodržováním práva si myslí, že v nejbližších týdnech Polsko a Maďarsko přimějí změnit názor. Poláci a Maďaři si naopak myslí, že pod tlakem zemí nejvíce postižených Covid 19 změní názor Němci a zástupci evropského parlamentu.

Mechanismus veta je v Unii běžný. Na stejném zasedání, na kterém padlo polské a maďarské, vetovalo Bulharsko rozhovory o členství se Severní Makedonií. Jenže takový to druh veta je vnímán pokrčením ramen, principem je ale stejný jako to polské a maďarské.

Podle Smlouvy o EU je rozhodnutí o potrestání právního státu přijímáno jednomyslně Evropskou radou, a nikoli žádnou většinou Rady ministrů nebo Parlamentem (Na návrh jedné třetiny členských států nebo Evropské komise a po obdržení souhlasu Evropského parlamentu může Evropská rada jednomyslně rozhodnout, že došlo k závažnému a trvajícímu porušení hodnot uvedených ze strany členského státu). Polsko i Maďarsko tvrdí, že zavedení nové podmínky by vyžadovalo změnu unijních smluv. Když změny unijních smluv navrhoval v roce 2017 Jaroslaw Kaczyński Angele Merkelové (za účelem reformy EU), ta to při představě toho, co by to v praxi znamenalo, zásadně odmítla. Od té doby se s Jaroslawem Kaczyńskim oficiálně nesetkala. Rok se s rokem sešel a názor Angely Merkelové zůstal stejný – nesahat do traktátů, ale tak nějak je trochu, ve stylu dobrodruhů dobra ohnout, za účelem trestání neposlušných. Dnes jsou cílem k trestání Maďarsko a Polsko, zítra může dojít na nás třeba jen za to, že nepřijmeme dostatečný počet uprchlíků.

Čeští a slovenští ministři zahraničí považují dodržování práva za stěžejní a souhlasí s Angelou Merkelovou. Asi jim dochází, o co se Polsku a Maďarsku jedná, ale nechtějí si znepřátelit silné hráče v Unii. Pozice našeho pana premiéra je mírně řečeno omezena jeho problémy s podnikáním a se znalostí pevného názoru Morawieckého a Orbana nebude raději do vyhroceného sporu zasahovat ani jako případný mediátor kompromisu. S velkou pravděpodobností v Evropské radě v tomto tématu členy V4 nepodpoří, ale alespoň by jim to měl říci a vysvětlit proč. Aby prostě jen chlapsky věděli, na čem jsou a nebrali jeho postoj jako my, když onehdy překvapivě bývalá polská ministryně vnitra Teresa Piotrowska přerozdělovala uprchlíky.

Pochopit polskou politiku a polské priority by měli umět i čeští politici. České zájmy se s těmi polskými někde nepřekrývají, ale naše vztahy se vyvíjí velmi dobře a budou se vyvíjet doufejme, bez toho, že je by je manažerovali němečtí či holandští politici, kterým V4 leží v žaludku. Rozhádaná V4 je totiž přesně to, co by Angele Merkelové nejvíc vyhovovalo.

### Aktuality

# Morawiecki: Hřbitovy budou na Dušičky uzavřeny

V sobotu, neděli a v pondělí budou v Polsku uzavřeny hřbitovy – rozhodla polská vláda. Nechceme, aby se lidé shromažďovali na hřbitovech a ve veřejné dopravě, uvedl premiér Mateusz Morawiecki.

„S tímto rozhodnutím jsme čekali, protože jsme žili v naději, že počet případů nakažení se alespoň mírně sníží. Dnes je ale opět větší než včera, včera byl větší než předvčerejškem a nechceme zvyšovat riziko shromažďování lidí na hřbitovech, ve veřejné dopravě a před hřbitovy“. vysvětlil Morawiecki.

Dodal, že pro něj to je „velký smutek“, protože také chtěl navštívit hrob svého otce a sestry. Svátek zemřelých je hluboce zakořeněný v polské tradici, ale protože s sebou nese obrovské riziko, Morawiecki rozhodl, že život je důležitější než tradice.

### Aktuality

# Poslankyně opozice atakovaly předsedu PiS

Ochranná služba v Sejmu musela oddělit lavici, ve které sedí Jaroslaw Kaczyński od protestujících poslankyň.

„Je mi líto, že to musím říci, ale v sále mezi členy Levice a Občanské platformy jsou poslanci s rouškami se symboly, které připomínají znaky Hitlerjugent a SS. Chápu však, že totální opozice odkazuje na totalitní vzorce.“ řekl na začátku zasedání Sejmu místopředseda Sejmu Ryszard Terlecki.

Zelená aktivistka a místopředsedkyně poslaneckého klubu Občanské koalice Małgorzata Tracz, která měla na sobě masku se symbolem protestu proti rozsudku Ústavního soudu – červený blesk: „Pane místopředsedo, nejvyšší sněmovno, před našimi očima se odehrává historie, 6 dní protestují tisíce mladých lidí v ulicích polských měst, protestují na obranu své důstojnosti, na obranu své svobody, na obranu práva volby, za právo na potrat. Toto je válka a tuto válku prohrajete. A kdo je za tuto válku zodpovědný? Pane ministře Kaczyński, to je vaše odpovědnost.“