Introduction to cryptography

Transcription

1 Chapter 1 Introduction to cryptography 1.1 From caesar cipher to public key cryptography Cryptography: is the practical means for protecting information transmitted through public communications networks, such as those using telephone lines, microwaves, or satellites. Definition The information to be concealed is called plain text. Definition After transformation to a secret form, a message is called ciphertext. Definition The process of converting from plaintext to ciphertext is said to be encrypting ( or enciphering ). Definition The process of changing from ciphertext back to plaintext is called decrypting ( or deciphering ). Caesar cryptographic system: This cryptographic system depends in using dementary substitution cipher in which each letter of the alphabet is replaced by the letter that occurs three places down the alphabet, with the last three letters cycled back to the first three letters. If we write the ciphertext equivalent underneath the plaintext letter, the substitution alphabet for the caesar cipher is given by: plaintext: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z D E F G H I J K L M N O P Q R S T U V W X Y Z A B C 1

2 For example, the plaintext message to CAESER WAS GREAT (1) is transformed into the ciphertext FDHVDU ZDV JUHDW (2) We can describe the Caesar cipher by using congruences any plaintext is first expressed numerically by translating the characters of the text into digits by means of some correspondence such as the following: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z If p is the digital equivalent of alphabet and C is the digital equivalent of the corresponding ciphertext letter, then C p + 3(mod 26) The letters of the message in equation (1) are converted to their equivalents: using the congruence C p + 3(mod 26), this becomes the ciphertext To recover the plaintext, the procedure is simply reversed by means of the congruence p C 3 C + 23(mod 26) Poly alphabetic cipher: was published by the French cryptography Blaise de vigenure ( ). To implement this system, the communicating parties agree on an easily remembered word on phrase. with the standard alphabet numbered from A=00 to Z=25, the digital equivalent of the keyword is repeated as many as necessary beneath that of the plaintext message. The message is then enciphered by adding, modulo 26, each plaintext number to the one immediately beneath it. The process may be illustrated with the keyword READY, whose numerical version is Repetitions of the sequence are arranged below the numerical plaintext of the message ATTACK AT ONCE 2

3 to produce the array when the columns are added modulo 26, the plaintext message is encryted as or converted to letters, RXTDAB ET RLTI Notice that a given letter of plaintext is represented by different letters in ciphertext. The double T in the word ATTACK no longer appears as a double letter when ciphered, while the ciphertext letter R first corresponds to A and then to O in the original message. In general, any sequence of n letters with numerical equivalents b 1, b 2,..., b n (00 b i 25) will serve as the keyword. The plaintext message is expressed as successive blocks p 1 p 2 p n of n two-digit integers p i and then converted to ciphertext blocs C 1 C 2 C n by means of the congruences C i p i + b i (mod 26) 1 i n Decryption is carried out by using the relations p i C i b i (mod 26) 1 i n Note The continued repetition of the key word is called running key. Definition A modification of polyalphabetic cipher by using autokey: This method depends in the use of the plaintext message itself in constructing the encryption key. The idea is to start off the key word with a short seed ( generally a single letter ) followed by the plaintext, whose ending is truncated by the length of the seed. The auto key cipher enjoyed considerable popularity in the 16th and 17th centuries, since all it required of a legitimate pair of users was to remember the seed, which could easily be changed. Let us give a simple example of the method. 3

4 Example Assume that the message ONE IF BY DAWN is to be encrypted. Taking the letter k as the seed, the key word becomes KONE IF BY DAW When both the plaintext and keyword are converted to numerical form, we obtain the array plaintext keyword Adding the integers in matching positions modulo 26 yields the ciphertext or, changing back to letters YBR MN GZ BDWJ. Decipherment is achieved by returning to the numerical form of both the plaintext and its ciphertext. suppose that the plaintext has digital equivalents p 1 p 2 p n and the ciphertext C 1 C 2 C n If S indicates the seed, then the first plaintext number is p 1 = C 1 S = = 14(mod 26) Thus, the deciphering transformation becomes This recovers, for example, the integers p k C k p k 1 (mod 26) 2 k n. p = 13 13(mod 26) p 3 C 3 p 2 = 17 13(mod 26) 4(mod 26) where, to maintain the two-digit format, the 4 written on. Hill s cipher: This method if ciphering depends on dividing the plaintext message into blocks of n letters ( possibly filling out the last block by adding dummy letters such as X s ) and then to encrypt block by block using a system of n linear congruences in n variables. In its simplest form, when n = 2, the procedure takes two successive letters and transforms their numerical equivalents p 1 p 2 into a block C 1 C 2 of ciphertext numbers via the pair of congruences C 1 ap 1 + bp 2 (mod 26) C 2 cp 1 + dp 2 (mod 26) To permit decipherment, the four coefficients a, b, c, d must be selected so that the g.c.d(ad bc, 26) = 1. 4

5 Example Let us use the congruences C 1 2p 1 + 3p 2 (mod 26) C 2 5p 1 + 8p 2 (mod 26) g.c.d( , 26) = (1, 26) = 1 To encrypt the message BUY NOW. The first block BU of two letters is numerically equivalent to 01 20, This is replaced by 2(01) + 3(20) 62 10(mod26) 5(01) + 8(20) (mod26) Continuing two letters at a time, we find that the completed ciphertext is which can be expressed alphabetically as KJJ QQM. Decipherment requires solving the original system of congruences for p 1 and p 2 in terms of C 1 and C 2. If follows from the proof Th 4.9 that the plaintext block p 1 p 2 can be recovered from the cipherext block by C 1 C 2 means of the congruences p 1 8C 1 3C 2 (mod 26) p 2 5c 1 + 2C 2 (mod 26) For the block of ciphertext, we calculate p 1 8(10) 3(09) 53 01(mod 26) p 2 5(10) + 2(09) 32 20(mod 26) which is the same as the letter pair BU. The remaining plaintext restored in a similar manner. Definition The RSA enciphering method: The Name is taken from the inventors Rivests Shamir and Adleman. The security of messages enciphered by the method depends on the difficulty of factoring a large number n. primes p and q. This number will be the product of two large To be safe from unouthorized decipherment using a large computer, p and q should each be a bout 200 decimal digits long. making n = pq a number of about 400 too digits. To illustrate the method let n = pq 5

6 we choose a positive integer k g.c.d(k, φ(n)) = 1 Now we publish the numbers n and k. This means allowing the computer anyone who wants to communicate with us to use n and k to encrypt and send a message. (k is called enciphering exponent). We keep the numbers p and q secret to those from whom we expect secret messages. The RSA enciphering process: Suppose some one want to sent us a message no one else can read. We convert the message to numerical digits form M, by using the following table A = 00., = 26 2 = 31 B = 01.. = 27. C = 02 x = 23? = 28.. y = 24 0 = 29 9 = 38. z = 25 1 = 30! = 39 and the space with 99. Example The Message (The brown fox is quite) is transformed into the numerical values to M = It is assumed that M < n or We broke M into blocks of digits M 1, M 2,..., M s where M i < n for i = 1,..., s To encript M we rise it to the kth power and then reducing the result modulo n, that is M k r(mod n), so M is encripted to r The receiver to decipher the transmitted information he first find an integer j (j is called recovery exponent) kj 1(mod φ(n)) (1) 6

7 Since g.c.d(k, φ(n)) = 1, so the congruent (1) has a unique solution j modulo φ(n). In fact j is a solution x of the equation kx + φ(n)y = 1 The value of j can only be calculated by someone who knows both k and φ(n) = (p 1)(q 1) and, hence, knows the prime factors p and q of n. Thus j is secure from an illegitimate third party whose knowledge is limited to the public key (n, k). To retrive M from v, we calculate r j (mod n) since kj = 1 + φ(n)t for some t Z then r j (M k ) j M 1+φ(n)t M(M φ(n)t ) M 1 t M(mod n) whenever g.c.d(m, n) = 1. That is raising the ciphertext number to the jth power and reducing it modulo n recovers the original plaintext number M. Note The sender know n and k while the receiver of the message know the prime factors of n, p and q and the key k to decrypt the message. Example Let p = 29, q = 53, n = = 1537 and φ(n) = = 1456 since g.c.d(47, φ(n)) = (47, 1456) = 1, so k = 47 can be chosen to be enciphering exponent. The recovery exponent, is the unique solution j satisfying the congruent kj = 1(mod φ(n)) 47j 1(mod1456) which is j = 31 To encrypt the message No way we translate each letter into its digital equivalent so the plaintext number is M = we divide the plaintext into blocks each block less than n = So the blocks of M are 131,499,220,024 The first block M 1 = 131 encripted as ciphertext into the number c 1 = M 47 1 = (mod 1537). Since the recovery exponent is j = 31 so to decript c 1 = 570 into M 1 we compute (mod 1537) so M 1 = 131 The total ciphertext of our message is