Number Theory & Cryptography Chapter 4 PDF

Number Theory & Cryptography Chapter 4 With Question/Answer Animations  Chapter Motivation Number theory is the part of mathematics devoted to the study of the integers & their properties. Key ideas in number theory include divisibility & the primality of int...

Number Theory & Cryptography Chapter 4 With Question/Answer Animations  Chapter Motivation Number theory is the part of mathematics devoted to the study of the integers & their properties. Key ideas in number theory include divisibility & the primality of integers. Representations of integers, including binary & hexadecimal representations, are part of number theory. Number theory has long been studied because of the beauty of its We’ll use many ideas developed in Chapter 1 about proof methods ideas, its accessibility, & its wealth of open questions. & proof strategy in our exploration of number theory. Mathematicians have long considered number theory to be pure & cryptography studied in Sections 4.5 & 4.6. mathematics, but it has important applications to computer science  Chapter Summary Divisibility & Modular Arithmetic Integer Representations & Algorithms Primes & Greatest Common Divisors Solving Congruences Applications of Congruences Cryptography Divisibility & Modular Arithmetic Section 4.1  Section Summary Division Division Algorithm Modular Arithmetic  Division Definition: If a & b are integers with a ≠ 0, then a divides b if there exists an integer c such that b = ac. When a divides b we say that a is a factor or divisor of b & that b is a multiple of a. The notation a | b denotes that a divides b. If a | b, then b/a is an integer. If a does not divide b, we write a ∤ b. e.g., Determine whether 3 | 7 & whether 3 | 12.  Properties of Divisibility Theorem 1: Let a, b, & c be integers, where a ≠0. i. If a | b & a | c, then a | (b + c); ii. If a | b, then a | bc for all integers c; iii. If a | b & b | c, then a | c. Proof: (i) Suppose a | b & a | c, then it follows that there are integers s & t with b b + c = as + at = a(s + t). Hence, a | (b + c) = as & c = at. Hence, (Exercises 3 & 4 ask for proofs of parts (ii) & (iii).) Corollary: If a, b, & c be integers, where a ≠0, such that a | b & a | c, then a | mb + nc whenever m & n are integers. Can you show how it follows easily from from (ii) & (i) of Theorem 1?  Division Algorithm When an integer is divided by a positive integer, there is a quotient & a remainder. This is traditionally called the “Division Algorithm,” but is really a theorem. are unique integers q & r, with 0 ≤ r < d, such that a =Definitions Division Algorithm: If a is an integer & d a positive integer, then there in Section 5.2). dq + r (proved of Functions div & mod d is called the divisor. a is called the dividend. q = a div d q is called the quotient. r = a mod d r is called the remainder. e.g., What are the quotient & remainder when 101 is divided by 11? Solution: The quotient when 101 is divided by 11 is 9 = 101 div 11, & the remainder is 2 = 101 mod 11. What are the quotient & remainder when −11 is divided by 3?  Congruence Relation Definition: If a & b are integers & m is a positive integer, then a is congruent to b modulo m if m divides a – b. The notation a ≡ b (mod m) says that a is congruent to b modulo m. We say that a ≡ b (mod m) is a congruence & that m is its modulus. Two integers are congruent mod m if & only if they have the same remainder when divided by m. If a is not congruent to b modulo m, we write a ≢ b (mod m) e.g., Determine whether 17 is congruent to 5 modulo 6 & whether 24 & 14 are congruent modulo 6. Solution: 17 ≡ 5 (mod 6) because 6 divides 17 − 5 = 12. 24 ≢ 14 (mod 6) since 24 − 14 = 10 is not divisible by 6.  More on Congruences Theorem 4: Let m be a positive integer. The integers a & b are congruent modulo m if & only if there is an integer k such that a = b + km. Proof: If a ≡ b (mod m), then (by the definition of congruence) m | a – b. Hence, there is an integer k such that a – b = km & equivalently a = b + km. | a – b & a ≡ b (mod m). Conversely, if there is an integer k such that a = b + km, then km = a – b. Hence, m  The Relationship between (mod m) & mod m Notations The use of “mod” in a ≡ b (mod m) & a mod m = b are different. a ≡ b (mod m) is a relation on the set of integers. In a mod m = b, the notation mod denotes a function. The relationship between these notations is made clear in this theorem. Theorem 3: Let a & b be integers, & let m be a positive integer. Then a ≡ b (mod m) if & only if a mod m = b mod m. (Proof in the exercises)  Congruences of Sums & Products Theorem 5: Let m be a positive integer. If a ≡ b (mod m) & c ≡ d (mod m), then a + c ≡ b + d (mod m) & ac ≡ bd (mod m) Proof: Because a ≡ b (mod m) & c ≡ d (mod m), by Theorem 4 there are integers s & t with b = a + sm & d = c + tm. Therefore, b + d = (a + sm) + (c + tm) = (a + c) + m(s + t) and b d = (a + sm) (c + tm) = ac + m(at + cs + stm). Hence, a + c ≡ b + d (mod m) & ac ≡ bd (mod m). e.g., Because 7 ≡ 2 (mod 5) & 11 ≡ 1 (mod 5) , it follows from Theorem 5 that 18 = 7 + 11 ≡ 2 + 1 = 3 (mod 5) 77 = 7 ∙ 11 ≡ 2 ∙ 1 = 2 (mod 5)  Algebraic Manipulation of Congruences Multiplying both sides of a valid congruence by an integer preserves If a ≡ b (mod m) holds then c∙a ≡ c∙b (mod m), where c is any integer, validity. holds by Theorem 5 with d = c. Adding an integer to both sides of a valid congruence preserves If a ≡ b (mod m) holds then c + a ≡ c + b (mod m), where c is any integer, validity. holds by Theorem 5 with d = c. Dividing a congruence by an integer does not always produce a valid e.g., The congruence 14≡ 8 (mod 6) holds. But dividing both sides by congruence. 2 does not produce a valid congruence since 14/2 = 7 & 8/2 = 4, but 7≢4 (mod 6). See Section 4.3 for conditions when division is ok.  Computing the mod m Function of Products & Sums We use the following corollary to Theorem 5 to compute the remainder of the product or sum of 2 integers when divided by m from the remainders when each is divided by m. Corollary: Let m be a positive integer & let a & b be integers. Then (a + b) (mod m) = ((a mod m) + (b mod m)) mod m and ab mod m = ((a mod m) (b mod m)) mod m. (proof in text)  Arithmetic Modulo m Definitions: Let Zm be the set of nonnegative integers less than m: {0,1, …., m−1} The operation +m is defined as a +m b = (a + b) mod m. This is addition modulo m. The operation ∙m is defined as a ∙m b = (a ∙ b) mod m. This is multiplication modulo m. Using these operations is said to be doing arithmetic modulo m. e.g., Find 7 +11 9 & 7 ∙11 9. Solution: Using the definitions above: 7 +11 9 = (7 + 9) mod 11 = 16 mod 11 = 5 7 ∙11 9 = (7 ∙ 9) mod 11 = 63 mod 11 = 8  Arithmetic Modulo m The operations +m & ∙m satisfy many of the same properties as ordinary addition & multiplication. Closure: If a & b belong to Zm , then a +m b & a ∙m b belong to Zm. Associativity: If a, b, & c belong to Zm , then (a +m b) +m c = a +m (b +m c) & (a ∙m b) ∙m c = a ∙m (b ∙m c). Commutativity: If a & b belong to Zm , then a +m b = b +m a & a ∙m b = b ∙m a. Identity elements: The elements 0 & 1 are identity elements for addition & multiplication modulo If a belongs to Zm , then a +m 0 = a & a ∙m 1 = a. m, respectively. Additive inverses: If a≠ 0 belongs to Zm , then m− a is the additive inverse of a modulo m & 0 is a +m (m− a ) = 0 & 0 +m 0 = 0 its own additive inverse. Distributivity: If a, b, & c belong to Zm , then a ∙m (b +m c) = (a ∙m b) +m (a ∙m c) & (a +m b) ∙m c = (a ∙m c) +m (b ∙m c). Exercises 42-44 ask for proofs of these properties. Multiplicatative inverses have not been included since they do not always exist. e.g., there is no multiplicative inverse of 2 modulo 6. (optional) Using the terminology of abstract algebra, Zm with +m is a commutative group & Zm with +m & ∙m is a commutative ring. Applications of Congruences Section 4.5  Section Summary Hashing Functions Pseudorandom Numbers Check Digits  Hashing Functions Definition: A hashing function h assigns memory location h(k) to the record that has k as its key. A common hashing function is h(k) = k mod m, where m is the number of memory locations. Because this hashing function is onto, all memory locations are possible. e.g., Let h(k) = k mod 111. This hashing function assigns the records of customers with social security numbers as keys to memory locations in the following manner: h(064212848) = 064212848 mod 111 = 14 h(037149212) = 037149212 mod 111 = 65 h(107405723) = 107405723 mod 111 = 14, but since location 14 is already occupied, the record is assigned to the next available position, which is 15. The hashing function is not one-to-one as there are many more possible keys than memory locations. When more than 1 record is assigned to the same location, we say a collision occurs. Here a collision has been resolved by assigning the record to the first free location. For collision resolution, we can use a linear probing function: h(k,i) = (h(k) + i) mod m, where i runs from 0 to m − 1. There are many other methods of handling with collisions. You may cover these in a later CS course.  Pseudorandom Numbers Randomly chosen numbers are needed for many purposes, including computer simulations. Pseudorandom numbers are not truly random since they are generated by systematic methods. The linear congruential method is 1 commonly used procedure for generating pseudorandom numbers. x0, with 2 ≤ a < m, 0 ≤ c < m, 0 ≤ x0 < m. Four integers are needed: the modulus m, the multiplier a, the increment c, & seed We generate a sequence of pseudorandom c) mod m.{xn}, with 0 ≤ xn < m for all n, xn+1 = (axn +numbers by successively using the recursively defined function (an example of a recursive definition, discussed in Section 5.3) If psudorandom numbers between 0 & 1 are needed, then the generated numbers are divided by the modulus, xn /m.  Pseudorandom Numbers e.g., Find the sequence of pseudorandom numbers generated by the linear congruential method with modulus m = 9, multiplier a = 7, increment c = 4, & seed x0 = 3. Solution: Compute the terms of the sequence by successively using the congruence xn+1 = (7xn + 4) mod 9, with x0 = 3. x1 = 7x0 + 4 mod 9 = 7∙3 + 4 mod 9 = 25 mod 9 = 7, x2 = 7x1 + 4 mod 9 = 7∙7 + 4 mod 9 = 53 mod 9 = 8, x3 = 7x2 + 4 mod 9 = 7∙8 + 4 mod 9 = 60 mod 9 = 6, x4 = 7x3 + 4 mod 9 = 7∙6 + 4 mod 9 = 46 mod 9 = 1, x5 = 7x4 + 4 mod 9 = 7∙1 + 4 mod 9 = 11 mod 9 = 2, x6 = 7x5 + 4 mod 9 = 7∙2 + 4 mod 9 = 18 mod 9 = 0, x7 = 7x6 + 4 mod 9 = 7∙0 + 4 mod 9 = 4 mod 9 = 4, x8 = 7x7 + 4 mod 9 = 7∙4 + 4 mod 9 = 32 mod 9 = 5, x9 = 7x8 + 4 mod 9 = 7∙5 + 4 mod 9 = 39 mod 9 = 3. The sequence generated is 3,7,8,6,1,2,0,4,5,3,7,8,6,1,2,0,4,5,3,… It repeats after generating 9 terms. Commonly, computers use a linear congruential generator with increment c = 0. This is called a pure multiplicative generator. Such a generator with modulus 231 − 1 & multiplier 75 = 16,807 generates 231 − 2 numbers before repeating.  Check Digits: UPCs A common method of detecting errors in strings of digits is to add an extra digit at the end, which is evaluated using a function. If the final digit is not correct, then the string is assumed not to be correct. these have 12 decimal digits, the last one being the check digit. The check digit is e.g., Retail products are identified by their Universal Product Codes (UPCs). Usually determined by the congruence: 3x1 + x2 + 3x3 + x4 + 3x5 + x6 + 3x7 + x8 + 3x9 + x10 + 3x11 + x12 ≡ 0 (mod 10). a. Suppose that the first 11 digits of the UPC are 79357343104. What is the check digit? b. Is 041331021641 a valid UPC? Solution: c. 3∙7 + 9 + 3∙3 + 5 + 3∙7 + 3 + 3∙4 + 3 + 3∙1 + 0 + 3∙4 + x12 ≡ 0 (mod 10) 21 + 9 + 9 + 5 + 21 + 3 + 12+ 3 + 3 + 0 + 12 + x12 ≡ 0 (mod 10) 98 + x12 ≡ 0 (mod 10) x12 ≡ 2 (mod 10) So, the check digit is 2. b. 3∙0 + 4 + 3∙1 + 3 + 3∙3 + 1 + 3∙0 + 2 + 3∙1 + 6 + 3∙4 + 1 ≡ 0 (mod 10) 0 + 4 + 3 + 3 + 9 + 1 + 0+ 2 + 3 + 6 + 12 + 1 = 44 ≡ 4 ≢ 0 (mod 10) Hence, 041331021641 is not a valid UPC.  Check Digits: ISBNs Books are identified by an International Standard Book Number (ISBN-10), a 10 digit code. The first 9 digits identify the language, the publisher, & the book. The tenth digit is a check digit, which is determined by the following congruence The validity of an ISBN-10 number can be evaluated with the equivalent a. Suppose that the first 9 digits of the ISBN-10 are 007288008. What is the check digit? b. Is 084930149X a valid ISBN10? Solution: X is used X10 ≡ 1∙0 + 2∙0 + 3∙7 + 4∙2 + 5∙8 + 6∙8 + 7∙ 0 + 8∙0 + 9∙8 (mod 11). digit 10. for the a. X10 ≡ 0 + 0 + 21 + 8 + 40 + 48 + 0 + 0 + 72 (mod 11). X10 ≡ 189 ≡ 2 (mod 11). Hence, X10 = 2. b. 1∙0 + 2∙8 + 3∙4 + 4∙9 + 5∙3 + 6∙0 + 7∙ 1 + 8∙4 + 9∙9 + 10∙10 = 0 + 16 + 12 + 36 + 15 + 0 + 7 + 32 + 81 + 100 = 299 ≡ 2 ≢ 0 (mod 11) Hence, 084930149X is not a valid ISBN-10. A single error is an error in 1 digit of an identification number & a transposition error is the accidental interchanging of 2 digits. Both of these kinds of errors can be detected by the check digit for ISBN-10. (see text for more details) Cryptography Section 4.6  Section Summary Classical Cryptography Cryptosystems Public Key Cryptography RSA Cryptosystem Crytographic Protocols  Caesar Cipher Julius Caesar created secret messages by shifting each letter 3 letters forward in the alphabet (sending the last 3 letters to the first 3 letters.) e.g., the letter B is replaced by E & the letter X is replaced by A. This process of making a message secret is an example of encryption. Here is how the encryption process works: Replace each letter by an integer from Z26, that is an integer from 0 to 25 representing 1 less than its position in The encryption function is f(p) = (p + 3) mod 26. It replaces each integer p in the set {0,1,2,…,25} by f(p) in the the alphabet. Replace each integer p by the letter with the position p + 1 in the alphabet. set {0,1,2,…,25}. e.g., Encrypt the message “MEET YOU IN THE PARK” using the Caesar cipher. Solution: 12 4 4 19 24 14 20 8 13 19 7 4 15 0 17 10. Now replace each of these numbers p by f(p) = (p + 3) mod 26. 15 7 7 22 1 17 23 11 16 22 10 7 18 3 20 13. Translating the numbers back to letters produces the encrypted message “PHHW BRX LQ WKH SDUN.” 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 0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5  Caesar Cipher To recover the original message, use f−1(p) = (p−3) mod 26. So, each letter in the coded message is shifted back 3 letters in the alphabet, with the first 3 letters sent to the last 3 letters. This process of recovering the original message from the encrypted message is called decryption. Letters can be shifted by an integer k, with 3 being just 1 possibility. The Caesar cipher is 1 of a family of ciphers called shift ciphers. The encryption function is f(p) = (p + k) mod 26 and the decryption function is f−1(p) = (p−k) mod 26 The integer k is called a key.  Shift Cipher Example 1: Encrypt the message “STOP GLOBAL WARMING” using the shift cipher with k = 11. Solution: Replace each letter with the corresponding element of Z26. 18 19 14 15 6 11 14 1 0 11 22 0 17 12 8 13 6. Apply the shift f(p) = (p + 11) mod 26, yielding 3 4 25 0 17 22 25 12 11 22 7 11 2 23 19 24 17. Translating the numbers back to letters produces the ciphertext “DEZA RWZMLW HLCXTYR.” 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 0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5  Shift Cipher Example 2: Decrypt the message “LEWLYPLUJL PZ H NYLHA 7. ALHJOLY” that was encrypted using the shift cipher with k = Solution: Replace each letter with the corresponding element of Z26. 11 4 22 11 24 15 11 20 9 11 15 25 7 13 24 11 7 0 0 11 7 9 14 11 24. Shift each of the numbers by −k = −7 modulo 26, yielding 4 23 15 4 17 8 4 13 2 4 8 18 0 6 17 4 0 19 19 4 0 2 7 4 17. Translating the numbers back to letters produces the decrypted message A B C“EXPERIENCE D E F G H IS I JA GREAT TEACHER.” K L M N O P Q R S T U V W X Y Z 0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5  Affine Ciphers Shift ciphers are a special case of affine ciphers which use functions of f(p) = (ap + b) mod 26, the form where a & b are integers, chosen so that f is a bijection. The function is a bijection if & only if gcd(a,26) = 1. e.g., What letter replaces the letter K when the function f(p) = (7p + 3) mod 26 is used for encryption. Solution: Since 10 represents K, f(10) = (7∙10 + 3) mod 26 =21, which is then replaced by V. To decrypt a message encrypted by a shift cipher, the congruence c ≡ ap + b (mod 26) needs to be solved for p. Subtract b from both sides to obtain c− b ≡ ap (mod 26). Multiply both sides by the inverse of a modulo 26, which exists since gcd(a,26) = 1.  Cryptanalysis of Affine Ciphers The process of recovering plaintext from ciphertext without knowledge both of the encryption method & the key is known as cryptanalysis or breaking codes. frequencies of letters. The nine most common letters in the English texts are E 13%, T 9%, A 8%, An important tool for cryptanalyzing ciphertext produced with a affine ciphers is the relative O 8%, I 7%, N 7%, S 7%, H 6%, & R 6%. To analyze ciphertext: Find the frequency of the letters in the ciphertext. Hypothesize that the most frequent letter is produced by encrypting E. If the value of the shift from E to the most frequent letter is k, shift the ciphertext by −k & see if it makes sense. If not, try T as a hypothesis & continue. e.g., We intercepted the message “ZNK KGXRE HOXJ MKZY ZNK CUXS” that we know was produced by a shift cipher. Let’s try to cryptanalyze. since this would then map E to K. Shifting the entire message by −6 gives us “THE EARLY BIRD Solution: The most common letter in the ciphertext is K. So perhaps the letters were shifted by 6 GETS THE WORM.”  Block Ciphers Ciphers that replace each letter of the alphabet by another letter are called character or monoalphabetic ciphers. They are vulnerable to cryptanalysis based on letter frequency. Block ciphers avoid this problem, by replacing blocks of letters with other blocks of letters. A simple type of block cipher is called the transposition cipher. The key is a permutation σ of the set {1,2,…,m}, where m is an integer, that is a one-to-one function from {1,2,…,m} to itself. To encrypt a message, split the letters into blocks of size m, adding additional letters to fill out the final block. We encrypt p1,p2,…,pm as c1,c2,…,cm = pσ(1),pσ(2),…,pσ(m). To decrypt the c1,c2,…,cm transpose the letters using the inverse permutation σ−1. e.g., Using the transposition cipher based on the permutation σ of the set {1,2,3,4} with σ(1) = 3, σ(2) = 1, σ(3) = 4, σ(4) = 2, a. Encrypt the plaintext PIRATE ATTACK b. Decrypt the ciphertext message SWUE TRAEOEHS, which was encryted using the same cipher. Solution: c. Split into 4 blocks PIRA TEAT TACK. Apply the permutation σ giving IAPR ETTA AKTC. σ−1 : σ −1(1) = 2, σ −1(2) = 4, σ −1(3) = 1, σ −1(4) = 3. Apply the permutation σ−1 giving USEW ATER HOSE. b. Split into words to obtain USE WATER HOSE.  Cryptosystems Definition: A cryptosystem is a five-tuple (P,C,K,E,D), where P is the set of plainntext strings, C is the set of ciphertext strings, K is the keyspace (set of all possible keys), E is the set of encription functions, and D is the set of decryption functions. The encryption function in E corresponding to the key k is denoted by Ek & the decription function in D that decrypts cipher text enrypted using Ek is denoted by Dk. Therefore: Dk(Ek(p)) = p, for all plaintext strings p. e.g., Describe the family of shift ciphers as a cryptosystem. Solution: Assume the messages are strings consisting of elements in Z26. P is the set of strings of elements in Z26, C is the set of strings of elements in Z26, K = Z26, E consists of functions of the form Ek (p) = (p + k) mod 26 , and D is the same as E where Dk (p) = (p − k) mod 26.  Public Key Cryptography All classical ciphers, including shift & affine ciphers, are private key cryptosystems. Knowing the encryption key allows one to quickly determine the decryption key. All parties who wish to communicate using a private key cryptosystem must share the key & keep it a secret. In public key cryptosystems, first invented in the 1970s, knowing how to encrypt a message does not help one to decrypt the message. Therefore, everyone can have a publicly known encryption key. The only key that needs to be kept secret is the decryption key.  The RSA Cryptosystem (Born 1950) Clifford Cocks A public key cryptosystem, now known as the RSA system was introduced in 1976 by 3 researchers at MIT. Leonard Adi Shamir (Born 1948) Ronald Rivest Adelman 1952) (Born 1945) (Born It is now known that the method was discovered earlier by Clifford Cocks, working secretly for the UK government. large (200 digits) primes p & q, & an exponent e that is relatively prime to (p−1)(q The public encryption key is (n,e), where n = pq (the modulus) is the product of 2 −1). The 2 large primes can be quickly found using probabilistic primality tests, discussed earlier. But n = pq, with approximately 400 digits, cannot be factored in a reasonable length of time.  RSA Encryption To encrypt a message using RSA using a key (n,e) : Use 00 for A, 01 for B, etc. i. Translate the plaintext message M into sequences of 2 digit integers representing the letters. Divide this string into equally sized blocks of 2N digits where 2N is the largest even number ii. Concatenate the 2 digit integers into strings of digits. 2525…25 with 2N digits that does not exceed n. iii. Each block (an integer) is encrypted using the function C = Me mod n. iv. The plaintext message M is now a sequence of integers m1,m2,…,mk. v. e.g., Encrypt the message STOP using the RSA cryptosystem with key(2537,13). 2537 = 43∙ 59, p = 43 & q = 59 are primes & gcd(e,(p−1)(q −1)) = gcd(13, 42∙ 58) = 1. Solution: Translate the letters in STOP to their numerical equivalents 18 19 14 15. Divide into blocks of 4 digits (because 2525 < 2537 < 252525) to obtain 1819 1415. Encrypt each block using the mapping C = M13 mod 2537. Since 181913 mod 2537 = 2081 & 141513 mod 2537 = 2182, the encrypted message is 2081 2182.  RSA Decryption modulo (p−1)(q −1) is needed. The inverse exists since gcd(e,(p−1)(q −1)) = To decrypt a RSA ciphertext message, the decryption key d, an inverse of e gcd(13, 42∙ 58) = 1. With the decryption key d, we can decrypt each block with the computation M = Cd mod p∙q. (see text for full derivation) RSA works as a public key system since the only known method of finding d is based on a factorization of n into primes. There is currently no known feasible method for factoring large numbers into primes. e.g., The message 0981 0461 is received. What is the decrypted message if it was encrypted using the RSA cipher from the previous example. Solution: The message was encrypted with n = 43∙ 59 & exponent 13. An inverse of 13 modulo 42∙ 58 = 2436 (exercise 2 in Section 4.4) is d = 937. To decrypt a block C, M = C937 mod 2537. Since 0981937 mod 2537 = 0704 & 0461937 mod 2537 = 1115, the decrypted message is 0704 1115. Translating back to English letters, the message is HELP. RSA Algorithm: Example 2 RSA Algorithm  Cryptographic Protocols: Key Exchange Cryptographic protocols are exchanges of messages carried out by two or more parties to achieve a particular security goal. Key exchange is a protocol by which two parties can exchange a secret key over an insecure channel without having any past shared secret information. Here the Diffe-Hellman key agreement protocol is described by example. i. Suppose that Alice & Bob want to share a common key. ii. Alice & Bob agree to use a prime p & a primitive root a of p. iii. Alice chooses a secret integer k1 & sends ak1 mod p to Bob. iv. Bob chooses a secret integer k2 & sends ak2 mod p to Alice. v. Alice computes (ak2)k1 mod p. vi. Bob computes (ak1)k2 mod p. At the end of the protocol, Alice & Bob have their shared key (ak2)k1 mod p = (ak1)k2 mod p. To find the secret information from the public information would require the adversary to find k1 & k2 from ak1 mod p & ak2 mod p respectively. This is an instance of the discrete logarithm problem, considered to be computationally infeasible when p & a are sufficiently large.  Cryptographic Protocols: Digital Signatures Adding a digital signature to a message is a way of ensuring the recipient that the message came from the purported sender. Suppose that Alice’s RSA public key is (n,e) & her private key is d. Alice encrypts a plain text message x using E(n,e) (x)= xe mod n. She decrypts a ciphertext message y using D(n,e) (y)= yd mod n. Alice wants to send a message M so that everyone who receives the message knows that it came from her. 1. She translates the message to numerical equivalents & splits into blocks, just as in RSA encryption. 2. She then applies her decryption function D(n,e) to the blocks & sends the results to all intended recipients. 3. The recipients apply Alice’s encryption function & the result is the original plain text since E(n,e) (D(n,e) (x))= x. Everyone who receives the message can then be certain that it came from Alice. Because, if a recipient uses another person’s public key, say E (n’e’), then the result will be different.  Cryptographic Protocols: Digital Signatures key(2537,13), 2537 = 43∙ 59, p = 43 & q = 59 are primes & gcd(e,(p−1)(q −1)) = gcd(13, 42∙ e.g., Suppose Alice’s RSA cryptosystem is the same as in the earlier example with 58) = 1. Her decryption key is d = 937. She wants to send the message “MEET AT NOON” or say “MESSAGE FROM ALICE” to her friends so that they can be certain that the message is from her. Solution: Alice translates the message into blocks of digits 1204 0419 0019 1314 1413. 1. She then applies her decryption transformation D(2537,13) (x)= x937 mod 2537 to each block. 1204937 mod 2537 = 817, 419937 mod 2537 = 555 , 19937 mod 2537 = 1310, 1314937 mod 2537 2. She finds (using her laptop, programming skills, & knowledge of discrete mathematics) that = 2173, & 1413937 mod 2537 = 1026. 3. She sends 0817 0555 1310 2173 1026. When one of her friends receive the message, they apply Alice’s encryption transformation E(2537,13) to each block. They then obtain the original message which they translate back to English letters and get the signature of Alice which is “MEET AT NOON” or say “MESSAGE FROM ALICE”. Using any other person’s public key will NOT give “MEET AT NOON” or say “MESSAGE FROM ALICE”, which is indicate that the message was not sent from Alice.