RSA is a first successful public key cryptographic algorithm.It is also known as an asymmetric cryptographic algorithm because two different keys are used for encryption and decryption. What is the max integer that can be encrypted? Example 1 for RSA Algorithm • Let p = 13 and q = 19. Is there any changes in the answers, if we swap the values of p and q? Calculate n = p × × q. For this example we can use p = 5 & q = 7. Answer: n = p * q = 7 * 11 = 77 . Example. Let p = 7, q = 11, e = 13, and M = 5 (M: message). RSA algorithm is an asymmetric cryptography algorithm which means, there should be two keys involve while communicating, i.e., public key and private key. RSA Key Construction: Example Select two large primes: p, q, p ≠q p = 17, q = 11 n = p×q = 17×11 = 187 Calculate = (p-1)(q-1) = 16x10 = 160 Select e, such that gcd( , e) = 1; 0 < e < say, e = 7 Calculate d such that de mod = 1 Use Euclid’s algorithm to find d=e-1mod 160k+1 = 161, 321, 481, 641 The math needed to find the private exponent d given p q and e without any fancy notation would be as follows: (a) Describe how to generate the pair of public key and private key in RSA algorithm? 4.Description of Algorithm: 1.Most widely accepted and implemented general purpose approach to public key encryption developed by Rivest-Shamir and Adleman (RSA) at MIT university. This implies that n = p∊×q = 77 and f(n) = (p − 1)(q − 1) = 60. Computing the public and private key Once you have chosen k, computing d is easy by using the Euclidean Algorithm, an algorithm for integer division. Alice choose e=17, a relative prime to 60 private key is d=53 where e*d mod (n) =1; 17*53 mod 60 = 1 If we represent 07 as and 25 as Z, 26 as blank, then HELLO WORLD will be … Show details of the following. Practically, these values are very high). 7 Answer: n = p * q = 7 * 11 = 77 . &. In this article, we will discuss about RSA Algorithm. i.e n<2. Find appropriate exponents d and e. Assignment 02 (cont.) Asymmetric Encryption Algorithms- The famous asymmetric encryption algorithms are- RSA Algorithm; Diffie-Hellman Key Exchange . Find the multiplicative inverse of 45 mod 238. ... RSA Example Key Setup 1 Select primes p 17 q 11 2 Compute n pq 17 x 11187 3. RSA Example Key Setup 1 Select primes p 17 q 11 2 Compute n pq 17 x 11187 3 from IS 493 at King Saud University. Generating the public key. RSA algorithm is asymmetric cryptography algorithm. Public key cryptography: The RSA algorithm After seeing several examples of \classical" cryptography, where the encoding procedure has to be kept secret (because otherwise it would be easy to design the decryption procedure), we turn to more modern methods, in which one can make the encryption procedure public, Choose p = 3 and q = 11 ; Compute n = p * q = 3 * 11 = 33 ; Compute φ(n) = (p - 1) * (q - 1) = 2 * 10 = 20 ; Choose e such that 1 ; e φ(n) and e and φ (n) are coprime. Let p=7 and q = 11. n=77 and (n) =60. RSA is an asymmetric cryptographic algorithm which is used for encryption purposes so that only the required sources should know the text and no third party should be allowed to decrypt the text as it is encrypted. Let e = 7 5) Compute a value for d such that (d e) % p(n) =1. RSA Algorithm Example . RSA is named after Rivest, Shamir and Adleman the three inventors of RSA algorithm. The RSA Encryption Scheme is often used to encrypt and then decrypt electronic communications. The term RSA is an acronym for Rivest-Shamir-Adleman who brought out the algorithm in 1977. Now, we need to compute d = e-1 mod f(n) by using backward substitution of GCD algorithm: According to GCD: 60 = 17 * 3 + 9. Asymmetric actually means that it works on two different keys i.e. RSA algorithm is asymmetric cryptography algorithm. They decided to use the public key cryptology algorithm RSA. With the above background, we have enough tools to describe RSA and show how it works. RSA: Confidentiality Example Encrypted using Alices Public key. The RSA Encryption Scheme Suppose Alice wants her friends to encrypt email messages before sending them to her. Public Key and Private Key. l a� � � � � � x x $$If a$gd- z kdX $$If �l � �0 ��� T � �      w n n n n n �h^�hgdPY{ Consider the following textbook RSA example. For simplicity I choose two small primes for p and q. p=3 q=11 n=33 Φ(n)=20 Now we need to find the public key e, which has to be coprime with Φ(n). An example of asymmetric cryptography : Learn about RSA algorithm in Java with program example. This example uses small integers because it is for understanding, it is for our study. A valid choice for the public exponent is e = 13. Putting the message digest algorithm at the beginning of the message enables the recipient to compute the message digest on the fly while reading the message. In an RSA cryptosystem, p = 7 and q = 11. Problem 5 (Chapter 8 , problem 8 - 6 points) - Consider RSA with p=5 and q=11. In our examples: Alice chooses two prime numbers p and q. 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Q: 9.2 Perform encryption and decryption using the RSA algorithm, as in Figure 9.6, for the following: 1. p = 3; q = 11, e = 7; M = 5 2. p Q_9.2 Perform encryption and decryption using the RSA algorithm, as in Figure 9.6 for the following: p=3; q=11; e=7; M=5 Answer: n = p * q = 3 * 11 = 33 f(n) = (p-1) * (q-1) = 2 * 10 = 20 Now, we need to compute d = e-1 mod f(n) by using backward substitution of GCD algorithm: According to GCD: 20 = 7 * 2 + 6 7 = 6 * 1 + 1 6 = 1 * 6 + 0 Therefore, we have: 1 = 7 � 6 = 7 � (20 � 7 * 2) = 7 � 20 + 7 * 2 = -20 + 7 * 3 Hence, we get d = e-1 mod f(n) = e-1 mod 20 = 3 mod 30 = 3 So, the public key is {7, 33} and the private key is {3, 33}, RSA encryption and decryption is following: SHAPE \* MERGEFORMAT p=5; q=11; e=3; M=9 Answer: n = p * q = 5 * 11 = 55 f(n) = (p-1) * (q-1) = 4 * 10 = 40 Now, we need to compute d = e-1 mod f(n) by using backward substitution of GCD algorithm: According to GCD: 40 = 3 * 13 + 1 13 = 1 * 13 + 0 Therefore, we have: 1 = 40 � 3 * 13 Hence, we get d = e-1 mod f(n) = e-1 mod 40 = -13 mod 40 = (27 � 40) mod 40 = 27 So, the public key is {3, 55} and the private key is {27, 55}, RSA encryption and decryption is following: SHAPE \* MERGEFORMAT p=7; q=11; e=17; M=8 Answer: n = p * q = 7 * 11 = 77 f(n) = (p-1) * (q-1) = 6 * 10 = 60 Now, we need to compute d = e-1 mod f(n) by using backward substitution of GCD algorithm: According to GCD: 60 = 17 * 3 + 9 17 = 9 * 1 + 8 9 = 8 * 1 + 1 8 = 1 * 8 + 0 Therefore, we have: 1 = 9 � 8 = 9 � (17 � 9) = 9 � (17 � (60 � 17 * 3)) = 60 � 17*3 � (17 � 60 + 17*3) = 60 � 17 *3 + 60 � 17*4 = 60*2 � 17*7 Hence, we get d = e-1 mod f(n) = e-1 mod 60 = -7 mod 60 = (53-60) mod 60 = 53 So, the public key is {17, 77} and the private key is {53, 77}, RSA encryption and decryption is following: SHAPE \* MERGEFORMAT p=11; q=13; e=11; M=7 Answer: n = p * q = 11 * 13 = 143 f(n) = (p-1) * (q-1) = 10 * 12 = 120 Now, we need to compute d = e-1 mod f(n) by using backward substitution of GCD algorithm: According to GCD: 120 = 11 * 10 + 10 11 = 10 * 1 + 1 10 = 1 * 10 + 0 Therefore, we have: 1 = 11 � 10 = 11 � (120 � 11 * 10) = 11 � 120 + 11 * 10 = -120 + 11 * 11 Hence, we get d = e-1 mod f(n) = e-1 mod 120 = 11 mod 120 = 11 So, the public key is {11, 143} and the private key is {11, 143}, RSA encryption and decryption is following: SHAPE \* MERGEFORMAT p=17; q=31; e=7; M=2 n = p * q = 17 * 31 = 527 f(n) = (p-1) * (q-1) = 16 * 30 = 480 Now, we need to compute d = e-1 mod f(n) by using backward substitution of GCD algorithm: According to GCD: 480 = 7 * 68 + 4 7 = 4 * 1 + 3 4 = 3 * 1 + 1 3 = 1 * 3 + 0 Therefore, we have: 1 = 4 � 3 = 4 � (7 � 4) = 4 � (7 � (480 � 7*68)) = 4 � (7 � 480 + 7*68) = 480 � 7*68 � 7 + 480 � 7*68 = 480*2 � 7*137 Hence, we get d = e-1 mod f(n) = e-1 mod 480 = -137 mod 480 = (343 � 480) mod 480 =343 So, the public key is {7, 527} and the private key is {343, 527}, RSA encryption and decryption is following: SHAPE \* MERGEFORMAT PR= (3, 33) Plaintext 5 ciphertext 14 PU= (7, 33) Plaintext 5 5 Decryption 143 Mod 33 = 5 Encryption 57 Mod 33= 14 93 Mod 55= 14 Encryption 1427 Mod 55 = 9 Decryption Plaintext 9 5 PU= (3, 55) ciphertext 14 Plaintext 9 PR= (27, 55) 817 Mod 77= 57 Encryption 5753 Mod 77 = 8 Decryption Plaintext 8 5 PU= (17, 77) ciphertext 57 Plaintext 8 PR= (53, 77) 711 Mod 143 = 106 Encryption 10611 Mod 143 = 8 Decryption Plaintext 7 5 PU= (11, 143) ciphertext 106 Plaintext 7 PR= (11, 143) 27 Mod 527 = 128 Encryption 128343 Mod 527 = 2 Decryption Plaintext 2 5 PU= (7, 527) ciphertext 128 Plaintext 2 PR= (343, 527) � � � � � � � � � � � � ! " An example of generating RSA Key pair is given below. or this This makes e “co-prime” to t. 13. Show all work. 120-126, Feb1978 • Security relies on the difficulty of factoring large composite numbers In our example, Alice . Sample of RSA Algorithm. λ(701,111) = 349,716. p = 5 & q = 7. l a� � � � � � x x $$If a$gd- z kd� $$If �l � �0 ��� T � RSA Algorithm • Invented in 1978 by Ron Rivest, AdiShamir and Leonard Adleman – Published as R. L. Rivest, A. Shamir, L. Adleman, "On Digital Signatures and Public Key Cryptosystems", Communications of the ACM, vol. t ��0 � � � � � � � 6� � � � �� � �� � �� � �4� 4� • RSA-640 bits, Factored Nov. 2 2005 • RSA-200 (663 bits) factored in May 2005 • RSA-768 has 232 decimal digits and was factored on December 12, 2009, latest. RSA Algorithm Example . An example of asymmetric cryptography : We randomly pick two prime numbers, p = 7 and q = 11. l a� � � � " W X � � � � Hardware fault-base attack - this involves inducing hardware faults in the processor that is generating digital signature. t ��0 � � � � � � � 6� � � � �� � �� � �� � �4� 4� To achieve this goal Sr2Jr organized the textbook’s question and answers. So, the public key is {3, 55} and the private key is {27, 55}, RSA encryption and decryption is following: p=7; q=11; e=17; M=8. Example-1: Step-1: Choose two prime number and Lets take and ; Step-2: Compute the value of and It is given as, Example: \(\phi(7) = \left|\{1,2,3,4,5,6\}\right| = 6\) 2.. RSA . ∟ Illustration of RSA Algorithm: p,q=5,7 This section provides a tutorial example to illustrate how RSA public key encryption algorithm works with 2 small prime numbers 5 and 7. The question and answers posted will be available free of cost to all. RSA is actually a set of two algorithms: Key Generation: A key generation algorithm. 1.Most widely accepted and implemented general purpose approach to public key encryption developed by Rivest-Shamir and Adleman (RSA) at MIT university. The Extended Euclidean Algorithm takes p, q, and e as input and gives d as output. For simplicity I choose two small primes for p and q. p=3 q=11 n=33 Φ(n)=20 Now we need to find the public key e, which has to be coprime with Φ(n). 5. With the above background, we have enough tools to describe RSA and show how it works. 18. the encrypted version to recover the original plaintext message. RSA algorithm in the SSL. RSA Algorithm- Let-Public key of the receiver = (e , n) Private key of the receiver = (d , n) Then, RSA Algorithm works in the following steps- Step-01: At sender side, Let be p = 7, q = 11 and e = 3. As the name describes that the Public Key is given to everyone and Private key is kept private. Illustration of the RSA algorithm Clark U. For this example we can use. f(n) = (p-1) * (q-1) = 6 * 10 = 60. � 12.2 The Rivest-Shamir-Adleman (RSA) Algorithm for 8 Public-Key Cryptography — The Basic Idea 12.2.1 The RSA Algorithm — Putting to Use the Basic Idea 12 12.2.2 How to Choose the Modulus for the RSA Algorithm 14 12.2.3 Proof of the RSA Algorithm 17 12.3 Computational Steps for Key Generation in RSA 21 The RSA Encryption Scheme is often used to encrypt and then decrypt electronic communications. f(n) = (p-1) * (q-1) = 6 * 10 = 60. Let e = 7 Compute a value for d such that (d * e) % φ(n) = 1. One solution is d = 3 [(3 * … 4) A worked example of RSA public key encryption Let’s suppose that Alice and Bob want to communicate, using RSA technology (It’s always For example, it is easy to check that 31 and 37 multiply to 1147, but trying to find the factors of 1147 is a much longer process. ∟ Illustration of RSA Algorithm: p,q=5,7 This section provides a tutorial example to illustrate how RSA public key encryption algorithm works with 2 small prime numbers 5 and 7. very big number. 5. Give a general algorithm for calculating d and run such algorithm with the above inputs. 5 Using RSA, choose p = 3 and q = 11, and encode the word “dog” by Then n = p * q = 5 * 7 = 35. � To better understand how RSA works, let us consider an example involving small numbers. t ��0 � � � � � � � 6� � � � �� � �� � �� � �4� 4� a. Now that we have Carmichael’s totient of our prime numbers, it’s time to figure out our public key. The algorithm was introduced in the year 1978. -Sr2Jr. RSA is a first successful public key cryptographic algorithm.It is also known as an asymmetric cryptographic algorithm because two different keys are used for encryption and decryption. • Solution: • The value of n = p*q = 13*19 = 247 • (p-1)*(q-1) = 12*18 = 216 • Choose the encryption key e = 11, which is relatively prime to 216 = (p-1)*(q-1). l a� $$If a$gd- gd- $a$gd- � � �� � � � � � x x $$If a$gd- z kdd $$If �l � �0 ��� T � l a� � � � � � x x $$If a$gd- z kd, $$If �l � �0 ��� T � Operation of RSA Algorithm The RSA algorithm involves three operations: Ø Key generation Ø Encryption Ø Decryption. Calculate F (n): F (n): = (p-1)(q-1) = 4 * 6 = 24 Choose e & d: d & n must be relatively prime (i.e., gcd(d,n) … 3. Apply the decryption algorithm to the encrypted version to recover the original plaintext message. 4.Description of Algorithm: p =17, q = 11 n = 187, e= 7 & d = 23 After sufring on internet i found this command to generate the public,private key pair : ... it already has an example for constructing an RSA key. RSA is named after Rivest, Shamir and Adleman the three inventors of RSA algorithm. t ��0 � � � � � � � 6� � � � �� � �� � �� � �4� 4� � As the name describes that the Public Key is given to everyone and Private key is kept private. CIS341 . Encrypt m= 3: EA(m) meA 37 42 (mod 143) c Eli Biham - May 3, 2005 389 Tutorial on Public Key Cryptography { RSA (14) Why is this an acceptable choice for e? 2.RSA scheme is block cipher in which the plaintext and ciphertext are integers between 0 and n-1 for same n. 3.Typical size of n is 1024 bits. Calculates m = (p 1)(q 1): Chooses numbers e and d so that ed has a remainder of 1 when divided by m. Publishes her public key (n;e). I have doubts about this question Consider the following textbook RSA example. Choose n: Start with two prime numbers, p and q. What is the running time of your algorithm as a function of n? I want to know about the explanation of RSA, here is the example . General Alice’s Setup: Chooses two prime numbers. Repeat part (a) but now encrypt “dog” as one message m. Take e=9, since 9 and 20 have no common factors  and d=29, since 9.29-1(that is, e.d-1) is exactly divisible by 20. Sr2Jr is community based and need your support to fill the question and answers. The approved answer by Thilo is incorrect as it uses Euler's totient function instead of Carmichael's totient function to find d.While the original method of RSA key generation uses Euler's function, d is typically derived using Carmichael's function instead for reasons I won't get into. 17 = 9 * 1 + 8. It is public key cryptography as one of the keys involved is made public. Step two, get n where n = pq: n = 7 * 11: n = 77: Step three, get "phe" where phe(n) = (p - 1)(q - 1) phe(77) = (7 - 1)(11 - 1) phe(77) = 60: Step four, select e such that e is relatively prime to phe(n); gcd(phe(n), e) = 1 where 1 < e < phe(n) i.e n<2. Find the encryption and decryption keys. Let two primes be p = 7 and q = 13. Let be p = 7, q = 11 and e = 3. 2.RSA scheme is block cipher in which the plaintext and ciphertext are integers between 0 and n-1 for same n. 3.Typical size of n is 1024 bits. t ��0 � � � � � � � 6� � � � �� � �� � �� � �4� 4� Taking a Crack at Asymmetric Cryptosystems Part 1 (RSA) Take for example: p=3 q=5 n=15 t=8 e=7. RSA Algorithm- Let-Public key of the receiver = (e , n) Private key of the receiver = (d , n) Then, RSA Algorithm works in the following steps- Step-01: At sender side, s p = 7 and q = 13., Sample of RSA Algorithm. Putting the message digest algorithm at the beginning of the message enables the recipient to compute the message digest on the fly while reading the message. RSA algorithm is an asymmetric cryptographic algorithm as it creates 2 different keys for the purpose of encryption and decryption. RSA is an encryption algorithm, used to securely transmit messages over the internet. So, the encrypting the each letter “dog” by RSA encryption, e=9, n=33. RSA { Encryption/Decryption { Example The encryption algorithm E: Everybody can encrypt messages m(0 m (7, 33) 17 = 9 * 1 + 8. equal. Exercise 1.33 page 41 DPV! p=3, q=11, e=3, M=9 And can you also please help me perform the signature generation and verification using RSA algorithm with the following parameters (hash algorithm must not be considered)? 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