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, ... RSA Example Key Setup 1 Select primes p 17 q 11 2 Compute n pq 17 x 11187 3. Let be p = 7, q = 11 and e = 3. 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. a. Public Key and Private Key. RSA Algorithm Example 1) Choose p 3 and q 11 2) Compute n p*q =3* 11 = 33 3) Compute p(n) = (p - 1) * (q - 1) = 2 * 10 = 20 4) Choose e such that 1 < e (7, 33) Post the discussion to improve the above solution. As the name describes that the Public Key is given to everyone and Private key is kept private. 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. I have doubts about this question Consider the following textbook RSA example. As the name describes that the Public Key is given to everyone and Private key is kept private. The following table decryption algorithm to the encrypted version to recover the original plaintext message by using RSA: Consider the above part (a), now have to encrypt “dog” as one message m: So, the encrypted message “dog” as one message m is 402. 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. With the above background, we have enough tools to describe RSA and show how it works. 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. 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. 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. The Link Layer: Links,access Networks, And Lans, Computer Networking : A Top-down Approach. RSA is an encryption algorithm, used to securely transmit messages over the internet. 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. (For ease of understanding, the primes p & q taken here are small values. Viewed 537 times 0. Give an efficient algorithm to compute the least common multiple of two n-bit numbers x and y, that is, the smallest number divisible by both x and y. With the above background, we have enough tools to describe RSA and show how it works. (b) 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's review the RSA algorithm operation with an example, plugging in numbers. encrypting each letter separately. To demonstrate the RSA public key encryption algorithm, let's start it with 2 smaller prime numbers 5 and 7. t ��0 � � � � � � � 6� � � � �� � �� � �� � �4� 4� Can you please help me how to perform encryption and decryption using the RSA algorithm with the following parameters? 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. l a� � � � � � x x $$If a$gd- z kd, $$If �l � �0 ��� T � 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) (a) Using RSA, choose p = 3 and q = 11, and encode the word “dog” by encrypting each letter separately. ��ࡱ� > �� { } ���� z �������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� @ �� � bjbj5*5* .^ W@ W@ � � �� �� �� � � � � � " " " 6 �, �, �, �, \ �, � 6 �_ � �- ^ �- �- �- �- �W �W �W ^_ `_ `_ `_ `_ `_ `_ $ �` R #c V �_ " �W �V � �W �W �W �_ � � �- �- �( �_ ;[ ;[ ;[ �W � � �- " �- ^_ ;[ �W ^_ ;[ ;[ �] R � " ^ �- �- ��P:.� �, �W � �] �^ � �_ 0 �_ �] , yc 9X � yc ^ 6 6 � � � � yc " ^ � �W �W ;[ �W �W �W �W �W �_ �_ 6 6 d$ �, +[ 6 6 �, Assignment #3 My full name written in lower case is �yufei xu�. Exercise 1.33 page 41 DPV! �      w n n n n n �h^�hgdPY{ t ��0 � � � � � � � 6� � � � �� � �� � �� � �4� 4� This implies that n = p∊×q = 77 and f(n) = (p − 1)(q − 1) = 60. � Let e = 7 5) Compute a value for d such that (d e) % p(n) =1. 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 … Sr2Jr is community based and need your support to fill the question and answers. t ��0 � � � � � � � 6� � � � �� � �� � �� � �4� 4� Find the encryption and decryption keys. Choose your encryption key to be at least 10. Calculate ϕ ϕ (n) = (p - 1) (q - 1). An example of asymmetric cryptography : • Three most effective algorithms are – quadratic sieve – elliptic curve factoring algorithm – number field sieve 25 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, 641 120-126, Feb1978 • Security relies on the difficulty of factoring large composite numbers Choose n: Start with two prime numbers, p and q. 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 Let c denote the corresponding ciphertext. 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. To demonstrate the RSA public key encryption algorithm, let's start it with 2 smaller prime numbers 5 and 7. The RSA Encryption Scheme is often used to encrypt and then decrypt electronic communications. 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). They decided to use the public key cryptology algorithm RSA. 4.Description of Algorithm: 5.1. 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. CIS341 . 5. 5. Find the multiplicative inverse of 45 mod 238. Chosen ciphertext attacks - this type of attack exploits properties of RSA algorithm. 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, 17 = 9 * 1 + 8. RSA algorithm is asymmetric cryptography algorithm. Example. equal. p=3, q=11, e=13, d=17, M=2 Let two primes be p = 7 and q = 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. Hardware fault-base attack - this involves inducing hardware faults in the processor that is generating digital signature. For this example we can use p = 5 & q = 7. very big number. f(n) = (p-1) * (q-1) = 6 * 10 = 60. λ(701,111) = 349,716. What is the running time of your algorithm as a function of n? The Extended Euclidean Algorithm takes p, q, and e as input and gives d as output. Example-1: Step-1: Choose two prime number and Lets take and ; Step-2: Compute the value of and It is given as, What is the max integer that can be encrypted? ... RSA Example Key Setup 1 Select primes p 17 q 11 2 Compute n pq 17 x 11187 3. 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 actually a set of two algorithms: Key Generation: A key generation algorithm. ∟ 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. Let p=7 and q = 11. n=77 and (n) =60. Let e = 7 Compute a value for d such that (d * e) % φ(n) = 1. or this This makes e “co-prime” to t. 13. (b) Repeat part (a) but now encrypt “dog” as one message m. Show details of the following. General Alice’s Setup: Chooses two prime numbers. Example: \(\phi(7) = \left|\{1,2,3,4,5,6\}\right| = 6\) 2.. RSA . Example: \(\phi(7) = \left|\{1,2,3,4,5,6\}\right| = 6\) 2.. RSA . 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. 2. Give a general algorithm for calculating d and run such algorithm with the above Example 1 for RSA Algorithm • Let p = 13 and q = 19. • 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). Apply the decryption algorithm to I want to know about the explanation of RSA, here is the example . Asymmetric Encryption Algorithms- The famous asymmetric encryption algorithms are- RSA Algorithm; Diffie-Hellman Key Exchange . RSA: Confidentiality Example Encrypted using Alices Public key. View doc 1.docx from ICTN 2750 at East Carolina University. equal. 18. Computers represent text as long numbers (01 for \A", 02 for \B" and so on), so an email message is just a very big number. 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-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 . Based on this principle, the RSA encryption algorithm uses prime factorization as the To start with, Sr2Jr’s first step is to reduce the expenses related to education. 1.Most widely accepted and implemented general purpose approach to public key encryption developed by Rivest-Shamir and Adleman (RSA) at MIT university. p=7; q=11, e=17; M=8. Find appropriate exponents d and e. Assignment 02 (cont.) The term RSA is an acronym for Rivest-Shamir-Adleman who brought out the algorithm in 1977. An example of generating RSA Key pair is given below. 17 = 9 * 1 + 8. Taking a Crack at Asymmetric Cryptosystems Part 1 (RSA) Take for example: p=3 q=5 n=15 t=8 e=7. 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) � � � � � � � � � � � � ! 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