Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
expand_more
expand_more
format_list_bulleted
Question
Chapter 31.7, Problem 3E
Program Plan Intro
To prove that the RSA is a multiplicative for the below given expression,
And, to calculate the probability of encrypting a message with high success rate.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
One of the one-way functions used in public key cryptography is the discrete logarithm. Computing r = gº mod p from
g, e, and p is easy. But given only r, g and p, recovering e is hard.
Suppose p
1597,
2 and r =
128.
What is the smallest positive integer e such that r = ge mod p?
Suppose that Alice and Bob communicate using
ElGamal cipher and f (p. 9. Z) is common public
values. Bob generates his private key d ER Z and
then computes the corresponding and public
public key y=g" (mod p). To save time, Bob uses the
same number r each time he encrypts a plaintext
message m (ie., r is a fixed nonce of Bob, and it
is not randomly generated each time encryption
is performed). Assume that Alice compute the
ciphertext for the message m as (cc) = (g mod p, mxy
mod p). and for the message m as (1,2)=(g" mod p,
xy' mod p). Show how an adversary who possesses a
plaintext-ciphertext pair (m. (c.ca)) can decrypt (1, 2)
without knowing the private key d of Bob.
Messages are to be encoded using the RSA method, and the primes chosen are p = 13
and q = 23, so that n = pq = 299. The encryption exponent is e = 13. Thus, the public
key is (299, 13).
(a) Use the repeated squaring algorithm to find the encrypted form c of the message
84.
m =
(b) Show that the decryption exponent d (the private key) is 61.
(c) Verify that you obtain the original message after decryption. Use the repeated
squaring algorithm.
Chapter 31 Solutions
Introduction to Algorithms
Ch. 31.1 - Prob. 1ECh. 31.1 - Prob. 2ECh. 31.1 - Prob. 3ECh. 31.1 - Prob. 4ECh. 31.1 - Prob. 5ECh. 31.1 - Prob. 6ECh. 31.1 - Prob. 7ECh. 31.1 - Prob. 8ECh. 31.1 - Prob. 9ECh. 31.1 - Prob. 10E
Ch. 31.1 - Prob. 11ECh. 31.1 - Prob. 12ECh. 31.1 - Prob. 13ECh. 31.2 - Prob. 1ECh. 31.2 - Prob. 2ECh. 31.2 - Prob. 3ECh. 31.2 - Prob. 4ECh. 31.2 - Prob. 5ECh. 31.2 - Prob. 6ECh. 31.2 - Prob. 7ECh. 31.2 - Prob. 8ECh. 31.2 - Prob. 9ECh. 31.3 - Prob. 1ECh. 31.3 - Prob. 2ECh. 31.3 - Prob. 3ECh. 31.3 - Prob. 4ECh. 31.3 - Prob. 5ECh. 31.4 - Prob. 1ECh. 31.4 - Prob. 2ECh. 31.4 - Prob. 3ECh. 31.4 - Prob. 4ECh. 31.5 - Prob. 1ECh. 31.5 - Prob. 2ECh. 31.5 - Prob. 3ECh. 31.5 - Prob. 4ECh. 31.6 - Prob. 1ECh. 31.6 - Prob. 2ECh. 31.6 - Prob. 3ECh. 31.7 - Prob. 1ECh. 31.7 - Prob. 2ECh. 31.7 - Prob. 3ECh. 31.8 - Prob. 1ECh. 31.8 - Prob. 2ECh. 31.8 - Prob. 3ECh. 31.9 - Prob. 1ECh. 31.9 - Prob. 2ECh. 31.9 - Prob. 3ECh. 31.9 - Prob. 4ECh. 31 - Prob. 1PCh. 31 - Prob. 2PCh. 31 - Prob. 3PCh. 31 - Prob. 4P
Knowledge Booster
Similar questions
- Use a coding matrix A of your choice. Use a graphing utility to find the multiplicative inverse of your coding matrix. Write a cryptogram for each message. C perform all necessary matrix multiplications. DGE OU T AHEAD BR 2 18 4 75 0 15 21 20 0 1 8 5 1 4 .... 1 1 -12 32 -15 Using the coding matrix the coded message is.····☐☐☐☐☐☐☐☐☐☐ 2 2 1 5 4 4 4 7 (Simplify your answers.) 61arrow_forwardNote: The notation from this problem is from Understanding Cryptography by Paar and Pelzl. We conduct a known-plaintext attack against an LFSR. Through trial and error we have determined that the number of states is m = 4. The plaintext given by 01100101 —D ХоXјX2XҙX4X5X6X7 when encrypted by the LFSR produced the ciphertext 10010100 — Уo У1 У2 Уз Уз У5 У6 Ут- What are the tap bits of the LFSR? Please enter your answer as unspaced binary digits (e.g. 0101 to represent P3 = 0, p2 = 1, Pi = 0, po = 1).arrow_forwardThe answer above is NOT correct. Note: The notation from this problem is from Understanding Cryptography by Paar and Pelzl. We conduct a known-plaintext attack against an LFSR. Through trial and error we have determined that the number of states is m = 4. The plaintext given by 11100010 — ХоXјX2XҙX4XsX6X7 when encrypted by the LFSR produced the ciphertext 11010110 — Уo У1 У2 Уз Уз У5 У6 Ут. What are the tap bits of the LFSR? Please enter your answer as unspaced binary digits (e.g. 0101 to represent p3 = 0, p2 = 1, p1 = 0, po = 1). 0010arrow_forward
- Alice wants Bob (the Bank) to sign a cheque, c. She has Bob's public key e and modulus N, and chooses a random number r such that gcd(r,N) = 1. She computes re (mod N) and then multiplies the cheque, c, by this number (mod N) and sends the resulting message off to Bob to be signed. When she receives his reply, m, what should she do with it to produce the signed cheque? Select one: a. Divide the reply, m, by rd. b. Divide the reply, m, by r. c. Divide the reply, m, by re. d. Nothing - the reply is the signed chequearrow_forwardNote: The notation from this problem is from Understanding Cryptography by Paar and Pelzl. We conduct a known-plaintext attack against an LFSR. Through trial and error we have determined that the number of states is m = 4. The plaintext given by 11000101 — ХоXјX2XҙХ4X5X6X7 when encrypted by the LFSR produced the ciphertext 10010000 Уo У1 У2 Уз Уз У5 У6 Ут . What are the tap bits of the LFSR? Please enter your answer as unspaced binary digits (e.g. 0101 to represent P3 = 0, p2 = 1, p1 = 0, po = 1).arrow_forwardAlice wants Bob (the Bank) to sign a cheque, c. She has Bob's public key e and modulus N, and chooses a random number r such that gcd(r,N) = 1. She computes re (mod N) and then multiplies the cheque, c, by this number (mod N) and sends the resulting message off to Bob to be signed. When she receives his reply, m, what should she do with it to produce the signed cheque? Select one: a. Divide the reply, m, by rd. b. Divide the reply, m, by r. c. Divide the reply, m, by re. No - that won't do it. d. Nothing - the reply is the signed chequearrow_forward
- We consider ElGamal cryptosystem with public prime p = 53 and generator g = 2. Let h = 16 be Bob’s public-key (his private key x is unknown, h = g^x ). Alice encrypted message m with Bob’s public key and sent the corresponding ciphertext to Bob: (C1 ,C2 ) = (15, 50). You intercepted this ciphertext. Can you break the encryption and recover the original message m? Hint: Both g and h are very small numbers, can you deduce private key x?arrow_forwardDetermine the appropriate criteria for a cryptographic hash function h. We can find different data x and y where h(x)=h(y). For a given digest z, it is impossible to reconstruct the data x where h(x) = z. For a given data x where h(x) = z, we can find a data y different from x where h(x)=h(y)= z. For a given data x and a key k, it should be easy to compute h(x) = z using the key k.arrow_forwardCryptography Algorithm Q: Say that x^2 = y^2 mod n, but x != y mod n and x != −y mod n. (a) Show that we found a nontrivial factor of narrow_forward
- 1. Let (01, 02, 03,..., 026!} be the set of permutations of the alphabet A = {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}, and consider the simple substitution cryptosystem with ok = A B C D E F G H I J K L M X MTAK ZQB NO LES N O P Q R S T U V W X Y Z IF G R V JH CY UDP W and key k. (a) Compute C = e(AUBUR NUNIV ERSIT YATMO NTGOM ERY, k). (b) Compute M = d(NIZFV SXHNF IHBKF VPXIA KIHVF GP, k).arrow_forwardConsider a very simple symmetric block encryption algorithm in which 32-bits blocks of plaintext are encrypted using a 64-bit key. Encryption is defined as C = (PK₁) K₁ where C = ciphertext, K = secret key, Ko = leftmost 64 bits of K, K₁ = rightmost 64 bits of K,+ = bitwise exclusive OR, and is addition mod 264. a. Show the decryption equation. That is, show the equation for P as a function of C, Ko, and K₁. b. Suppose and adversary has access to two sets of plaintexts and their correspond- ing ciphertexts and wishes to determine K. We have the two equations: C = (PK) K₁; C = (PK) K₁ First, derive an equation in one unknown (e.g., Ko). Is it possible to proceed fur- ther to solve for Ko?arrow_forwardAn old encryption system uses 24-bit keys. A cryptanalyst, who wants to brute-force attack the encryption system, is working on a computer system with a performance rate N keys per second. How many possible keys will be available in the above encryption system? What will be the maximum number of keys per second (N), that the computer system is working with, if the amount of time needed to brute-force all the possible keys was 8192 milli seconds?arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Database System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill Education
Starting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSON
Digital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSON
C How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSON
Database Systems: Design, Implementation, & Manag...Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage Learning
Programmable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education
Database System Concepts
Computer Science
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:9780134444321
Author:Tony Gaddis
Publisher:PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:9780132737968
Author:Thomas L. Floyd
Publisher:PEARSON
C How to Program (8th Edition)
Computer Science
ISBN:9780133976892
Author:Paul J. Deitel, Harvey Deitel
Publisher:PEARSON
Database Systems: Design, Implementation, & Manag...
Computer Science
ISBN:9781337627900
Author:Carlos Coronel, Steven Morris
Publisher:Cengage Learning
Programmable Logic Controllers
Computer Science
ISBN:9780073373843
Author:Frank D. Petruzella
Publisher:McGraw-Hill Education