3.9. A decryption exponent for an RSA public key (N, e) is an integer d with the property that ade = a (mod N) for all integers a that are relatively prime to N. (a) Suppose that Eve has a magic box that creates decryption exponents for (N, e) for a fixed modulus N and for a large number of different encryption expo- nents e. Explain how Eve can use her magic box to try to factor N. (b) Let N = 38749709. Eve's magic box tells her that the encryption exponent e = 10988423 has decryption exponent d = 16784693 and that the encryp- tion exponent e = 25910155 has decryption exponent d = 11514115. Use this information to factor N. 178 Exercises (c) Let N = 225022969. Eve's magic box tells her the following three encryp- tion/decryption pairs for N: (70583995, 4911157), (173111957, 7346999), (180311381, 29597249). Use this information to factor N.

I need help with the following problems in cryptography. A. Encrypt the message “Freedom” using the Playfair matrix M F H I K U N O P Q Z V W X Y E L A R G D S T B C T B. Construct the Playfair matrix with the keyword “LARGEST” and encrypt the message “Freedom” again. C. Compare the results obtained in A and B. How do you account for these results?

Question 4. A sealed-bid auction is a process where a number of partiescalled bidders submit sealed bids for buying some item. Once all sealed bidshave been submitted, the bids are revealed and the highest bidder is the onethat buys the item.Suppose that the following cryptographic primitives are available:• A fully secure public-key encryption scheme (e.g., RSA-OAEP) thatconsists of the key-generation algorithm PKE.Gen, the encryption algorithm Enc, and the decryption algorithm Dec.• An existentially unforgeable digital signature scheme (e.g., RSA-FDH)that consists of the key-generation algorithm DS.Gen, the signing algorithm Sign, and the verification algorithm Vrfy.• A trusted certificate authority (CA) that can be used to issue publickey certificates for any party.Using the cryptographic primitives above, design and analyse an electronic sealed-bid auction protocol according to the following guidelines:• The parties involved in the auction are: (i) an auction administrator(AA) and…

Consider 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?