Alice, Bob and Carol use the the elliptic curves encryption scheme to communicate, and use the...

 

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Alice, Bob and Carol use the the elliptic curves encryption scheme to communicate, and use the elliptic curve digital signature algorithm (ECDSA) to sign all messages they send. They decided on the following for their system: For sending messages according to the elliptic curve encryption schema they will use the elliptic curve group G= [32423423, 0, 111111756372541867431527]. The (base) point relative to which the rest of their public key will be chosen for encryption purposes is g:=[133213, 42904742913851518927444]. The public point b, to be used for encryption, are as follows: Alice: [77012625572753817011635, 107177820655281658062820] Bob: [63567155336615021877739, 36007178347937208049611] Carol:[104225718616102818923406, 48710727653703914714853] For signing messages according to the ECDSA they will use the elliptic curve [0, 98766778, 12357627687623542975694261] and the base point g= [2539662136869871783297850, 1414894921445613162605806]. It is known that the order of g is 620176035713316419537. The public point b, to be used for signature, are as follows: Alice: [5246859784562186496635824, 5457612259949488195005309] Bob: [5278780115068892215096216, 6857888556017625502158912] Carol:[11500587084934239979269346, 4630177618473717713077113] The tolerance parameter for their system is 45. One of them sent a message to another member of the group, with a digital signature attached. The ciphertext is [64822910729849959453845, 54324913307643112006572] and [40965317570056334177487, 76589794174623293315486]. The signature is (979332919684899048601692, 426149012637180793991). Determine (a) Everyone's private keys. (b) The message, for whom it was encrypted, and by whom it was signed. (c) The random number used in signing the message. (d) The random number used in encrypting the message. Alice, Bob and Carol use the the elliptic curves encryption scheme to communicate, and use the elliptic curve digital signature algorithm (ECDSA) to sign all messages they send. They decided on the following for their system: For sending messages according to the elliptic curve encryption schema they will use the elliptic curve group G= [32423423, 0, 111111756372541867431527]. The (base) point relative to which the rest of their public key will be chosen for encryption purposes is g:=[133213, 42904742913851518927444]. The public point b, to be used for encryption, are as follows: Alice: [77012625572753817011635, 107177820655281658062820] Bob: [63567155336615021877739, 36007178347937208049611] Carol:[104225718616102818923406, 48710727653703914714853] For signing messages according to the ECDSA they will use the elliptic curve [0, 98766778, 12357627687623542975694261] and the base point g= [2539662136869871783297850, 1414894921445613162605806]. It is known that the order of g is 620176035713316419537. The public point b, to be used for signature, are as follows: Alice: [5246859784562186496635824, 5457612259949488195005309] Bob: [5278780115068892215096216, 6857888556017625502158912] Carol:[11500587084934239979269346, 4630177618473717713077113] The tolerance parameter for their system is 45. One of them sent a message to another member of the group, with a digital signature attached. The ciphertext is [64822910729849959453845, 54324913307643112006572] and [40965317570056334177487, 76589794174623293315486]. The signature is (979332919684899048601692, 426149012637180793991). Determine (a) Everyone's private keys. (b) The message, for whom it was encrypted, and by whom it was signed. (c) The random number used in signing the message. (d) The random number used in encrypting the message.

Answer rating: 100% (QA)

a Private Keys We can use the fact that the private key d corresponding to a public key b is the int... View the full answer