EXAMPLE: Let n( Alice) = 95, e( Alice) = 59, d( Alice) = 11, n( Bob) = 77, e( Bo

Question

EXAMPLE: Let n(Alice) = 95, e(Alice) = 59, d(Alice) = 11, n(Bob) = 77, e(Bob) = 53, and d(Bob) = 17. Alice and Bob have 26 possible contracts, from which they are to select and sign one. Alice first asks Bob to sign contract F:

She then asks him to sign contract R:

Alice now computes 05 x 17 mod 77 = 08. She then claims that Bob agreed to contract I, and as evidence presents the signature 3 x 19 mod 77 = 57. Judge Janice is called, and she computes

Naturally, she concludes that Bob is lying, because his public key deciphers the signature. So Alice has successfully tricked Bob. Give another example of the same trick explained in this example. Please explain your steps clearly.

Explanation / Answer

Example 1:

demonstrates that messages that are both enciphered and signed should be signed first,then enciphered. Suppose Alice is sending Bob her signature on a confidential contract m. She enciphers it first,then signs it

C=(meBob mod nBob)dAlice mod nAlice

and sends the result to Bob. However, Bob wants to claim that Alice sent him the contract M. Bob computes a number r such that Mr    mod nBob=m    He then republishes his public key as(reBob , nBob) Note that the modulus does not change. Now, he claims that Alice sent him

M. The judge verifies this using his current public key. Thesimplest way to fix this is to require all users to use the same exponent but vary the moduli.

Smarting from Alice's trick, Bob seeks revenge. He and Alice agree to sign the contract G(06). Alice first enciphers it, then signs it

(0653 mod 77)11 mod 95=63

and sends it to Bob. Bob, however, wants the contract to be N (13). He computes an r such that 13r mod 77 = 6; one such r is r=59 . He then computes a new public key

reBob mod f(nBob)=59*53 mod 60=7

He replaces his current public key with (7, 77), and resets his private key to 43. He nowclaims that Alice sent him contract N, signed by her.Judge Janice is called. She takes the message 63 and deciphers it :

(6359 mod 95 )43 mod 77=13

and concludes that Bob is correct.

Example2: Alice, Bob communicating

Signature validated; Bob is toast!

Example3

Solution: sign first and then enciher!!

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