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Chapter 1: Q12E (page 48)
What is ?
The solution is .
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RSA and digital signatures. Recall that in the RSA public-key cryptosystem, each user has a public key P=(N,e) and a secret key d. In a digital signature scheme, there are two algorithms, sign and verify. The sign procedure takes a message and a secret key, then outputs a signature . The verify procedure takes a public key , a signature , and a message M, then returns 鈥渢rue鈥� if could have been created by sign (when called with message M and the secret key (N, e) corresponding to the public key ); 鈥渇alse鈥� otherwise.
(a)Why would we want digital signatures?
(b) An RSA signature consists of sign, where d is a secret key and N is part of the public key . Show that anyone who knows the public key can perform verify , i.e., they can check that a signature really was created by the private key. Give an implementation and prove its correctness.
(c) Generate your own RSA modulus, N=pq public key e, and private key d (you don鈥檛 need to use a computer). Pick p and q so you have a 4-digit modulus and work by hand. Now sign your name using the private exponent of this RSA modulus. To do this you will need to specify some one-to-one mapping from strings to integers in . Specify any mapping you like. Give the mapping from your name to numbers then sign the first number by giving the value , and finally show that .
(d) Alice wants to write a message that looks like it was digitally signed by Bob. She notices that Bob鈥檚 public RSA key is . To what exponent should she raise her message?
Is the difference of a multiple of ?
Determine necessary and sufficient conditions on so that the following holds: for any if , then .
Show that if is a nontrivial square root of 1 modulo N , that is if but , then must be composite. (For instance,; thus 4 is a nontrivial square root of 1 modulo 15.)
Consider an RSA key set with p = 17 , q = 23, N = 23 and e = 3 (as in Figure 1.9). What value of d should be used for the secret key? What is the encryption of the message M = 41 ?
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