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Chapter 1: Q20E (page 49)
Find the inverse of:.
Inverse of the given numbers is obtained.
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Calculate using any method you choose. (Hint: 127 is prime.)
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?
Digital signatures, continued.Consider the signature scheme of Exercise .
(a) Signing involves decryption, and is therefore risky. Show that if Bob agrees to sign anything he is asked to, Eve can take advantage of this and decrypt any message sent by Alice to Bob.
(b) Suppose that Bob is more careful, and refuses to sign messages if their signatures look suspiciously like text. (We assume that a randomly chosen messagethat is, a random number in the range is very unlikely to look like text.) Describe a way in which Eve can nevertheless still decrypt messages from Alice to Bob, by getting Bob to sign messages whose signatures look random.
What is ?
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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