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Chapter 1: Q39E (page 51)
Give a polynomial-time algorithm for computing, given a,b,c, and prime p.
An Algorithm:
Use Fermat鈥檚 little theorem
Apply on with twice modular exponentiation operation
Return a,b,c,p
The algorithm runs in time.
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Show that if a has a multiplicative inverse modulo N, then this inverse is unique (modulo N).
Prove or disprove: If a has an inverse modulo b, then b has an inverse modulo a.
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?
Prove that the grade-school multiplication algorithm (page 24), when applied to binary numbers, always gives the right answer.
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