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Chapter 1: Q17E (page 49)

Consider the problem of computing x y for given integers x and y: we want the whole answer, not modulo a third integer. We know two algorithms for doing this: the iterative algorithm which performs y 鈭� 1 multiplications by x; and the recursive algorithm based on the binary expansion of y. Compare the time requirements of these two algorithms, assuming that the time to multiply an n-bit number by an m-bit number is O(mn).

Short Answer

Expert verified

Time complexity of both the algorithms are compared.

Step by step solution

01

Explain Time Complexity

Time complexity is defined as the total amount of time taken to run and complete the function. It is being observed step by step at each statement of the function.Two integers x and y are considered and the exponential is found out by iterative and recursive approach and by comparing their time complexities.

02

Iterative algorithm

Iterative Algorithm is defined as:

defiterative(x,y):Input=x,yOutput=xyFinal=xForiinrange(1,y):Final脳=xReturnFinal

Total time complexity of the given algorithm is O(2n).

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Most popular questions from this chapter

In an RSA cryptosystem, p = 7and q = 11(as in Figure 1.9). Find appropriate exponents and .

A positive integer N is a power if it is of the formqk , where q,role="math" localid="1658399000008" k are positive integers and k>1.

(a) Give an efficient algorithm that takes as input a number and determines whether it is a square, that is, whether it can be written asq2 for some positive integer q. What is the running time of your algorithm?

(b) Show that if N=qk (with role="math" localid="1658399171717" N,q , andk all positive integers), then either role="math" localid="1658399158890" k鈮�logNorN=1.

(c) Give an efficient algorithm for determining whether a positive integerN is a power. Analyze its running time.

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 蟽. The verify procedure takes a public key (N,e), a signature 蟽, and a message M, then returns 鈥渢rue鈥� if 蟽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, (M,d)=Md(modN)where d is a secret key and N is part of the public key . Show that anyone who knows the public key (N,e)can perform verify ((N,e),Md,M), 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 [0,N-1]. Specify any mapping you like. Give the mapping from your name to numbers m1,m2,...mk,then sign the first number by giving the value md1(modN), and finally show that .

(md1)e=m1(modN)

(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 (17,391). To what exponent should she raise her message?

Is the difference of 530,000and6123,456a multiple of31 ?

Quadratic residues. Fix a positive integer N. We say that a is a quadratic residue modulo N ifthere exists a such that a鈮�x2modN.
(a) Let N be an odd prime and be a non-zero quadratic residue modulo N. Show that there are exactly two values in{0,1,....,N-1} satisfying x2鈮�amodN.
(b) Show that if N is an odd prime, there are exactly(N+1)2 quadratic residues in {0,1,...,N-1}.
(c) Give an example of positive integers a and N such thatx2鈮�amodNhas more than two solutions in {0,1,...,N-1}.
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