Cochin University of Science and Techology (CUST) 2008 M.C.A Computer Aplications NUMBER THEORY - Question Paper

MCA DEGREE II SEMESTER EXAMINATION, APRIL 2008

CAS 2202 NUMBER THEORY

MCA.2/08.2(a)

Time: 3 Hours    Maximum marks: SO

FART A

(Answer ALL questions)

(AH questions cany EQUAL marks)

(15x2-30)

1 a. Show that the number of primes is infinite.

b.    Prove that the necessary and sufficient condition that a number is a multiple of 11 is that the difference between the sum of odd digits and the sum of even digits is a multiple of II.

c.    If m is relatively prime to n, show that j(mn)    .

I. a. Solve 49x 47(mod 81).

b.    Show that every prime of the form 4k->-l can be represented uniquely as the sum of two squares.

c.    Explain the term rational cubic curve with examples.

III. a. With the help of suitable examples explain the terms quadratic residues and quadratic non residues of a prime p.

u    j ,

Define Gauss sum G and show that <7 =(1) * H-

_t ( 595

Evaluate -

1,7657 /

IV.    h. Define a pseudo prime to a base and find all non trivial bases for which 15 is a pseudo prime.

b.    Using Fermat factorization, factorize 809009.

c.    Find the continued fraction representation of the rational number .

89

V.    a. On the elliptic curve y* x* - 36x, take P (3.9) and Q (-2.8). Find and 2P.

b.    What do you mean by a zero knowledge protocol, explain?

c.    Explain how public key cryptography can be used for digital signatures.

PART B

(All questions carry EQUAL marks)

(5 x 4 - 20)

VI.    A State and prove Permats little theorem.

OR

B. Find the smallest positive integer containing 4 digits such that the remainder is 1 when it is divided by 11; the remainder is 3 when it is divided by 13; and the remainder is 7 when it is divided by 17.

(Turn ovtr)

VII.    A Find all solutions of 2x³ + 3x² +5x+ 4 =0 (mod 140).

OR

B. Let p, d be positive integers, p>n70, prove that p is prime if and only if (n- ])!(/> -n)l = (-1)" (mod p)

(

VIII.    A. Define Legendre symbol I I for an integer a and a prime p>2. Also show that

j = a (modp).

OR

(m\ ,    V(n\

B. For any two positive odd integers m and n prove that I I = (-1) ^/A I I.

IX.    A. Write notes on:

(i)    Quadratic Sueve method

(ii)    Factor base algorithm

OR

B. If a real number x>l has continued fraction expansion with convergents bijci, prove that [if,³ - x²c* ] < 2x for all /.

X.    A. Describe any one of the classical encryption techniques.

OR

B. Distinguish between public key and private key encryption techniques. Also point out the merits and demerits of both.

Attachment: cochin-university-of-science-and-techology--cust--2008-m-c-a-computer-aplications-37423-1.JPG cochin-university-of-science-and-techology--cust--2008-m-c-a-computer-aplications-37423-2.JPG

Earning:   Approval pending.