Deck 8: More Number Theory

Question 1

The first assertion of the CRT,concerning arithmetic operations, follows from the rules for modular arithmetic.

Answer 1

False

Question 2

Miller's test will return __________ if it fails to detect that n is not prime.

A)rejected
B)inconclusive
C)composite
D)discrete

Answer 2

B

Question 3

The Miller-Rabin test can determine if a number is not prime but cannot determine if a number is prime.

Answer 3

True

Question 4

If p is prime and a is a positive integer,then ap = amod p)is an alternative form of _________ theorem.

A)Rijndael's
B)Vignere's
C)Euler's
D)Fermat's

Question 5

With ordinary positive real numbers the logarithm function is the inverse of exponentiation.

Question 6

Discrete logarithms are not fundamental to public-key algorithms.

Question 7

All integers have primitive roots.

Question 8

One of the useful features of the Chinese remainder theorem is that it provides a way to manipulate potentially very large numbers mod M in terms of tuples of smaller numbers.

Question 9

The Chinese Remainder Theorem is believed to have been discovered by the Chinese mathematician Agrawal in 100 A.D.

Question 10

The number 37 is prime so therefore all of the positive integers from 1 to 36 are relatively prime to 37.

Question 11

A prime number can have a remainder when divided by positive or negative values of itself.

Question 12

The logarithm of a number is defined to be the power to which some positive base except 1)must be raised in order to equal the number.

Question 13

Prime numbers play a _________ role in number theory.

A)minor
B)nonessential
C)critical
D)abbreviated

Question 14

The primitive roots for the prime number 19 are 2,3,10,13,14 and 15.

Question 15

A _________ number can only be divided by +/- values of itself and 1 and cannot have a remainder.

A)prime
B)composite
C)indexed
D)positive

Question 16

An important requirement in a number of cryptographic algorithms is the ability to choose a large prime number.

Question 17

An area of ongoing research is the development of efficient algorithms for determining if a randomly chosen large integer is a prime number.

Question 18

Discrete logarithms are analogous to ordinary logarithms but are defined using modular arithmetic.

Question 19

Prime numbers play a very small role in cryptography.

Question 20

An important quantity in number theory referred to as __________ ,is defined as the number of positive integers less than n and relatively prime to n.

A)CRT
B)Miller-Rabin
C)Euler's totient function
D)Fermat's theorem

Question 21

The _________ algorithm is typically used to test a large number for primality.

A)Rijndael
B)Fermat
C)Miller-Rabin
D)Euler

Question 22

Discrete logarithms are fundamental to a number of public-key algorithms including __________ key exchange and the DSA.

A)Diffie-Hellman
B)Rijndael-Fadiman
C)Fermat-Euler
D)Miller-Rabin

Question 23

Two theorems that play important roles in public-key cryptography are Fermat's theorem and __________ theorem.

Question 24

Two numbers are relatively prime if they have _________ prime factors in common.

A)some
B)zero
C)multiple
D)all

Question 25

Discrete logarithms are fundamental to the digital signature algorithm and the _________ algorithm.

Question 26

A one-to-one correspondence is called __________ .

A)a bijection
B)an inclusive
C)a composite
D)an index

Question 27

Two numbers are relatively prime if they have ________ prime factors in common.

A)zero
B)two
C)several
D)one

Question 28

The __________ theorem states that it is possible to reconstruct integers in a certain range from their residues modulo a set of pairwise relatively prime moduli.

Question 29

Two numbers are __________ if their greatest common divisor is 1.

Question 30

The mapping of the CRT equation is a one-to-one correspondence called a _________ between Zm and the Cartesian product Zm1 X Zm2 X ...X Zmk.

Question 31

Discrete logarithms are fundamental to the ____________ .

A)Euler algorithm
B)digital signature algorithm
C)Miller-Rabin algorithm
D)Rijndael algorithm

Question 32

The number of positive integers less than n and relatively prime to n is referred to as __________ function.

Question 33

The procedure TEST takes a candidate integer n as input and returns the result __________ if n may or may not be a prime.

A)discrete
B)composite
C)inconclusive
D)primitive

Question 34

For any integer b and a primitive root a of prime number p we can find a unique exponent i .This exponent i is referred to as the ___________ .

A)order
B)discrete logarithm
C)bijection
D)primitive root

Question 35

The _________ of a number is defined to be the power to which some positive base except 1)must be raised in order to equal the number.

Question 36

Discrete logarithms are analogous to ordinary logarithms but are defined using __________ arithmetic.

Question 37

The procedure TEST takes a candidate integer n as input and returns the result __________ if n is definitely not a prime.

A)discrete
B)composite
C)inconclusive
D)primitive

Question 38

__________ theorem states the following: If p is prime and a is a positive integer not divisible by p,then ap-1 = 1mod p).

Question 39

If a number is the highest possible exponent to which a number can belong,it is referred to as a _________ of n.

A)primitive root
B)composite
C)discrete logarithm
D)bijection

Question 40

A __________ number is an integer that can only be divided by positive and negative values of itself and 1 without having a remainder.

Question 41

Although it does not appear to be as efficient as the Miller-Rabin algorithm,in 2002 a relatively simple deterministic algorithm that efficiently determines whether a given large number is a prime was developed.This algorithm is known as the _________ algorithm.

Question 42

The _________ of integers a and b,expressed gcd a,b),is an integer c that divides both a and b without remainder and that any divisor of a and b is a divisor of c.

Question 43

To determine whether an odd integer n is prime with a reasonable degree of confidence repeatedly invoke TEST n)using randomly chosen values for a.If,at any point,TEST returns _________ then n is determined to be nonprime.

Question 44

Two numbers are relatively prime if their greatest common divisor is _________ .

Question 45

An integer p > 1 is a __________ number if and only if its only divisors are + 1 and + 1.