Download CH04 Testbank Crypto6e... Cryptography and Network Security: Principles and Practice, 6th Edition, by William Stallings CHAPTER 4: BASIC CONCEPTS IN NUMBER THEORY AND FINITE FIELDS TRUE OR FALSE T F 1. Finite fields play a crucial role in several areas of cryptography. T F 2. Unlike ordinary addition, there is not an additive inverse to each integer in modular arithmetic. T F 3. The scheme where you can find the greatest common divisor of two integers by repetitive application of the division algorithm is known as the Brady algorithm. T F 4. Two integers a and b are said to be congruent modulo n, if (a mod n) = (b mod n). T F 5. Cryptographic algorithms do not rely on properties of finite fields. T F 6. Finite fields of order p can be defined using arithmetic mod p. T F 7. The Advanced Encryption Standard uses infinite fields. T F 8. The rules for ordinary arithmetic involving addition, subtraction, and multiplication carry over into modular arithmetic. T F 9. A cyclic group is always commutative and may be finite or infinite. T F 10. A field is a set in which we can do addition, subtraction, multiplication and division without leaving the set. T F 11. It is easy to find the multiplicative inverse of an element in g(p) for large values of p by constructing a multiplication table, however for small values of p this approach is not practical. T F 12. Polynomial arithmetic includes the operations of addition, subtraction and multiplication. T F 13. If we attempt to perform polynomial division over a coefficient set that is not a field, we find that division is not always defined. T F 14. The euclidean algorithm cannot be adapted to find the multiplicative inverse of a polynomial. T F 15. As a congruence relation, mod expresses that two arguments have Cryptography and Network Security: Principles and Practice, 6th Edition, by William Stallings the same remainder with respect to a given modulus. MULTIPLE CHOICE 1. The greatest common divisor of two integers is the largest positive integer that exactly _________ both integers. A) multiplies B) exponentially multiplies C) squares D) divides 2. Two integers are __________ if their only common positive integer factor is 1. A) relatively prime C) polynomials B) congruent modulo D) residual 3. The __________ of two numbers is the largest integer that divides both numbers. A) greatest common divisor B) prime polynomial C) lowest common divisor D) integral divisor 4. A ring is said to be _________ if it satisfies the condition ab = ba for all a, b in R. A) cyclic B) commutative C) abelian D) infinite 5. A _________ is a set of elements on which two arithmetic operations have been defined and which has the properties of ordinary arithmetic, such as closure, associativity, commutativity, distributivity, and having both additive and multiplicative inverses. A) field B) modulus C) group D) ring 6. A _________ is a field with a finite number of elements. A) finite group B) finite order Cryptography and Network Security: Principles and Practice, 6th Edition, by William Stallings C) finite field D) finite ring 7. If b|a, we say that b is a __________ of a. A) residue B) group C) divisor D) modulus 8. For given integers a and b, the extended __________ algorithm not only calculates the greatest common divisor d but also two additional integers x and y. A) modular C) associative B) Euclidean D) cyclic 9. A group is said to be _________ if it satisfies the condition a * b = b * a for all a, b in G. A) abelian B) infinite C) cyclic D) commutative 10. In the context of abstract algebra we are usually not interested in evaluating a polynomial for a particular value of x. To emphasize this point the variable x is sometimes referred to as the __________ . A) monic B) constant C) indeterminate D) coefficient 11. With the understanding that remainders are allowed, we can say that polynomial division is possible if the coefficient set is a __________ . A) ring B) field C) factor D) divisor 12. By analogy to integers, an irreducible polynomial is also called a __________ . A) constant polynomial B) monic polynomial C) polynomial ring D) prime polynomial Cryptography and Network Security: Principles and Practice, 6th Edition, by William Stallings 13. The congruence relation is used to define __________ . A) finite groups C) lowest common divisor B) greatest common divisor D) residue classes 14. As a _________ relation, mod expresses that two arguments have the same remainder with respect to a given modulus. A) finite B) monic C) congruence D) cyclic 15. The order of a finite field must be of the form pn where p is a prime and n is a __ . A) identity element B) positive integer C) commutative ring D) associative SHORT ANSWER 1. The remainder r in the division algorithm is often referred to as a __________ . 2. One of the basic techniques of number theory is the __________ algorithm which is a simple procedure for determining the greatest common divisor of two positive integers. 3. If a is an integer and n is a positive integer, we define a mod n to be the remainder when a is divided by n. The integer n is called the __________ . 4. An nth-degree polynomial is said to be a _________ polynomial if an = 1. 5. __________ arithmetic is a kind of integer arithmetic that reduces all numbers to one of a fixed set [0,..., n - 1]. 6. Elliptic curve cryptography and the _________ are two cryptographic algorithms that rely heavily on properties of finite fields. 7. Let S be the set of integers, positive, negative, and 0, under the usual operations of addition and multiplication. S is an __________ domain. Cryptography and Network Security: Principles and Practice, 6th Edition, by William Stallings 8. GF stands for __________ field in honor of the mathematician who first studied finite fields. 9. Two integers are relatively _________ if their only common positive integer factor is 1. 10. A zero-degree polynomial is called a __________ polynomial and is simply an element of the set of coefficients. 11. A polynomial f(x) over a field F is called __________ if and only if f(x) cannot be expressed as a product of two polynomials, both over F, and both of degree lower than that of f(x). 12. The polynomial c(x) is said to be the __________ of a(x) and b(x) if c(x) divides both a(x) and b(x) and any divisor of a(x) and b(x) is a divisor of c(x). 13. If a is an integer and n is a nonzero integer, we define a mod n to be the remainder when a is divided by n. The integer n is called the __________ and the remainder is called the residue. 14. A __________ g of a finite field F or order q is an element whose first q - 1 powers generate all the nonzero elements of F. 15. Consider a field F defined by a polynomial f(x). An element b contained in F is called a __________ of the polynomial if f(b) = 0. |
