* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download TRUE/FALSE. Write `T` if the statement is true and `F` if the
Quartic function wikipedia , lookup
Root of unity wikipedia , lookup
Birkhoff's representation theorem wikipedia , lookup
Gröbner basis wikipedia , lookup
Modular representation theory wikipedia , lookup
Horner's method wikipedia , lookup
Group (mathematics) wikipedia , lookup
Cayley–Hamilton theorem wikipedia , lookup
Commutative ring wikipedia , lookup
Field (mathematics) wikipedia , lookup
System of polynomial equations wikipedia , lookup
Polynomial greatest common divisor wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Factorization wikipedia , lookup
Polynomial ring wikipedia , lookup
Eisenstein's criterion wikipedia , lookup
Algebraic number field wikipedia , lookup
Factorization of polynomials over finite fields wikipedia , lookup
TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. 1) Finite fields play a crucial role in several areas of cryptography. 1) _______ 2) Unlike ordinary addition, there is not an additive inverse to each integer in modular arithmetic. 2) _______ 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. 3) _______ 4) Two integers a and b are said to be congruent modulo n, if (a mod n) = (b mod n). 4) _______ 5) Cryptographic algorithms do not rely on properties of finite fields. 5) _______ 6) Finite fields of order p can be defined using arithmetic mod p. 6) _______ 7) The Advanced Encryption Standard uses infinite fields. 7) _______ 8) The rules for ordinary arithmetic involving addition, subtraction, and multiplication carry over into modular arithmetic. 8) _______ 9) A cyclic group is always commutative and may be finite or infinite. 9) _______ 10) A field is a set in which we can do addition, subtraction, multiplication and division without leaving the set. 10) ______ 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. 11) ______ 12) Polynomial arithmetic includes the operations of addition, subtraction and multiplication. 12) ______ 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. 13) ______ 14) The euclidean algorithm cannot be adapted to find the multiplicative inverse of a polynomial. 14) ______ 15) As a congruence relation, mod expresses that two arguments have the same remainder with respect to a given modulus. 15) ______ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 16) The greatest common divisor of two integers is the largest positive 16) ______ integer that exactly _________ both integers. A) squares B) multiplies C) exponentially multiplies D) divides 17) Two integers are __________ if their only common positive integer factor is 1. 17) ______ A) congruent modulo C) residual B) relatively prime D) polynomials 18) The __________ of two numbers is the largest integer that divides both numbers. A) prime polynomial B) lowest common divisor C) integral divisor D) greatest common divisor 18) ______ 19) A ring is said to be _________ if it satisfies the condition ab = ba for all a, b in R. A) cyclic B) commutative C) infinite D) abelian 19) ______ 20) 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) modulus B) field C) group D) ring 20) ______ 21) A _________ is a field with a finite number of elements. A) finite group B) finite order C) finite ring D) finite field 21) ______ 22) If b|a, we say that b is a __________ of a. A) residue B) modulus C) divisor 22) ______ D) group 23) 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) Euclidean B) associative C) modular D) cyclic 23) ______ 24) A group is said to be _________ if it satisfies the condition a * b = b * a for all a, b in G. A) cyclic B) infinite C) abelian D) commutative 24) ______ 25) 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) coefficient B) monic C) constant D) indeterminate 25) ______ 26) With the understanding that remainders are allowed, we can say that polynomial division is possible if the coefficient set is a __________ . A) field B) ring C) factor D) divisor 26) ______ 27) By analogy to integers, an irreducible polynomial is also called a __________ . A) constant polynomial B) polynomial ring C) prime polynomial D) monic polynomial 27) ______ 28) The congruence relation is used to define __________ . A) residue classes B) greatest common divisor C) finite groups D) lowest common divisor 28) ______ 29) As a _________ relation, mod expresses that two arguments have the same remainder with respect to a given modulus. A) cyclic B) congruence C) monic D) finite 29) ______ 30) 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) commutative ring C) positive integer D) associative 30) ______ SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 31) The remainder r in the division algorithm is often referred to as a 31) _____________ __________ . 32) 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. 32) _____________ 33) 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 __________ . 33) _____________ 34) An nth-degree polynomial is said to be a _________ polynomial if an = 1. 34) _____________ 35) __________ arithmetic is a kind of integer arithmetic that reduces all number to one of a fixed set [0,..., n - 1]. 35) _____________ 36) Elliptic curve cryptography and the _________ are two cryptographic algorithms that rely heavily on properties of finite fields. 36) _____________ 37) Let S be the set of integers, positive, negative, and 0, under the usual operations of addition and multiplication. S is an __________ domain. 37) _____________ 38) GF stands for __________ field in honor of the mathematician who first studied finite fields. 38) _____________ 39) Two integers are relatively _________ if their only common positive integer factor is 1. 39) _____________ 40) A zero-degree polynomial is called a __________ polynomial and is simply an element of the set of coefficients. 40) _____________ 41) 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). 41) _____________ 42) 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). 42) _____________ 43) 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. 43) _____________ 44) 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. 44) _____________ 45) 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. 45) _____________ 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) TRUE FALSE FALSE TRUE FALSE TRUE FALSE TRUE FALSE TRUE FALSE TRUE TRUE FALSE TRUE D B D B B D C A C D A C A B C residue Euclidean modulus monic Modular Advanced Encryption Standard (AES) integral Galois prime constant irreducible greatest common divisor modulus generator root