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1997 by CRC Press, Inc.

Chapter 2
Mathematical Background
Contents in Brief
2.1 Probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2 Information theory . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.3 Complexity theory . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.4 Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.5 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.6 Finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.7 Notes and further references . . . . . . . . . . . . . . . . . . . . 85
This chapter is a collection of basic material on probability theory, information the-
ory, complexity theory, number theory, abstract algebra, and finite fields that will be used
throughout this book. Further background and proofs of the facts presented here can be
found in the references given in §2.7. The following standard notation will be used through-
out:
1. Z denotes the set of integers; that is, the set {. . . , −2, −1, 0, 1, 2, . . .}.
2. Q denotes the set of rational numbers; that is, the set {a
b | a, b ∈ Z, b = 0}.
3. R denotes the set of real numbers.
4. π is the mathematical constant; π ≈ 3.14159.
5. e is the base of the natural logarithm; e ≈ 2.71828.
6. [a, b] denotes the integers x satisfying a ≤ x ≤ b.
7. x is the largest integer less than or equal to x. For example, 5.2 = 5 and
−5.2 = −6.
8. x
is the smallest integer greater than or equal to x. For example, 5.2
= 6 and
−5.2
= −5.
9. If A is a finite set, then |A| denotes the numberof elements in A, called the cardinality
of A.
10. a ∈ A means that element a is a member of the set A.
11. A ⊆ B means that A is a subset of B.
12. A ⊂ B means that A is a proper subset of B; that is A ⊆ B and A = B.
13. The intersection of sets A and B is the set A ∩ B = {x | x ∈ A and x ∈ B}.
14. The union of sets A and B is the set A ∪ B = {x | x ∈ A or x ∈ B}.
15. The difference of sets A and B is the set A − B = {x | x ∈ A and x ∈ B}.
16. The Cartesian product of sets A and B is the set A × B = {(a, b) | a ∈ A and b ∈
B}. For example, {a1, a2} × {b1, b2, b3} = {(a1, b1), (a1, b2), (a1, b3), (a2, b1),
(a2, b2), (a2, b3)}.
49

50 Ch. 2 Mathematical Background
17. A function or mapping f : A −→ B is a rule which assigns to each element a in A
precisely one element b in B. If a ∈ A is mapped to b ∈ B then b is called the image
of a, a is called a preimage of b, and this is written f(a) = b. The set A is called the
domain of f, and the set B is called the codomain of f.
18. A function f : A −→ B is 1 − 1 (one-to-one) or injective if each element in B is the
image of at most one element in A. Hence f(a1) = f(a2) implies a1 = a2.
19. A function f : A −→ B is onto or surjective if each b ∈ B is the image of at least
one a ∈ A.
20. A function f : A −→ B is a bijection if it is both one-to-one and onto. If f is a
bijection between finite sets A and B, then |A| = |B|. If f is a bijection between a
set A and itself, then f is called a permutation on A.
21. ln x is the natural logarithm of x; that is, the logarithm of x to the base e.
22. lg x is the logarithm of x to the base 2.
23. exp(x) is the exponential function ex
.
24.
n
i=1 ai denotes the sum a1 + a2 + · · · + an.
25.
n
i=1 ai denotes the product a1 · a2 · · · · · an.
26. For a positive integer n, the factorial function is n! = n(n − 1)(n − 2) · · · 1. By
convention, 0! = 1.
2.1 Probability theory
2.1.1 Basic definitions
2.1 Definition An experiment is a procedure that yields one of a given set of outcomes. The
individual possible outcomes are called simple events. The set of all possible outcomes is
called the sample space.
This chapter only considers discrete sample spaces; that is, sample spaces with only
finitely many possible outcomes. Let the simple events of a sample space S be labeled
s1, s2, . . . , sn.
2.2 Definition A probability distribution P on S is a sequence of numbers p1, p2, . . . , pn that
are all non-negativeand sum to 1. The numberpi is interpretedas the probability of si being
the outcome of the experiment.
2.3 Definition An event E is a subset of the sample space S. The probability that event E
occurs, denoted P(E), is the sum of the probabilities pi of all simple events si which belong
to E. If si ∈ S, P({si}) is simply denoted by P(si).
2.4 Definition If E is an event, the complementary event is the set of simple events not be-
longing to E, denoted E.
2.5 Fact Let E ⊆ S be an event.
(i) 0 ≤ P(E) ≤ 1. Furthermore, P(S) = 1 and P(∅) = 0. (∅ is the empty set.)
(ii) P(E) = 1 − P(E).
c
1997 by CRC Press, Inc. — See accompanying notice at front of chapter.

§2.1 Probability theory 51
(iii) If the outcomes in S are equally likely, then P(E) = |E|
|S| .
2.6 Definition Two events E1 and E2 are called mutually exclusive if P(E1 ∩E2) = 0. That
is, the occurrence of one of the two events excludes the possibility that the other occurs.
2.7 Fact Let E1 and E2 be two events.
(i) If E1 ⊆ E2, then P(E1) ≤ P(E2).
(ii) P(E1 ∪ E2) + P(E1 ∩ E2) = P(E1) + P(E2). Hence, if E1 and E2 are mutually
exclusive, then P(E1 ∪ E2) = P(E1) + P(E2).
2.1.2 Conditional probability
2.8 Definition Let E1 and E2 be two events with P(E2) 0. The conditional probability of
E1 given E2, denoted P(E1|E2), is
P(E1|E2) =
P(E1 ∩ E2)
P(E2)
.
P(E1|E2) measures the probability of event E1 occurring, given that E2 has occurred.
2.9 Definition Events E1 and E2 are said to be independent if P(E1 ∩ E2) = P(E1)P(E2).
Observe that if E1 and E2 are independent,then P(E1|E2) = P(E1) andP(E2|E1) =
P(E2). That is, the occurrence of one event does not influence the likelihood of occurrence
of the other.
2.10 Fact (Bayes’ theorem) If E1 and E2 are events with P(E2) 0, then
P(E1|E2) =
P(E1)P(E2|E1)
P(E2)
.
2.1.3 Random variables
Let S be a sample space with probability distribution P.
2.11 Definition A random variable X is a function from the sample space S to the set of real
numbers; to each simple event si ∈ S, X assigns a real number X(si).
Since S is assumed to be finite, X can only take on a finite number of values.
2.12 Definition Let X be a randomvariable on S. The expected value or mean of X is E(X) =

si∈S X(si)P(si).
2.13 Fact Let X be a random variable on S. Then E(X) =

x∈R x · P(X = x).
2.14 Fact If X1, X2, . . . , Xm are random variables on S, and a1, a2, . . . , am are real numbers,
then E(
m
i=1 aiXi) =
m
i=1 aiE(Xi).
2.15 Definition The variance of a random variable X of mean μ is a non-negative number de-
fined by
Var(X) = E((X − μ)2
).
The standard deviation of X is the non-negative square root of Var(X).
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.

If a random variable has small variance then large deviations from the mean are un-
likely to be observed. This statement is made more precise below.
2.16 Fact (Chebyshev’s inequality) Let X be a random variable with mean μ = E(X) and
variance σ2
= Var(X). Then for any t 0,
P(|X − μ| ≥ t) ≤
σ2
t2
.
2.1.4 Binomial distribution
2.17 Definition Let n and k be non-negative integers. The binomial coefficient
n
k

is the num-
ber of different ways of choosing k distinct objects from a set of n distinct objects, where
the order of choice is not important.
2.18 Fact (properties of binomial coefficients) Let n and k be non-negative integers.
(i)
n
k

= n!
k!(n−k)! .
(ii)
n
k

=
n
n−k

.
(iii)
n+1
k+1

=
n
k

+
n
k+1

.
2.19 Fact (binomial theorem) For any real numbers a, b, and non-negativeinteger n, (a+b)n
=
n
k=0
n
k

ak
bn−k
.
2.20 Definition A Bernoulli trial is an experiment with exactly two possible outcomes, called
success and failure.
2.21 Fact Suppose that the probability of success on a particular Bernoulli trial is p. Then the
probability of exactly k successes in a sequence of n such independent trials is

n
k

pk
(1 − p)n−k
, for each 0 ≤ k ≤ n. (2.1)
2.22 Definition The probability distribution (2.1) is called the binomial distribution.
2.23 Fact The expected number of successes in a sequence of n independent Bernoulli trials,
with probability p of success in each trial, is np. The variance of the number of successes
is np(1 − p).
2.24 Fact (law of large numbers) Let X be the random variable denoting the fraction of suc-
cesses in n independent Bernoulli trials, with probability p of success in each trial. Then
for any 0,
P(|X − p| ) −→ 0, as n −→ ∞.
In other words, as n gets larger, the proportion of successes should be close to p, the
probability of success in each trial.
c

2.1.5 Birthday problems
2.25 Definition
(i) For positive integers m, n with m ≥ n, the number m(n)
is defined as follows:
m(n)
= m(m − 1)(m − 2) · · · (m − n + 1).
(ii) Let m, n be non-negative integers with m ≥ n. The Stirling number of the second
kind, denoted
m
n

, is

m
n

=
1
n!
n

k=0
(−1)n−k

n
k

km
,
with the exception that
0
0

= 1.
The symbol
m
n

counts the number of ways of partitioning a set of m objects into n
non-empty subsets.
2.26 Fact (classical occupancy problem) An urn has m balls numbered 1 to m. Suppose that n
balls are drawn from the urn one at a time, with replacement, and their numbers are listed.
The probability that exactly t different balls have been drawn is
P1(m, n, t) =

n
t

m(t)
mn
, 1 ≤ t ≤ n.
The birthday problem is a special case of the classical occupancy problem.
2.27 Fact (birthday problem) An urn has m balls numbered 1 to m. Suppose that n balls are
drawn from the urn one at a time, with replacement, and their numbers are listed.
(i) The probability of at least one coincidence (i.e., a ball drawn at least twice) is
P2(m, n) = 1 − P1(m, n, n) = 1 −
m(n)
mn
, 1 ≤ n ≤ m. (2.2)
If n = O(
√
m) (see Definition 2.55) and m −→ ∞, then
P2(m, n) −→ 1 − exp

−
n(n − 1)
2m
+ O

1
√
m

≈ 1 − exp

−
n2
2m

.
(ii) As m −→ ∞, the expected number of draws before a coincidence is

πm
2 .
The following explains why probability distribution (2.2) is referred to as the birthday
surprise or birthday paradox. The probability that at least 2 people in a room of 23 people
have the same birthday is P2(365, 23) ≈ 0.507, which is surprisingly large. The quantity
P2(365, n) also increases rapidly as n increases; for example, P2(365, 30) ≈ 0.706.
A different kind of problem is considered in Facts 2.28, 2.29, and 2.30 below. Suppose
that there are two urns, one containing m white balls numbered 1 to m, and the other con-
taining m red balls numbered 1 to m. First, n1 balls are selected from the first urn and their
numbers listed. Then n2 balls are selected from the second urn and their numbers listed.
Finally, the number of coincidences between the two lists is counted.
2.28 Fact (model A) If the balls from both urns are drawn one at a time, with replacement, then
the probability of at least one coincidence is
P3(m, n1, n2) = 1 −
1
mn1+n2

t1,t2
m(t1+t2)

n1
t1

n2
t2

,

where the summation is over all 0 ≤ t1 ≤ n1, 0 ≤ t2 ≤ n2. If n = n1 = n2, n = O(
√
m)
and m −→ ∞, then
P3(m, n1, n2) −→ 1 − exp

−
n2
m

1 + O

1
√
m

≈ 1 − exp

−
n2
m

.
2.29 Fact (model B) If the balls from both urns are drawn without replacement, then the prob-
ability of at least one coincidence is
P4(m, n1, n2) = 1 −
m(n1+n2)
m(n1)m(n2)
.
If n1 = O(
√
m), n2 = O(
√
m), and m −→ ∞, then
P4(m, n1, n2) −→ 1 − exp

−
n1n2
m

1 +
n1 + n2 − 1
2m
+ O

1
m

.
2.30 Fact (model C) If the n1 white balls are drawn one at a time, with replacement, and the n2
red balls are drawn without replacement, then the probability of at least one coincidence is
P5(m, n1, n2) = 1 −

1 −
n2
m
n1
.
If n1 = O(
√
m), n2 = O(
√
m), and m −→ ∞, then
P5(m, n1, n2) −→ 1 − exp

−
n1n2
m

1 + O

1
√
m

≈ 1 − exp

−
n1n2
m

.
2.1.6 Random mappings
2.31 Definition Let Fn denote the collection of all functions (mappings) from a finite domain
of size n to a finite codomain of size n.
Models where random elements of Fn are considered are called random mappings
models. In this section the only randommappingsmodel consideredis where every function
from Fn is equally likely to be chosen; such models arise frequently in cryptography and
algorithmic number theory. Note that |Fn| = nn
, whence the probability that a particular
function from Fn is chosen is 1/nn
.
2.32 Definition Let f be a function in Fn with domain and codomain equal to {1, 2, . . . , n}.
The functional graph of f is a directed graph whose points (or vertices) are the elements
{1, 2, . . . , n} and whose edges are the ordered pairs (x, f(x)) for all x ∈ {1, 2, . . . , n}.
2.33 Example (functionalgraph)Consider the functionf : {1, 2, . . . , 13} −→ {1, 2, . . . , 13}
defined by f(1) = 4, f(2) = 11, f(3) = 1, f(4) = 6, f(5) = 3, f(6) = 9, f(7) = 3,
f(8) = 11, f(9) = 1, f(10) = 2, f(11) = 10, f(12) = 4, f(13) = 7. The functional
graph of f is shown in Figure 2.1.
As Figure 2.1 illustrates, a functional graph may have several components (maximal
connected subgraphs), each component consisting of a directed cycle and some directed
trees attached to the cycle.
2.34 Fact As n tends to infinity, the following statements regarding the functional digraph of a
random function f from Fn are true:
(i) The expected number of components is 1
2 ln n.
c

13
7
5
3
12 4
1 9
6 8
11
2
10
Figure 2.1: A functional graph (see Example 2.33).
(ii) The expected number of points which are on the cycles is

πn/2.
(iii) The expected number of terminal points (points which have no preimages) is n/e.
(iv) The expected number of k-th iterate image points (x is a k-th iterate image point if
x = f(f(· · · f

k times
(y) · · · )) for some y) is (1 − τk)n, where the τk satisfy the recurrence
τ0 = 0, τk+1 = e−1+τk
for k ≥ 0.
2.35 Definition Let f be a random function from {1, 2, . . . , n} to {1, 2, . . . , n} and let u ∈
{1, 2, . . . , n}. Consider the sequence of points u0, u1, u2, . . . defined by u0 = u, ui =
f(ui−1) for i ≥ 1. In terms of the functional graph of f, this sequence describes a path that
connects to a cycle.
(i) The number of edges in the path is called the tail length of u, denoted λ(u).
(ii) The number of edges in the cycle is called the cycle length of u, denoted μ(u).
(iii) The rho-length of u is the quantity ρ(u) = λ(u) + μ(u).
(iv) The tree size of u is the number of edges in the maximal tree rooted on a cycle in the
component that contains u.
(v) The component size of u is the number of edges in the component that contains u.
(vi) The predecessors size of u is the number of iterated preimages of u.
2.36 Example The functional graph in Figure 2.1 has 2 components and 4 terminal points. The
point u = 3 has parameters λ(u) = 1, μ(u) = 4, ρ(u) = 5. The tree, component, and
predecessors sizes of u = 3 are 4, 9, and 3, respectively.
2.37 Fact As n tends to infinity, the following are the expectations of some parameters associ-
ated with a random point in {1, 2, . . . , n} and a random function from Fn: (i) tail length:

πn/8 (ii) cycle length:

πn/8 (iii) rho-length:

πn/2 (iv) tree size: n/3 (v) compo-
nent size: 2n/3 (vi) predecessors size:

πn/8.
2.38 Fact As n tends to infinity, the expectations of the maximum tail, cycle, and rho lengths in
a random function from Fn are c1
√
n, c2
√
n, and c3
√
n, respectively, where c1 ≈ 0.78248,
c2 ≈ 1.73746, and c3 ≈ 2.4149.
Facts 2.37 and 2.38 indicate that in the functional graph of a random function, most
points are grouped together in one giant component, and there is a small number of large
trees. Also, almost unavoidably, a cycle of length about
√
n arises after following a path of
length
√
n edges.

2.2 Information theory
2.2.1 Entropy
Let X be a random variable which takes on a finite set of values x1, x2, . . . , xn, with prob-
ability P(X = xi) = pi, where 0 ≤ pi ≤ 1 for each i, 1 ≤ i ≤ n, and where
n
i=1 pi = 1.
Also, let Y and Z be random variables which take on finite sets of values.
The entropy of X is a mathematical measure of the amount of information provided by
an observation of X. Equivalently, it is the uncertainity about the outcome before an obser-
vation of X. Entropy is also useful for approximating the average number of bits required
to encode the elements of X.
2.39 Definition The entropy or uncertainty of X is defined to be H(X) = −
n
i=1 pi lg pi =
n
i=1 pi lg

1
pi

where, by convention, pi · lg pi = pi · lg

1
pi

= 0 if pi = 0.
2.40 Fact (properties of entropy) Let X be a random variable which takes on n values.
(i) 0 ≤ H(X) ≤ lg n.
(ii) H(X) = 0 if and only if pi = 1 for some i, and pj = 0 for all j = i (that is, there is
no uncertainty of the outcome).
(iii) H(X) = lg n if and only if pi = 1/n for each i, 1 ≤ i ≤ n (that is, all outcomes are
equally likely).
2.41 Definition The joint entropy of X and Y is defined to be
H(X, Y ) = −

x,y
P(X = x, Y = y) lg(P(X = x, Y = y)),
where the summation indices x and y range over all values of X and Y , respectively. The
definition can be extended to any number of random variables.
2.42 Fact If X and Y are random variables, then H(X, Y ) ≤ H(X) + H(Y ), with equality if
and only if X and Y are independent.
2.43 Definition If X, Y are random variables, the conditional entropy of X given Y = y is
H(X|Y = y) = −

x
P(X = x|Y = y) lg(P(X = x|Y = y)),
where the summation index x ranges over all values of X. The conditional entropy of X
given Y , also called the equivocation of Y about X, is
H(X|Y ) =

y
P(Y = y)H(X|Y = y),
where the summation index y ranges over all values of Y .
2.44 Fact (properties of conditional entropy) Let X and Y be random variables.
(i) The quantity H(X|Y ) measures the amount of uncertainty remaining about X after
Y has been observed.
c

§2.3 Complexity theory 57
(ii) H(X|Y ) ≥ 0 and H(X|X) = 0.
(iii) H(X, Y ) = H(X) + H(Y |X) = H(Y ) + H(X|Y ).
(iv) H(X|Y ) ≤ H(X), with equality if and only if X and Y are independent.
2.2.2 Mutual information
2.45 Definition The mutual information or transinformation of random variables X and Y is
I(X; Y ) = H(X) − H(X|Y ). Similarly, the transinformation of X and the pair Y , Z is
defined to be I(X; Y, Z) = H(X) − H(X|Y, Z).
2.46 Fact (properties of mutual transinformation)
(i) The quantity I(X; Y ) can be thought of as the amount of information that Y reveals
about X. Similarly, the quantity I(X; Y, Z) can be thought of as the amount of in-
formation that Y and Z together reveal about X.
(ii) I(X; Y ) ≥ 0.
(iii) I(X; Y ) = 0 if and only if X and Y are independent (that is, Y contributes no in-
formation about X).
(iv) I(X; Y ) = I(Y ; X).
2.47 Definition The conditional transinformation of the pair X, Y given Z is defined to be
IZ(X; Y ) = H(X|Z) − H(X|Y, Z).
2.48 Fact (properties of conditional transinformation)
(i) The quantity IZ(X; Y ) can be interpreted as the amount of information that Y pro-
vides about X, given that Z has already been observed.
(ii) I(X; Y, Z) = I(X; Y ) + IY (X; Z).
(iii) IZ(X; Y ) = IZ(Y ; X).
2.3 Complexity theory
2.3.1 Basic definitions
The main goal of complexity theory is to providemechanisms for classifying computational
problems according to the resources needed to solve them. The classification should not
depend on a particular computational model, but rather should measure the intrinsic dif-
ficulty of the problem. The resources measured may include time, storage space, random
bits, number of processors, etc., but typically the main focus is time, and sometimes space.
2.49 Definition An algorithm is a well-defined computational procedure that takes a variable
input and halts with an output.

Of course, the term “well-defined computational procedure”is not mathematically pre-
cise. It can be made so by using formal computational models such as Turing machines,
random-access machines, or boolean circuits. Rather than get involved with the technical
intricacies of these models, it is simpler to think of an algorithm as a computer program
written in some specific programming language for a specific computer that takes a vari-
able input and halts with an output.
It is usually of interest to find the most efficient (i.e., fastest) algorithm for solving a
givencomputationalproblem. The time that an algorithmtakes to halt dependson the “size”
of the probleminstance. Also, the unit of time used shouldbe madeprecise, especially when
comparing the performance of two algorithms.
2.50 Definition The size of the input is the total number of bits needed to represent the input
in ordinary binary notation using an appropriate encoding scheme. Occasionally, the size
of the input will be the number of items in the input.
2.51 Example (sizes of some objects)
(i) The number of bits in the binary representation of a positive integer n is 1 + lg n
bits. For simplicity, the size of n will be approximated by lg n.
(ii) If f is a polynomial of degree at most k, each coefficient being a non-negativeinteger
at most n, then the size of f is (k + 1) lg n bits.
(iii) If A is a matrix with r rows, s columns, and with non-negative integer entries each
at most n, then the size of A is rs lg n bits.
2.52 Definition The running time of an algorithm on a particular input is the number of prim-
itive operations or “steps” executed.
Often a step is taken to mean a bit operation. For some algorithms it will be more con-
venient to take step to mean something else such as a comparison, a machine instruction, a
machine clock cycle, a modular multiplication, etc.
2.53 Definition The worst-case running time of an algorithm is an upper bound on the running
time for any input, expressed as a function of the input size.
2.54 Definition The average-case running time of an algorithm is the average running time
over all inputs of a fixed size, expressed as a function of the input size.
2.3.2 Asymptotic notation
It is often difficult to derive the exact running time of an algorithm. In such situations one
is forced to settle for approximations of the running time, and usually may only derive the
asymptotic running time. That is, one studies how the running time of the algorithm in-
creases as the size of the input increases without bound.
In what follows, the only functions considered are those which are defined on the posi-
tive integers and take on real values that are always positive from some point onwards. Let
f and g be two such functions.
2.55 Definition (order notation)
(i) (asymptotic upper bound) f(n) = O(g(n)) if there exists a positive constant c and a
positive integer n0 such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0.
c

(ii) (asymptotic lower bound) f(n) = Ω(g(n)) if there exists a positive constant c and a
positive integer n0 such that 0 ≤ cg(n) ≤ f(n) for all n ≥ n0.
(iii) (asymptotic tight bound) f(n) = Θ(g(n)) if there exist positive constants c1 and c2,
and a positive integer n0 such that c1g(n) ≤ f(n) ≤ c2g(n) for all n ≥ n0.
(iv) (o-notation) f(n) = o(g(n)) if for any positive constant c 0 there exists a constant
n0 0 such that 0 ≤ f(n) cg(n) for all n ≥ n0.
Intuitively, f(n) = O(g(n)) means that f grows no faster asymptotically than g(n) to
within a constant multiple, while f(n) = Ω(g(n)) means that f(n) grows at least as fast
asymptotically as g(n) to within a constant multiple. f(n) = o(g(n)) means that g(n) is an
upper bound for f(n) that is not asymptotically tight, or in other words, the function f(n)
becomes insignificant relative to g(n) as n gets larger. The expression o(1) is often used to
signify a function f(n) whose limit as n approaches ∞ is 0.
2.56 Fact (properties of order notation) For any functions f(n), g(n), h(n), and l(n), the fol-
lowing are true.
(i) f(n) = O(g(n)) if and only if g(n) = Ω(f(n)).
(ii) f(n) = Θ(g(n)) if and only if f(n) = O(g(n)) and f(n) = Ω(g(n)).
(iii) If f(n) = O(h(n)) and g(n) = O(h(n)), then (f + g)(n) = O(h(n)).
(iv) If f(n) = O(h(n)) and g(n) = O(l(n)), then (f · g)(n) = O(h(n)l(n)).
(v) (reflexivity) f(n) = O(f(n)).
(vi) (transitivity) If f(n) = O(g(n)) and g(n) = O(h(n)), then f(n) = O(h(n)).
2.57 Fact (approximations of some commonly occurring functions)
(i) (polynomial function) If f(n) is a polynomial of degree k with positive leading term,
then f(n) = Θ(nk
).
(ii) For any constant c 0, logc n = Θ(lg n).
(iii) (Stirling’s formula) For all integers n ≥ 1,
√
2πn
n
e
n
≤ n! ≤
√
2πn
n
e
n+(1/(12n))
.
Thus n! =
√
2πn
n
e
n
1 + Θ( 1
n )

. Also, n! = o(nn
) and n! = Ω(2n
).
(iv) lg(n!) = Θ(n lg n).
2.58 Example (comparative growth rates of some functions) Let and c be arbitrary constants
with 0 1 c. The following functions are listed in increasing order of their asymp-
totic growth rates:
1 ln ln n ln n exp(
√
ln n ln ln n) n
nc
nln n
cn
nn
ccn
.
2.3.3 Complexity classes
2.59 Definition A polynomial-time algorithm is an algorithm whose worst-case running time
function is of the form O(nk
), where n is the input size and k is a constant. Any algorithm
whose running time cannot be so bounded is called an exponential-time algorithm.
Roughly speaking, polynomial-time algorithms can be equated with good or efficient
algorithms, while exponential-time algorithms are considered inefficient. There are, how-
ever, some practical situations when this distinction is not appropriate. When considering
polynomial-timecomplexity, the degree of the polynomialis significant. For example, even

though an algorithm with a running time of O(nln ln n
), n being the input size, is asymptot-
ically slower that an algorithm with a running time of O(n100
), the former algorithm may
be faster in practice for smaller values of n, especially if the constants hidden by the big-O
notation are smaller. Furthermore, in cryptography, average-case complexity is more im-
portant than worst-case complexity — a necessary condition for an encryption scheme to
be considered secure is that the corresponding cryptanalysis problem is difficult on average
(or more precisely, almost always difficult), and not just for some isolated cases.
2.60 Definition A subexponential-time algorithm is an algorithm whose worst-case running
time function is of the form eo(n)
, where n is the input size.
A subexponential-timealgorithm is asymptotically faster than an algorithm whose run-
ning time is fully exponential in the input size, while it is asymptotically slower than a
polynomial-time algorithm.
2.61 Example (subexponential running time) Let A be an algorithm whose inputs are either
elements of a finite field Fq (see §2.6), or an integer q. If the expected running time of A is
of the form
Lq[α, c] = O

exp

(c + o(1))(ln q)α
(ln ln q)1−α

, (2.3)
where c is a positive constant, and α is a constant satisfying 0 α 1, then A is a
subexponential-time algorithm. Observe that for α = 0, Lq[0, c] is a polynomial in ln q,
while for α = 1, Lq[1, c] is a polynomial in q, and thus fully exponential in ln q.
For simplicity, the theory of computational complexity restricts its attention to deci-
sion problems, i.e., problems which have either YES or NO as an answer. This is not too
restrictive in practice, as all the computational problems that will be encountered here can
be phrased as decision problems in such a way that an efficient algorithm for the decision
problem yields an efficient algorithm for the computational problem, and vice versa.
2.62 Definition The complexity class P is the set of all decision problems that are solvable in
polynomial time.
2.63 Definition The complexity class NP is the set of all decision problems for which a YES
answer can be verified in polynomialtime given some extra information, called a certificate.
2.64 Definition The complexity class co-NP is the set of all decision problems for which a NO
answer can be verified in polynomial time using an appropriate certificate.
It must be emphasized that if a decision problem is in NP, it may not be the case that the
certificate of a YES answer can be easily obtained; what is asserted is that such a certificate
does exist, and, if known, can be used to efficiently verify the YES answer. The same is
true of the NO answers for problems in co-NP.
2.65 Example (problem in NP) Consider the following decision problem:
COMPOSITES
INSTANCE: A positive integer n.
QUESTION: Is n composite? That is, are there integers a, b 1 such that n = ab?
COMPOSITES belongs to NP because if an integer n is composite, then this fact can be
verified in polynomial time if one is given a divisor a of n, where 1 a n (the certificate
in this case consists of the divisor a). It is in fact also the case that COMPOSITES belongs
to co-NP. It is still unknown whether or not COMPOSITES belongs to P.
c

2.66 Fact P ⊆ NP and P ⊆ co-NP.
The following are among the outstanding unresolved questions in the subject of com-
plexity theory:
1. Is P = NP?
2. Is NP = co-NP?
3. Is P = NP ∩ co-NP?
Most experts are of the opinionthat the answer to each of the three questions is NO, although
nothing along these lines has been proven.
The notion of reducibility is useful when comparing the relative difficulties of prob-
lems.
2.67 Definition Let L1 and L2 be two decision problems. L1 is said to polytime reduce to L2,
written L1 ≤P L2, if there is an algorithm that solves L1 which uses, as a subroutine, an
algorithm for solving L2, and which runs in polynomial time if the algorithm for L2 does.
Informally, if L1 ≤P L2, then L2 is at least as difficult as L1, or, equivalently, L1 is
no harder than L2.
2.68 Definition Let L1 and L2 be two decision problems. If L1 ≤P L2 and L2 ≤P L1, then
L1 and L2 are said to be computationally equivalent.
2.69 Fact Let L1, L2, and L3 be three decision problems.
(i) (transitivity) If L1 ≤P L2 and L2 ≤P L3, then L1 ≤P L3.
(ii) If L1 ≤P L2 and L2 ∈ P, then L1 ∈ P.
2.70 Definition A decision problem L is said to be NP-complete if
(i) L ∈ NP, and
(ii) L1 ≤P L for every L1 ∈ NP.
The class of all NP-complete problems is denoted by NPC.
NP-complete problems are the hardest problems in NP in the sense that they are at
least as difficult as every other problem in NP. There are thousands of problems drawn from
diverse fields such as combinatorics, number theory, and logic, that are known to be NP-
complete.
2.71 Example (subset sum problem) The subset sum problem is the following: given a set of
positive integers {a1, a2, . . . , an} and a positive integer s, determine whether or not there
is a subset of the ai that sum to s. The subset sum problem is NP-complete.
2.72 Fact Let L1 and L2 be two decision problems.
(i) If L1 is NP-complete and L1 ∈ P, then P = NP.
(ii) If L1 ∈ NP, L2 is NP-complete, and L2 ≤P L1, then L1 is also NP-complete.
(iii) If L1 is NP-complete and L1 ∈ co-NP, then NP = co-NP.
By Fact 2.72(i), if a polynomial-time algorithm is found for any single NP-complete
problem, then it is the case that P = NP, a result that would be extremely surprising. Hence,
a proof that a problem is NP-complete provides strong evidence for its intractability. Fig-
ure 2.2 illustrates what is widely believed to be the relationship between the complexity
classes P, NP, co-NP, and NPC.
Fact 2.72(ii) suggests the following procedure for proving that a decision problem L1
is NP-complete:

co-NP
NPC
NP
P
NP ∩ co-NP
Figure 2.2: Conjectured relationship between the complexity classes P, NP, co-NP, and NPC.
1. Prove that L1 ∈ NP.
2. Select a problem L2 that is known to be NP-complete.
3. Prove that L2 ≤P L1.
2.73 Definition A problem is NP-hard if there exists some NP-completeproblemthat polytime
reduces to it.
Note that the NP-hard classification is not restricted to only decision problems. Ob-
serve also that an NP-complete problem is also NP-hard.
2.74 Example (NP-hard problem) Given positive integers a1, a2, . . . , an and a positive inte-
ger s, the computational version of the subset sum problem would ask to actually find a
subset of the ai which sums to s, provided that such a subset exists. This problem is NP-
hard.
2.3.4 Randomized algorithms
The algorithms studied so far in this section have been deterministic; such algorithms fol-
low the same execution path (sequence of operations) each time they execute with the same
input. By contrast, a randomized algorithm makes random decisions at certain points in
the execution; hence their execution paths may differ each time they are invoked with the
same input. The random decisions are based upon the outcome of a random number gen-
erator. Remarkably, there are many problems for which randomized algorithms are known
that are more efficient, both in terms of time and space, than the best known deterministic
algorithms.
Randomized algorithms for decision problems can be classified according to the prob-
ability that they return the correct answer.
2.75 Definition Let A be a randomized algorithm for a decision problem L, and let I denote
an arbitrary instance of L.
(i) A has 0-sided error if P(A outputs YES | I’s answer is YES ) = 1, and
P(A outputs YES | I’s answer is NO ) = 0.
(ii) A has 1-sided error if P(A outputs YES | I’s answer is YES ) ≥ 1
2 , and
P(A outputs YES | I’s answer is NO ) = 0.
c

§2.4 Number theory 63
(iii) A has 2-sided error if P(A outputs YES | I’s answer is YES ) ≥ 2
3 , and
P(A outputs YES | I’s answer is NO ) ≤ 1
3 .
The number 1
2 in the definition of 1-sided error is somewhat arbitrary and can be re-
placed by any positive constant. Similarly, the numbers 2
3 and 1
3 in the definition of 2-sided
error, can be replaced by 1
2 + and 1
2 − , respectively, for any constant , 0 1
2 .
2.76 Definition The expected running time of a randomized algorithm is an upper bound on the
expected running time for each input (the expectation being over all outputs of the random
number generator used by the algorithm), expressed as a function of the input size.
The important randomized complexity classes are defined next.
2.77 Definition (randomized complexity classes)
(i) The complexity class ZPP (“zero-sided probabilistic polynomial time”) is the set of
all decision problems for which there is a randomized algorithm with 0-sided error
which runs in expected polynomial time.
(ii) The complexity class RP (“randomized polynomial time”) is the set of all decision
problems for which there is a randomized algorithm with 1-sided error which runs in
(worst-case) polynomial time.
(iii) The complexity class BPP (“bounded error probabilistic polynomial time”) is the set
of all decision problems for which there is a randomized algorithm with 2-sided error
which runs in (worst-case) polynomial time.
2.78 Fact P ⊆ ZPP ⊆ RP ⊆ BPP and RP ⊆ NP.
2.4 Number theory
2.4.1 The integers
The set of integers {. . . , −3, −2, −1, 0, 1, 2, 3, . . .} is denoted by the symbol Z.
2.79 Definition Let a, b be integers. Then a divides b (equivalently: a is a divisor of b, or a is
a factor of b) if there exists an integer c such that b = ac. If a divides b, then this is denoted
by a|b.
2.80 Example (i) −3|18, since 18 = (−3)(−6). (ii) 173|0, since 0 = (173)(0).
The following are some elementary properties of divisibility.
2.81 Fact (properties of divisibility) For all a, b, c ∈ Z, the following are true:
(i) a|a.
(ii) If a|b and b|c, then a|c.
(iii) If a|b and a|c, then a|(bx + cy) for all x, y ∈ Z.
(iv) If a|b and b|a, then a = ±b.

2.82 Definition (division algorithm for integers) If a and b are integers with b ≥ 1, then or-
dinary long division of a by b yields integers q (the quotient) and r (the remainder) such
that
a = qb + r, where 0 ≤ r b.
Moreover, q and r are unique. The remainder of the division is denoted a mod b, and the
quotient is denoted a div b.
2.83 Fact Let a, b ∈ Z with b = 0. Then a div b = a/b and a mod b = a − ba/b.
2.84 Example If a = 73, b = 17, then q = 4 and r = 5. Hence 73 mod 17 = 5 and
73 div 17 = 4.
2.85 Definition An integer c is a common divisor of a and b if c|a and c|b.
2.86 Definition A non-negative integer d is the greatest common divisor of integers a and b,
denoted d = gcd(a, b), if
(i) d is a common divisor of a and b; and
(ii) whenever c|a and c|b, then c|d.
Equivalently, gcd(a, b) is the largest positive integer that divides both a and b, with the ex-
ception that gcd(0, 0) = 0.
2.87 Example The commondivisors of 12 and 18 are {±1, ±2, ±3, ±6},and gcd(12, 18) = 6.

2.88 Definition A non-negative integer d is the least common multiple of integers a and b, de-
noted d = lcm(a, b), if
(i) a|d and b|d; and
(ii) whenever a|c and b|c, then d|c.
Equivalently, lcm(a, b) is the smallest non-negative integer divisible by both a and b.
2.89 Fact If a and b are positive integers, then lcm(a, b) = a · b/ gcd(a, b).
2.90 Example Since gcd(12, 18) = 6, it follows that lcm(12, 18) = 12 · 18/6 = 36.
2.91 Definition Two integers a and b are said to be relatively prime or coprime if gcd(a, b) = 1.
2.92 Definition An integer p ≥ 2 is said to be prime if its only positive divisors are 1 and p.
Otherwise, p is called composite.
The following are some well known facts about prime numbers.
2.93 Fact If p is prime and p|ab, then either p|a or p|b (or both).
2.94 Fact There are an infinite number of prime numbers.
2.95 Fact (prime number theorem) Let π(x) denote the number of prime numbers ≤ x. Then
lim
x→∞
π(x)
x/ ln x
= 1.
c

This means that for large values of x, π(x) is closely approximated by the expres-
sion x/ ln x. For instance, when x = 1010
, π(x) = 455, 052, 511, whereas x/ ln x =
434, 294, 481. A more explicit estimate for π(x) is given below.
2.96 Fact Let π(x) denote the number of primes ≤ x. Then for x ≥ 17
π(x)
x
ln x
and for x 1
π(x) 1.25506
x
lnx
.
2.97 Fact (fundamental theorem of arithmetic) Every integer n ≥ 2 has a factorization as a
product of prime powers:
n = pe1
1 pe2
2 · · · pek
k ,
where the pi are distinct primes, and the ei are positive integers. Furthermore, the factor-
ization is unique up to rearrangement of factors.
2.98 Fact If a = pe1
1 pe2
2 · · · pek
k , b = pf1
1 pf2
2 · · · pfk
k , where each ei ≥ 0 and fi ≥ 0, then
gcd(a, b) = p
min(e1,f1)
1 p
min(e2,f2)
2 · · · p
min(ek,fk)
k
and
lcm(a, b) = p
max(e1,f1)
1 p
max(e2,f2)
2 · · · p
max(ek,fk)
k .
2.99 Example Let a = 4864 = 28
· 19, b = 3458 = 2 · 7 · 13 · 19. Then gcd(4864, 3458) =
2 · 19 = 38 and lcm(4864, 3458) = 28
· 7 · 13 · 19 = 442624.
2.100 Definition For n ≥ 1, let φ(n) denote the number of integers in the interval [1, n] which
are relatively prime to n. The function φ is called the Euler phi function (or the Euler totient
function).
2.101 Fact (properties of Euler phi function)
(i) If p is a prime, then φ(p) = p − 1.
(ii) The Euler phi function is multiplicative. That is, if gcd(m, n) = 1, then φ(mn) =
φ(m) · φ(n).
(iii) If n = pe1
1 pe2
2 · · · pek
k is the prime factorization of n, then
φ(n) = n

1 −
1
p1

1 −
1
p2

· · ·

1 −
1
pk

.
Fact 2.102 gives an explicit lower bound for φ(n).
2.102 Fact For all integers n ≥ 5,
φ(n)
n
6 ln ln n
.

2.4.2 Algorithms in Z
Let a and b be non-negative integers, each less than or equal to n. Recall (Example 2.51)
that the number of bits in the binary representation of n is lg n + 1, and this number is
approximated by lg n. The number of bit operations for the four basic integer operations of
addition, subtraction, multiplication, and division using the classical algorithms is summa-
rized in Table 2.1. These algorithms are studied in more detail in §14.2. More sophisticated
techniques for multiplication and division have smaller complexities.
Operation Bit complexity
Addition a + b O(lg a + lg b) = O(lg n)
Subtraction a − b O(lg a + lg b) = O(lg n)
Multiplication a · b O((lg a)(lg b)) = O((lg n)2
)
Division a = qb + r O((lg q)(lg b)) = O((lg n)2
)
Table 2.1: Bit complexity of basic operations in Z.
The greatest common divisor of two integers a and b can be computed via Fact 2.98.
However, computing a gcd by first obtaining prime-power factorizations does not result in
an efficient algorithm, as the problem of factoring integers appears to be relatively diffi-
cult. The Euclidean algorithm (Algorithm 2.104) is an efficient algorithm for computing
the greatest common divisor of two integers that does not require the factorization of the
integers. It is based on the following simple fact.
2.103 Fact If a and b are positive integers with a b, then gcd(a, b) = gcd(b, a mod b).
2.104 Algorithm Euclidean algorithm for computing the greatest common divisor of two integers
INPUT: two non-negative integers a and b with a ≥ b.
OUTPUT: the greatest common divisor of a and b.
1. While b = 0 do the following:
1.1 Set r←a mod b, a←b, b←r.
2. Return(a).
2.105 Fact Algorithm 2.104 has a running time of O((lg n)2
) bit operations.
2.106 Example (Euclidean algorithm) The following are the division steps of Algorithm 2.104
for computing gcd(4864, 3458) = 38:
4864 = 1 · 3458 + 1406
3458 = 2 · 1406 + 646
1406 = 2 · 646 + 114
646 = 5 · 114 + 76
114 = 1 · 76 + 38
76 = 2 · 38 + 0.
c

The Euclidean algorithm can be extended so that it not only yields the greatest common
divisor d of two integers a and b, but also integers x and y satisfying ax + by = d.
2.107 Algorithm Extended Euclidean algorithm
INPUT: two non-negative integers a and b with a ≥ b.
OUTPUT: d = gcd(a, b) and integers x, y satisfying ax + by = d.
1. If b = 0 then set d←a, x←1, y←0, and return(d,x,y).
2. Set x2←1, x1←0, y2←0, y1←1.
3. While b 0 do the following:
3.1 q←a/b, r←a − qb, x←x2 − qx1, y←y2 − qy1.
3.2 a←b, b←r, x2←x1, x1←x, y2←y1, and y1←y.
4. Set d←a, x←x2, y←y2, and return(d,x,y).
) bit operations.
2.109 Example (extended Euclidean algorithm) Table 2.2 shows the steps of Algorithm 2.107
with inputs a = 4864 and b = 3458. Hence gcd(4864, 3458) = 38 and (4864)(32) +
(3458)(−45) = 38.
q r x y a b x2 x1 y2 y1
− − − − 4864 3458 1 0 0 1
1 1406 1 −1 3458 1406 0 1 1 −1
2 646 −2 3 1406 646 1 −2 −1 3
2 114 5 −7 646 114 −2 5 3 −7
5 76 −27 38 114 76 5 −27 −7 38
1 38 32 −45 76 38 −27 32 38 −45
2 0 −91 128 38 0 32 −91 −45 128
Table 2.2: Extended Euclidean algorithm (Algorithm 2.107) with inputs a = 4864, b = 3458.
Efficient algorithms for gcd and extended gcd computations are further studied in §14.4.
2.4.3 The integers modulo n
Let n be a positive integer.
2.110 Definition If a and b are integers, then a is said to be congruent to b modulo n, written
a ≡ b (mod n), if n divides (a−b). The integer n is called the modulus of the congruence.
2.111 Example (i) 24 ≡ 9 (mod 5) since 24 − 9 = 3 · 5.
(ii) −11 ≡ 17 (mod 7) since −11 − 17 = −4 · 7.
2.112 Fact (properties of congruences) For all a, a1, b, b1, c ∈ Z, the following are true.
(i) a ≡ b (mod n) if and only if a and b leave the same remainder when divided by n.
(ii) (reflexivity) a ≡ a (mod n).
(iii) (symmetry) If a ≡ b (mod n) then b ≡ a (mod n).

(iv) (transitivity) If a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n).
(v) If a ≡ a1 (mod n) and b ≡ b1 (mod n), then a + b ≡ a1 + b1 (mod n) and
ab ≡ a1b1 (mod n).
The equivalence class of an integer a is the set of all integers congruent to a modulo
n. From properties (ii), (iii), and (iv) above, it can be seen that for a fixed n the relation of
congruence modulo n partitions Z into equivalence classes. Now, if a = qn + r, where
0 ≤ r n, then a ≡ r (mod n). Hence each integer a is congruent modulo n to a unique
integer between 0 and n−1, called the least residue of a modulo n. Thus a and r are in the
same equivalence class, and so r may simply be used to represent this equivalence class.
2.113 Definition The integers modulo n, denoted Zn, is the set of (equivalence classes of) in-
tegers {0, 1, 2, . . . , n − 1}. Addition, subtraction, and multiplication in Zn are performed
modulo n.
2.114 Example Z25 = {0, 1, 2, . . . , 24}. In Z25, 13 + 16 = 4, since 13 + 16 = 29 ≡ 4
(mod 25). Similarly, 13 · 16 = 8 in Z25.
2.115 Definition Let a ∈ Zn. The multiplicative inverse of a modulo n is an integer x ∈ Zn
such that ax ≡ 1 (mod n). If such an x exists, then it is unique, and a is said to be invert-
ible, or a unit; the inverse of a is denoted by a−1
.
2.116 Definition Let a, b ∈ Zn. Division of a by b modulo n is the product of a and b−1
modulo
n, and is only defined if b is invertible modulo n.
2.117 Fact Let a ∈ Zn. Then a is invertible if and only if gcd(a, n) = 1.
2.118 Example The invertible elements in Z9 are 1, 2, 4, 5, 7, and 8. For example, 4−1
= 7
because 4 · 7 ≡ 1 (mod 9).
The following is a generalization of Fact 2.117.
2.119 Fact Let d = gcd(a, n). The congruence equation ax ≡ b (mod n) has a solution x if
and only if d divides b, in which case there are exactly d solutions between 0 and n − 1;
these solutions are all congruent modulo n/d.
2.120 Fact (Chinese remainder theorem, CRT) If the integers n1, n2, . . . , nk are pairwise rela-
tively prime, then the system of simultaneous congruences
x ≡ a1 (mod n1)
x ≡ a2 (mod n2)
.
.
.
x ≡ ak (mod nk)
has a unique solution modulo n = n1n2 · · · nk.
2.121 Algorithm (Gauss’s algorithm) The solution x to the simultaneous congruences in the
Chinese remainder theorem (Fact 2.120) may be computed as x =
k
i=1 aiNiMi mod n,
where Ni = n/ni and Mi = N−1
i mod ni. These computations can be performed in
O((lg n)2
) bit operations.
c

Another efficient practical algorithm for solving simultaneous congruences in the Chinese
remainder theorem is presented in §14.5.
2.122 Example The pair of congruences x ≡ 3 (mod 7), x ≡ 7 (mod 13) has a unique solu-
tion x ≡ 59 (mod 91).
2.123 Fact If gcd(n1, n2) = 1, then the pair of congruencesx ≡ a (mod n1), x ≡ a (mod n2)
has a unique solution x ≡ a (mod n1n2).
2.124 Definition The multiplicative group of Zn is Z∗
n = {a ∈ Zn | gcd(a, n) = 1}. In
particular, if n is a prime, then Z∗
n = {a | 1 ≤ a ≤ n − 1}.
2.125 Definition The order of Z∗
n is defined to be the number of elements in Z∗
n, namely |Z∗
n|.
It follows from the definition of the Euler phi function (Definition 2.100) that |Z∗
n| =
φ(n). Note also that if a ∈ Z∗
n and b ∈ Z∗
n, then a · b ∈ Z∗
n, and so Z∗
n is closed under
multiplication.
2.126 Fact Let n ≥ 2 be an integer.
(i) (Euler’s theorem) If a ∈ Z∗
n, then aφ(n)
≡ 1 (mod n).
(ii) If n is a product of distinct primes, and if r ≡ s (mod φ(n)), then ar
≡ as
(mod n)
for all integers a. In other words, when working modulo such an n, exponents can
be reduced modulo φ(n).
A special case of Euler’s theorem is Fermat’s (little) theorem.
2.127 Fact Let p be a prime.
(i) (Fermat’s theorem) If gcd(a, p) = 1, then ap−1
≡ 1 (mod p).
(ii) If r ≡ s (mod p − 1), then ar
≡ as
(mod p) for all integers a. In other words,
when working modulo a prime p, exponents can be reduced modulo p − 1.
(iii) In particular, ap
≡ a (mod p) for all integers a.
2.128 Definition Let a ∈ Z∗
n. The order of a, denoted ord(a), is the least positive integer t such
that at
≡ 1 (mod n).
2.129 Fact If the order of a ∈ Z∗
n is t, and as
≡ 1 (mod n), then t divides s. In particular,
t|φ(n).
2.130 Example Let n = 21. Then Z∗
21 = {1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20}. Note that
φ(21) = φ(7)φ(3) = 12 = |Z∗
21|. The orders of elements in Z∗
21 are listed in Table 2.3.
a ∈ Z∗
21 1 2 4 5 8 10 11 13 16 17 19 20
order of a 1 6 3 6 2 6 6 2 3 6 6 2
Table 2.3: Orders of elements in Z∗
21.
2.131 Definition Let α ∈ Z∗
n. If the order of α is φ(n), then α is said to be a generator or a
primitive element of Z∗
n. If Z∗
n has a generator, then Z∗
n is said to be cyclic.

2.132 Fact (properties of generators of Z∗
n)
(i) Z∗
n has a generator if and only if n = 2, 4, pk
or 2pk
, where p is an odd prime and
k ≥ 1. In particular, if p is a prime, then Z∗
p has a generator.
(ii) If α is a generator of Z∗
n, then Z∗
n = {αi
mod n | 0 ≤ i ≤ φ(n) − 1}.
(iii) Suppose that α is a generator of Z∗
n. Then b = αi
mod n is also a generator of Z∗
n
if and only if gcd(i, φ(n)) = 1. It follows that if Z∗
n is cyclic, then the number of
generators is φ(φ(n)).
(iv) α ∈ Z∗
n is a generator of Z∗
n if and only if αφ(n)/p
≡ 1 (mod n) for each prime
divisor p of φ(n).
2.133 Example Z∗
21 is not cyclic since it does not contain an element of order φ(21) = 12 (see
Table 2.3); note that 21 does not satisfy the condition of Fact 2.132(i). On the other hand,
Z∗
25 is cyclic, and has a generator α = 2.
2.134 Definition Let a ∈ Z∗
n. a is said to be a quadratic residue modulo n, or a square modulo
n, if there exists an x ∈ Z∗
n such that x2
≡ a (mod n). If no such x exists, then a is called
a quadratic non-residue modulo n. The set of all quadratic residues modulo n is denoted
by Qn and the set of all quadratic non-residues is denoted by Qn.
Note that by definition 0 ∈ Z∗
n, whence 0 ∈ Qn and 0 ∈ Qn.
2.135 Fact Let p be an odd prime and let α be a generator of Z∗
p. Then a ∈ Z∗
p is a quadratic
residue modulo p if and only if a = αi
mod p, where i is an even integer. It follows that
|Qp| = (p − 1)/2 and |Qp| = (p − 1)/2; that is, half of the elements in Z∗
p are quadratic
residues and the other half are quadratic non-residues.
2.136 Example α = 6 is a generator of Z∗
13. The powers of α are listed in the following table.
i 0 1 2 3 4 5 6 7 8 9 10 11
αi
mod 13 1 6 10 8 9 2 12 7 3 5 4 11
Hence Q13 = {1, 3, 4, 9, 10, 12} and Q13 = {2, 5, 6, 7, 8, 11}.
2.137 Fact Let n be a product of two distinct odd primes p and q, n = pq. Then a ∈ Z∗
n is a
quadratic residue modulo n if and only if a ∈ Qp and a ∈ Qq. It follows that |Qn| =
|Qp| · |Qq| = (p − 1)(q − 1)/4 and |Qn| = 3(p − 1)(q − 1)/4.
2.138 Example Let n = 21. Then Q21 = {1, 4, 16} and Q21 = {2, 5, 8, 10, 11, 13, 17, 19, 20}.

2.139 Definition Let a ∈ Qn. If x ∈ Z∗
n satisfies x2
≡ a (mod n), then x is called a square
root of a modulo n.
2.140 Fact (number of square roots)
(i) If p is an odd prime and a ∈ Qp, then a has exactly two square roots modulo p.
(ii) More generally, let n = pe1
1 pe2
2 · · · pek
k where the pi are distinct odd primes and ei ≥
1. If a ∈ Qn, then a has precisely 2k
distinct square roots modulo n.
2.141 Example The square roots of 12 modulo 37 are 7 and 30. The square roots of 121 modulo
315 are 11, 74, 101, 151, 164, 214, 241, and 304.
c

2.4.4 Algorithms in Zn
Let n be a positive integer. As before, the elements of Zn will be represented by the integers
{0, 1, 2, . . . , n − 1}.
Observe that if a, b ∈ Zn, then
(a + b) mod n =

a + b, if a + b n,
a + b − n, if a + b ≥ n.
Hence modular addition (and subtraction) can be performed without the need of a long di-
vision. Modular multiplication of a and b may be accomplished by simply multiplying a
and b as integers, and then taking the remainder of the result after division by n. Inverses
in Zn can be computed using the extended Euclidean algorithm as next described.
2.142 Algorithm Computing multiplicative inverses in Zn
INPUT: a ∈ Zn.
OUTPUT: a−1
mod n, provided that it exists.
1. Use the extendedEuclideanalgorithm (Algorithm2.107)to find integersx and y such
that ax + ny = d, where d = gcd(a, n).
2. If d 1, then a−1
mod n does not exist. Otherwise, return(x).
Modular exponentiation can be performed efficiently with the repeated square-and-
multiply algorithm (Algorithm 2.143), which is crucial for many cryptographic protocols.
One version of this algorithm is based on the following observation. Let the binary repre-
sentation of k be
t
i=0 ki2i
, where each ki ∈ {0, 1}. Then
ak
=
t

i=0
aki2i
= (a20
)k0
(a21
)k1
· · · (a2t
)kt
.
2.143 Algorithm Repeated square-and-multiply algorithm for exponentiation in Zn
INPUT: a ∈ Zn, and integer 0 ≤ k n whose binary representation is k =
t
i=0 ki2i
.
OUTPUT: ak
mod n.
1. Set b←1. If k = 0 then return(b).
2. Set A←a.
3. If k0 = 1 then set b←a.
4. For i from 1 to t do the following:
4.1 Set A←A2
mod n.
4.2 If ki = 1 then set b←A · b mod n.
5. Return(b).
2.144 Example (modularexponentiation)Table 2.4 shows the steps involvedin the computation
of 5596
mod 1234 = 1013.
The number of bit operations for the basic operations in Zn is summarized in Table 2.5.
Efficient algorithms for performing modular multiplication and exponentiation are further
examined in §14.3 and §14.6.

i 0 1 2 3 4 5 6 7 8 9
ki 0 0 1 0 1 0 1 0 0 1
A 5 25 625 681 1011 369 421 779 947 925
b 1 1 625 625 67 67 1059 1059 1059 1013
Table 2.4: Computation of 5596
mod 1234.
Operation Bit complexity
Modular addition (a + b) mod n O(lg n)
Modular subtraction (a − b) mod n O(lg n)
Modular multiplication (a · b) mod n O((lg n)2
)
Modular inversion a−1
mod n O((lg n)2
)
Modular exponentiation ak
mod n, k n O((lg n)3
)
Table 2.5: Bit complexity of basic operations in Zn.
2.4.5 The Legendre and Jacobi symbols
The Legendre symbol is a useful tool for keeping track of whether or not an integer a is a
quadratic residue modulo a prime p.
2.145 Definition Let p be an odd prime and a an integer. The Legendre symbol
a
p

is defined
to be

a
p

=
⎧
⎨
⎩
0, if p|a,
1, if a ∈ Qp,
−1, if a ∈ Qp.
2.146 Fact (properties of Legendre symbol) Let p be an odd prime and a, b ∈ Z. Then the Leg-
endre symbol has the following properties:
(i)
a
p

≡ a(p−1)/2
(mod p). In particular,
1
p

= 1 and
−1
p

= (−1)(p−1)/2
. Hence
−1 ∈ Qp if p ≡ 1 (mod 4), and −1 ∈ Qp if p ≡ 3 (mod 4).
(ii)
ab
p

=
a
p
b
p

. Hence if a ∈ Z∗
p, then
a2
p

= 1.
(iii) If a ≡ b (mod p), then
a
p

=
b
p

.
(iv)
2
p

= (−1)(p2
−1)/8
. Hence
2
p

= 1 if p ≡ 1 or 7 (mod 8), and
2
p

= −1 if p ≡ 3
or 5 (mod 8).
(v) (law of quadratic reciprocity) If q is an odd prime distinct from p, then

p
q

=

q
p

(−1)(p−1)(q−1)/4
.
In other words,
p
q

=
q
p

unless both p and q are congruent to 3 modulo 4, in which
case
p
q

= −
q
p

.
The Jacobi symbol is a generalization of the Legendre symbol to integers n which are
odd but not necessarily prime.
c

2.147 Definition Let n ≥ 3 be odd with prime factorization n = pe1
1 pe2
2 · · · pek
k . Then the Jacobi
symbol
a
n

is defined to be

a
n

=

a
p1
e1

a
p2
e2
· · ·

a
pk
ek
.
Observe that if n is prime, then the Jacobi symbol is just the Legendre symbol.
2.148 Fact (properties of Jacobi symbol) Let m ≥ 3, n ≥ 3 be odd integers, and a, b ∈ Z. Then
the Jacobi symbol has the following properties:
(i)
a
n

= 0, 1, or − 1. Moreover,
a
n

= 0 if and only if gcd(a, n) = 1.
(ii)
ab
n

=
a
n
b
n

. Hence if a ∈ Z∗
n, then
a2
n

= 1.
(iii)
a
mn

=
a
m
a
n

.
(iv) If a ≡ b (mod n), then
a
n

=
b
n

.
(v)
1
n

= 1.
(vi)
−1
n

= (−1)(n−1)/2
. Hence
−1
n

= 1 if n ≡ 1 (mod 4), and
−1
n

= −1 if n ≡ 3
(mod 4).
(vii)
2
n

= (−1)(n2
−1)/8
. Hence
2
n

= 1 if n ≡ 1 or 7 (mod 8), and
2
n

= −1 if
n ≡ 3 or 5 (mod 8).
(viii)
m
n

=
n
m

(−1)(m−1)(n−1)/4
. In other words,
m
n

=
n
m

unless both m and n are
congruent to 3 modulo 4, in which case
m
n

= −
n
m

.
By properties of the Jacobi symbol it follows that if n is odd and a = 2e
a1 where a1
is odd, then

a
n

=

2e
n

a1
n

=

2
n
e
n mod a1
a1

(−1)(a1−1)(n−1)/4
.
This observation yields the following recursive algorithm for computing
a
n

, which does
not require the prime factorization of n.
2.149 Algorithm Jacobi symbol (and Legendre symbol) computation
JACOBI(a,n)
INPUT: an odd integer n ≥ 3, and an integer a, 0 ≤ a n.
OUTPUT: the Jacobi symbol
a
n

(and hence the Legendre symbol when n is prime).
1. If a = 0 then return(0).
2. If a = 1 then return(1).
3. Write a = 2e
a1, where a1 is odd.
4. If e is even then set s←1. Otherwise set s←1 if n ≡ 1 or 7 (mod 8), or set s← − 1
if n ≡ 3 or 5 (mod 8).
5. If n ≡ 3 (mod 4) and a1 ≡ 3 (mod 4) then set s← − s.
6. Set n1←n mod a1.
7. If a1 = 1 then return(s); otherwise return(s · JACOBI(n1,a1)).
) bit operations.

2.151 Remark (finding quadratic non-residues modulo a prime p) Let p denote an odd prime.
Even though it is known that half of the elements in Z∗
p are quadratic non-residues modulo
p (see Fact 2.135), there is no deterministic polynomial-time algorithm known for finding
one. A randomized algorithm for finding a quadratic non-residueis to simply select random
integers a ∈ Z∗
p until one is found satisfying
a
p

= −1. The expected number iterations
before a non-residueis found is 2, and hence the proceduretakes expected polynomial-time.
2.152 Example (Jacobi symbol computation) For a = 158 and n = 235, Algorithm 2.149 com-
putes the Jacobi symbol
158
235

as follows:

158
235

=

2
235

79
235

= (−1)

235
79

(−1)78·234/4
=

77
79

=

79
77

(−1)76·78/4
=

2
77

= −1.
Unlike the Legendre symbol, the Jacobi symbol
a
n

does not reveal whether or not a
is a quadratic residue modulo n. It is indeed true that if a ∈ Qn, then
a
n

= 1. However,
a
n

= 1 does not imply that a ∈ Qn.
2.153 Example (quadratic residues and non-residues) Table 2.6 lists the elements in Z∗
21 and
their Jacobi symbols. Recall from Example 2.138 that Q21 = {1, 4, 16}. Observe that
5
21

= 1 but 5 ∈ Q21.
a ∈ Z∗
21 1 2 4 5 8 10 11 13 16 17 19 20
a2
mod n 1 4 16 4 1 16 16 1 4 16 4 1
a
3

1 −1 1 −1 −1 1 −1 1 1 −1 1 −1
a
7

1 1 1 −1 1 −1 1 −1 1 −1 −1 −1
a
21

1 −1 1 1 −1 −1 −1 −1 1 1 −1 1
Table 2.6: Jacobi symbols of elements in Z∗
21.
2.154 Definition Let n ≥ 3 be an odd integer, and let Jn = {a ∈ Z∗
n |
a
n

= 1}. The set of
pseudosquares modulo n, denoted
Qn, is defined to be the set Jn − Qn.
2.155 Fact Let n = pq be a product of two distinct odd primes. Then |Qn| = |
Qn| = (p −
1)(q − 1)/4; that is, half of the elements in Jn are quadratic residues and the other half are
pseudosquares.
2.4.6 Blum integers
2.156 Definition A Blum integer is a composite integer of the form n = pq, where p and q are
distinct primes each congruent to 3 modulo 4.
2.157 Fact Let n = pq be a Blum integer, and let a ∈ Qn. Then a has precisely four square
roots modulo n, exactly one of which is also in Qn.
2.158 Definition Let n be a Blum integer and let a ∈ Qn. The unique square root of a in Qn is
called the principal square root of a modulo n.
c

§2.5 Abstract algebra 75
2.159 Example (Blum integer) For the Blum integer n = 21, Jn = {1, 4, 5, 16, 17, 20} and

Qn = {5, 17, 20}. The four square roots of a = 4 are 2, 5, 16, and 19, of which only 16 is
also in Q21. Thus 16 is the principal square root of 4 modulo 21.
2.160 Fact If n = pq is a Blum integer, then the function f : Qn −→ Qn defined by f(x) =
x2
mod n is a permutation. The inverse function of f is:
f−1
(x) = x((p−1)(q−1)+4)/8
mod n.
2.5 Abstract algebra
This section provides an overview of basic algebraic objects and their properties, for refer-
ence in the remainder of this handbook. Several of the definitions in §2.5.1 and §2.5.2 were
presented earlier in §2.4.3 in the more concrete setting of the algebraic structure Z∗
n.
2.161 Definition A binary operation ∗ on a set S is a mapping from S × S to S. That is, ∗ is a
rule which assigns to each ordered pair of elements from S an element of S.
2.5.1 Groups
2.162 Definition A group (G, ∗) consists of a set G with a binary operation ∗ on G satisfying
the following three axioms.
(i) The group operation is associative. That is, a∗(b∗c) = (a∗b)∗c for all a, b, c ∈ G.
(ii) There is an element 1 ∈ G, called the identity element, such that a ∗ 1 = 1 ∗ a = a
for all a ∈ G.
(iii) For each a ∈ G there exists an element a−1
∈ G, called the inverse of a, such that
a ∗ a−1
= a−1
∗ a = 1.
A group G is abelian (or commutative) if, furthermore,
(iv) a ∗ b = b ∗ a for all a, b ∈ G.
Note that multiplicative group notation has been used for the group operation. If the
group operation is addition, then the group is said to be an additive group, the identity ele-
ment is denoted by 0, and the inverse of a is denoted −a.
Henceforth, unless otherwise stated, the symbol ∗ will be omitted and the group oper-
ation will simply be denoted by juxtaposition.
2.163 Definition A group G is finite if |G| is finite. The number of elements in a finite group is
called its order.
2.164 Example The set of integers Z with the operation of addition forms a group. The identity
element is 0 and the inverse of an integer a is the integer −a.
2.165 Example The set Zn, with the operation of addition modulo n, forms a group of order
n. The set Zn with the operation of multiplication modulo n is not a group, since not all
elements have multiplicativeinverses. However, the set Z∗
n (see Definition 2.124)is a group
of order φ(n) under the operation of multiplication modulo n, with identity element 1.

2.166 Definition A non-empty subset H of a group G is a subgroup of G if H is itself a group
with respect to the operation of G. If H is a subgroup of G and H = G, then H is called a
proper subgroup of G.
2.167 Definition A group G is cyclic if there is an element α ∈ G such that for each b ∈ G there
is an integer i with b = αi
. Such an element α is called a generator of G.
2.168 Fact If G is a group and a ∈ G, then the set of all powers of a forms a cyclic subgroup of
G, called the subgroup generated by a, and denoted by a.
2.169 Definition Let G be a group and a ∈ G. The order of a is defined to be the least positive
integer t such that at
= 1, provided that such an integer exists. If such a t does not exist,
then the order of a is defined to be ∞.
2.170 Fact Let G be a group, and let a ∈ G be an element of finite order t. Then |a|, the size
of the subgroup generated by a, is equal to t.
2.171 Fact (Lagrange’s theorem) If G is a finite groupand H is a subgroupof G, then |H| divides
|G|. Hence, if a ∈ G, the order of a divides |G|.
2.172 Fact Every subgroup of a cyclic group G is also cyclic. In fact, if G is a cyclic group of
order n, then for each positive divisor d of n, G contains exactly one subgroup of order d.
2.173 Fact Let G be a group.
(i) If the order of a ∈ G is t, then the order of ak
is t/ gcd(t, k).
(ii) If G is a cyclic group of order n and d|n, then G has exactly φ(d) elements of order
d. In particular, G has φ(n) generators.
2.174 Example Consider the multiplicative group Z∗
19 = {1, 2, . . . , 18} of order 18. The group
is cyclic (Fact 2.132(i)), and a generator is α = 2. The subgroups of Z∗
19, and their gener-
ators, are listed in Table 2.7.
Subgroup Generators Order
{1} 1 1
{1, 18} 18 2
{1, 7, 11} 7, 11 3
{1, 7, 8, 11, 12, 18} 8, 12 6
{1, 4, 5, 6, 7, 9, 11, 16, 17} 4, 5, 6, 9, 16, 17 9
{1, 2, 3, . . . , 18} 2, 3, 10, 13, 14, 15 18
Table 2.7: The subgroups of Z∗
19.
2.5.2 Rings
2.175 Definition A ring (R, +, ×) consists of a set R with two binary operations arbitrarily de-
noted + (addition) and × (multiplication) on R, satisfying the following axioms.
(i) (R, +) is an abelian group with identity denoted 0.
c

(ii) The operation × is associative. That is, a × (b × c) = (a × b) × c for all a, b, c ∈ R.
(iii) There is a multiplicative identity denoted 1, with 1 = 0, such that 1×a = a×1 = a
for all a ∈ R.
(iv) The operation × is distributive over +. That is, a × (b + c) = (a × b) + (a × c) and
(b + c) × a = (b × a) + (c × a) for all a, b, c ∈ R.
The ring is a commutative ring if a × b = b × a for all a, b ∈ R.
2.176 Example The set of integers Z with the usual operations of addition and multiplication is
a commutative ring.
2.177 Example The set Zn with addition and multiplication performed modulo n is a commu-
tative ring.
2.178 Definition An element a of a ring R is called a unit or an invertible element if there is an
element b ∈ R such that a × b = 1.
2.179 Fact The set of units in a ring R forms a group under multiplication, called the group of
units of R.
2.180 Example The group of units of the ring Zn is Z∗
n (see Definition 2.124).
2.5.3 Fields
2.181 Definition A field is a commutative ring in which all non-zero elements have multiplica-
tive inverses.
2.182 Definition The characteristic of a field is 0 if
m times

1 + 1 + · · · + 1 is never equal to 0 for any
m ≥ 1. Otherwise, the characteristic of the field is the least positive integer m such that
m
i=1 1 equals 0.
2.183 Example The set of integers under the usual operations of addition and multiplication is
not a field, since the only non-zero integers with multiplicative inverses are 1 and −1. How-
ever, the rational numbers Q, the real numbers R, and the complex numbers C form fields
of characteristic 0 under the usual operations.
2.184 Fact Zn is a field (under the usual operations of addition and multiplication modulo n) if
and only if n is a prime number. If n is prime, then Zn has characteristic n.
2.185 Fact If the characteristic m of a field is not 0, then m is a prime number.
2.186 Definition A subset F of a field E is a subfield of E if F is itself a field with respect to
the operations of E. If this is the case, E is said to be an extension field of F.

2.5.4 Polynomial rings
2.187 Definition If R is a commutative ring, then a polynomial in the indeterminate x over the
ring R is an expression of the form
f(x) = anxn
+ · · · + a2x2
+ a1x + a0
where each ai ∈ R and n ≥ 0. The element ai is called the coefficient of xi
in f(x).
The largest integer m for which am = 0 is called the degree of f(x), denoted deg f(x);
am is called the leading coefficient of f(x). If f(x) = a0 (a constant polynomial) and
a0 = 0, then f(x) has degree 0. If all the coefficients of f(x) are 0, then f(x) is called the
zero polynomial and its degree, for mathematical convenience, is defined to be −∞. The
polynomial f(x) is said to be monic if its leading coefficient is equal to 1.
2.188 Definition If R is a commutative ring, the polynomial ring R[x] is the ring formed by the
set of all polynomials in the indeterminate x having coefficients from R. The two opera-
tions are the standard polynomial addition and multiplication, with coefficient arithmetic
performed in the ring R.
2.189 Example (polynomial ring) Let f(x) = x3
+ x + 1 and g(x) = x2
+ x be elements of
the polynomial ring Z2[x]. Working in Z2[x],
f(x) + g(x) = x3
+ x2
+ 1
and
f(x) · g(x) = x5
+ x4
+ x3
+ x.
For the remainder of this section, F will denote an arbitrary field. The polynomial ring
F[x] has many properties in common with the integers (more precisely, F[x] and Z are both
Euclidean domains, however, this generalization will not be pursued here). These similar-
ities are investigated further.
2.190 Definition Let f(x) ∈ F[x] be a polynomial of degree at least 1. Then f(x) is said to be
irreducible over F if it cannot be written as the product of two polynomials in F[x], each
of positive degree.
2.191 Definition (division algorithm for polynomials) If g(x), h(x) ∈ F[x], with h(x) = 0,
then ordinarypolynomiallong division of g(x) by h(x) yields polynomials q(x) and r(x) ∈
F[x] such that
g(x) = q(x)h(x) + r(x), where deg r(x) deg h(x).
Moreover, q(x) and r(x) are unique. The polynomial q(x) is called the quotient, while
r(x) is called the remainder. The remainder of the division is sometimes denoted g(x) mod
h(x), and the quotient is sometimes denoted g(x) div h(x) (cf. Definition 2.82).
2.192 Example (polynomialdivision) Consider the polynomialsg(x) = x6
+x5
+x3
+x2
+x+1
and h(x) = x4
+ x3
+ 1 in Z2[x]. Polynomial long division of g(x) by h(x) yields
g(x) = x2
h(x) + (x3
+ x + 1).
Hence g(x) mod h(x) = x3
+ x + 1 and g(x) div h(x) = x2
.
c

2.193 Definition If g(x), h(x) ∈ F[x] then h(x) divides g(x), written h(x)|g(x), if g(x) mod
h(x) = 0.
Let f(x) be a fixed polynomial in F[x]. As with the integers (Definition 2.110), one
can define congruences of polynomials in F[x] based on division by f(x).
2.194 Definition If g(x), h(x) ∈ F[x], then g(x) is said to be congruent to h(x) modulo f(x)
if f(x) divides g(x) − h(x). This is denoted by g(x) ≡ h(x) (mod f(x)).
2.195 Fact (properties of congruences) For all g(x), h(x), g1(x), h1(x), s(x) ∈ F[x], the fol-
lowing are true.
(i) g(x) ≡ h(x) (mod f(x)) if and only if g(x) and h(x) leave the same remainder
upon division by f(x).
(ii) (reflexivity) g(x) ≡ g(x) (mod f(x)).
(iii) (symmetry) If g(x) ≡ h(x) (mod f(x)), then h(x) ≡ g(x) (mod f(x)).
(iv) (transitivity) If g(x) ≡ h(x) (mod f(x)) and h(x) ≡ s(x) (mod f(x)), then
g(x) ≡ s(x) (mod f(x)).
(v) If g(x) ≡ g1(x) (mod f(x)) and h(x) ≡ h1(x) (mod f(x)), then g(x) + h(x) ≡
g1(x) + h1(x) (mod f(x)) and g(x)h(x) ≡ g1(x)h1(x) (mod f(x)).
Let f(x) be a fixed polynomial in F[x]. The equivalence class of a polynomial g(x) ∈
F[x] is the set of all polynomials in F[x] congruent to g(x) modulo f(x). From properties
(ii), (iii), and (iv) above, it can be seen that the relation of congruence modulo f(x) par-
titions F[x] into equivalence classes. If g(x) ∈ F[x], then long division by f(x) yields
unique polynomials q(x), r(x) ∈ F[x] such that g(x) = q(x)f(x) + r(x), where deg r(x)
deg f(x). Hence every polynomial g(x) is congruent modulo f(x) to a unique polyno-
mial of degree less than deg f(x). The polynomial r(x) will be used as representative of
the equivalence class of polynomials containing g(x).
2.196 Definition F[x]/(f(x)) denotes the set of (equivalence classes of) polynomials in F[x]
of degree less than n = deg f(x). Addition and multiplication are performed modulo f(x).
2.197 Fact F[x]/(f(x)) is a commutative ring.
2.198 Fact If f(x) is irreducible over F, then F[x]/(f(x)) is a field.
2.5.5 Vector spaces
2.199 Definition A vector space V over a field F is an abelian group (V, +), together with a
multiplication operation • : F × V −→ V (usually denoted by juxtaposition) such that for
all a, b ∈ F and v, w ∈ V , the following axioms are satisfied.
(i) a(v + w) = av + aw.
(ii) (a + b)v = av + bv.
(iii) (ab)v = a(bv).
(iv) 1v = v.
The elements of V are called vectors, while the elements of F are called scalars. The group
operation + is called vector addition, while the multiplication operation is called scalar
multiplication.

2.200 Definition Let V be a vector space overa field F. A subspace of V is an additive subgroup
U of V which is closed under scalar multiplication, i.e., av ∈ U for all a ∈ F and v ∈ U.
2.201 Fact A subspace of a vector space is also a vector space.
2.202 Definition Let S = {v1, v2, . . . , vn} be a finite subset of a vector space V over a field F.
(i) A linear combination of S is an expression of the form a1v1 + a2v2 + · · · + anvn,
where each ai ∈ F.
(ii) The span of S, denoted S, is the set of all linear combinations of S. The span of S
is a subspace of V .
(iii) If U is a subspace of V , then S is said to span U if S = U.
(iv) The set S is linearly dependent over F if there exist scalars a1, a2, . . . , an, not all
zero, such that a1v1 + a2v2 + · · · + anvn = 0. If no such scalars exist, then S is
linearly independent over F.
(v) A linearly independent set of vectors that spans V is called a basis for V .
2.203 Fact Let V be a vector space.
(i) If V has a finite spanning set, then it has a basis.
(ii) If V has a basis, then in fact all bases have the same number of elements.
2.204 Definition If a vector space V has a basis, then the number of elements in a basis is called
the dimension of V , denoted dim V .
2.205 Example If F is any field, then the n-fold Cartesian product V = F × F × · · · × F is a
vector space over F of dimension n. The standard basis for V is {e1, e2, . . . , en}, where
ei is a vector with a 1 in the ith
coordinate and 0’s elsewhere.
2.206 Definition Let E be an extension field of F. Then E can be viewed as a vector space
over the subfield F, where vector addition and scalar multiplication are simply the field
operations of addition and multiplication in E. The dimension of this vector space is called
the degree of E over F, and denoted by [E : F]. If this degree is finite, then E is called a
finite extension of F.
2.207 Fact Let F, E, and L be fields. If L is a finite extension of E and E is a finite extension
of F, then L is also a finite extension of F and
[L : F] = [L : E][E : F].
2.6 Finite fields
2.6.1 Basic properties
2.208 Definition A finite field is a field F which contains a finite number of elements. The order
of F is the number of elements in F.
c

§2.6 Finite fields 81
2.209 Fact (existence and uniqueness of finite fields)
(i) If F is a finite field, then F containspm
elements for some prime p and integer m ≥ 1.
(ii) For every prime power order pm
, there is a unique (up to isomorphism) finite field of
order pm
. This field is denoted by Fpm , or sometimes by GF(pm
).
Informally speaking, two fields are isomorphic if they are structurally the same, al-
though the representation of their field elements may be different. Note that if p is a prime
then Zp is a field, and hence every field of order p is isomorphic to Zp. Unless otherwise
stated, the finite field Fp will henceforth be identified with Zp.
2.210 Fact If Fq is a finite field of order q = pm
, p a prime, then the characteristic of Fq is p.
Moreover, Fq contains a copy of Zp as a subfield. Hence Fq can be viewed as an extension
field of Zp of degree m.
2.211 Fact (subfields of a finite field) Let Fq be a finite field of orderq = pm
. Then every subfield
of Fq has order pn
, for some n that is a positive divisor of m. Conversely, if n is a positive
divisor of m, then there is exactly one subfield of Fq of order pn
; an element a ∈ Fq is in
the subfield Fpn if and only if apn
= a.
2.212 Definition The non-zero elements of Fq form a group under multiplication called the mul-
tiplicative group of Fq, denoted by F∗
q.
2.213 Fact F∗
q is a cyclic group of order q − 1. Hence aq
= a for all a ∈ Fq.
2.214 Definition A generator of the cyclic group F∗
q is called a primitive element or generator
of Fq.
2.215 Fact If a, b ∈ Fq, a finite field of characteristic p, then
(a + b)pt
= apt
+ bpt
for all t ≥ 0.
2.6.2 The Euclidean algorithm for polynomials
Let Zp be the finite field of order p. The theory of greatest common divisors and the Eu-
clidean algorithm for integers carries over in a straightforward manner to the polynomial
ring Zp[x] (and more generally to the polynomial ring F[x], where F is any field).
2.216 Definition Let g(x), h(x) ∈ Zp[x], where not both are 0. Then the greatest common divi-
sor of g(x) and h(x), denoted gcd(g(x), h(x)), is the monic polynomial of greatest degree
in Zp[x] which divides both g(x) and h(x). By definition, gcd(0, 0) = 0.
2.217 Fact Zp[x] is a unique factorization domain. That is, every non-zero polynomial f(x) ∈
Zp[x] has a factorization
f(x) = af1(x)e1
f2(x)e2
· · · fk(x)ek
,
where the fi(x) are distinct monic irreducible polynomials in Zp[x], the ei are positive in-
tegers, and a ∈ Zp. Furthermore, the factorization is unique up to rearrangement of factors.
The following is the polynomial version of the Euclidean algorithm (cf. Algorithm 2.104).

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