section of this chapter contains an
introduction to the techniques of program
verification. This is a formal technique to
verify that procedures are correct.
Program verification serves as the basis
f
prove statements that assert that P (n) is
true for all positive integers n, where P
(n) is a propositional function. A proof by
mathematical Unfortunately, using the
terminology mathematical inductio
the ciphertext. 17. Construct a valid RSA
encryption key by finding two primes p
and q with 200 digits each and an integer
e > 1 relatively prime to (p 1)(q 1). 18.
Given a message and an integer n =
message secret decryption: the process of
returning a secret message to its original
form encryption key: a value that
determines which of a family of
encryption functions is to be used shift
cipher:
inverse of e = 13 modulo 42 58.) 28.
Suppose that (n, e) is an RSA encryption
key, with n = pq where p and q are large
primes and gcd(e, (p 1)(q 1) = 1.
Furthermore, suppose that d is an inverse
of e
(ii) of this theorem, and the inductive
hypothesis, we conclude that the first
term in this last sum, 7(7k+2 + 82k+1), is
divisible by 57. By part (ii) of this
theorem, the second term in this sum, 57
Use the autokey cipher to encrypt the
message NOW IS THE TIME TO DECIDE
(ignoring spaces) using a) the keystream
with seed X followed by letters of the
plaintext. b) the keystream with seed X
followed
there are infinitely many primes
(Theorem 3 in Section 4.3) to show that
are infinitely many primes in the
arithmetic progression 6k + 5, k = 1, 2,.
30. Explain why you cannot directly adapt
the proof
prove an extremely wide variety of
theorems, each of which is a statement of
this form. (Remember, many
mathematical assertions include an
implicit universal quantifier. The
statement if n is a positi
point needs to be made about
mathematical induction before we
commence a study of its use. The good
thing about mathematical induction is
that it can be used to prove P1: 1 CH05-7R
Rosen-2311T MHIA017
the definition of harmonic number = H2k
+ 1 2k + 1 + 1 2k+1 by the definition of
2kth harmonic number 1 + k 2 + 1 2k +
1 + 1 2k+1 by the inductive hypothesis
1 + k 2 + 2k 1 2k+1 because there are
2k
47, 7), dAlice = 1183 and (nBob, eBob) =
(3127, 21) = (59 53, 21), dBob = 1149.
First express your answers without
carrying out the calculations. Then, using
a computational aid, if available, perform
assume that If you are rusty simplifying
algebraic expressions, this is the time to
do some reviewing! 1 + 2 + k = k(k + 1)
2 . Under this assumption, it must be
shown that P (k + 1) is true, namely,
are almost certainly prime. Do such tests
have any potential drawbacks? 3. The
question of whether there are infinitely
many Carmichael numbers was solved
recently after being open for more than
75 ye
guessing the bits in its binary expansion.
13. Show that an integer is divisible by 9 if
and only if the sum of its decimal digits is
divisible by 9. P1: 1 CH04-7R Rosen2311T MHIA017-Rosen-v5.cls May
also be true. Remark: In a proof by
mathematical induction it is not assumed
that P (k) is true for all positive integers!
It is only shown that if it is assumed that P
(k) is true, then P (k + 1) is
Solution: To construct the proof, let P (n)
denote the proposition: n3 n is
divisible by 3. BASIS STEP: The statement
P (1) is true because 13 1 = 0 is divisible
by 3. This completes the basis step.
I
linear congruences modulo pairwise
relatively prime moduli, find the
simultaneous solution of these
congruences modulo the product of these
moduli. 11. Given a positive integer N, a
modulusm, a multip
interest to readers. The author is the
Series Editor of these books). Note that
there are many opportunities for errors in
induction proofs. We will describe some
incorrect proofs by mathematical
indu
encrypted using Alices key kAlice. (ii)
Cathy sends back to Alice a key
kAlice,Bob, which she generates,
encrypted using the key kAlice, followed
by this same key kAlice,Bob, encrypted
using Bobs key,
knocks the (k + 1)st domino overi.e., if
P (k) P (k + 1) is true for all positive
integers kthen all the dominoes are
knocked over. This is illustrated in Figure
2. Why Mathematical Induction is Valid
system of congruences x a1(mod m1)
and x a2 (mod m2), where a1, a2, m1,
and m2 are integers with m1 > 0 and m2
> 0, has a solution if and only if gcd(m1,
m2) | a1 a2. b) Show that if the system
in par
where p1, p2,.,pn are the n smallest
primes, for as many positive integer n as
possible. 4. Look for polynomials in one
variables whose values at long runs of
consecutive integers are all primes. 5.
F
proofs using mathematical induction, as
well as introducing the terminology
mathematical induction. Maurolicos
proofs were informal and he never used
the word induction. See [Gu11] to learn
more about
of factorial function. This shows that P (k
+ 1) is true when P (k) is true. This
completes the inductive step of the proof.
We have completed the basis step and the
inductive step. Hence, by mathemat
law (where the two sets are
k j = 1 Aj and Ak+1) = k j = 1 Aj
Ak+1 by the inductive hypothesis = k +1 j
= 1 Aj by the definition of union. This
completes the inductive step. Because we
have completed
digit. 9. Describe the Luhn algorithm for
finding the check digit of a credit card
number and discuss the types of errors
that can found using this check digit. 10.
Show how a congruence can be used t
proposition that a set with n elements has
2n subsets. BASIS STEP: P (0) is true,
because a set with zero elements, the
empty set, has exactly 20 = 1 subset,
namely, itself. INDUCTIVE STEP: For the
in