180
Exercises
(a) Compute the values of 1 (X) and 3 (X) for each of the following values of X.
(i) X = 10.
(ii) X = 25.
(iii) X = 100.
(b) Write a program to compute 1 (X) and 3 (X) and use it to compute their
values and the ratio 3 (X)/1 (X) for X = 100,
142
3. Integer Factorization and RSA
and we are searching for integers u1 , u2 , . . . , ur such that
e11 u1 + e21 u2 + + er1 ur 0
e12 u1 + e22 u2 + + er2 ur 0
.
.
.
.
(mod 2),
(mod 2),
e1t u1 + e2t u2 + + ert ur 0
(mod 2).
(3.17)
You have undoubtedly rec
96
2. Discrete Logarithms and DieHellman
Definition. Let R be a ring and let m R with m = 0. For any a R,
we write a for the set of all a R such that a a (mod m). The set a is
called the congruence class of a, and we denote the collection of all congruenc
118
3. Integer Factorization and RSA
17389d 1 (mod 63840),
where 63840 = (p 1)(q 1) = 228 280. The solution, using the method
described in Remark 1.15 or Exercise 1.12, is d 53509 (mod 63840). Then
Proposition 3.4 tells us that
x 4392753509 14458
(mod 643
114
3. Integer Factorization and RSA
power is not 1 modulo 15? A moments reflection shows that each of the
numbers 3, 5, 6, 9, 10, 12, 15 has a nontrivial factor in common with the modulus 15, while the numbers 1, 2, 4, 7, 8, 11, 13, 14 are relatively pri
108
Exercises
(a) Let eG be the identity element of G, let eH be the identity element of H, and
let g G. Prove that
(eG ) = eH
and
(g 1 ) = (g)1 .
(b) Let G be a commutative group. Prove that the map : G G defined
by (g) = g 2 is a homomorphism. Give an e
100
2. Discrete Logarithms and DieHellman
Proof. The existence of a factorization into irreducibles follows easily from the
fact that if a = bc, then deg a = deg b+deg c. (See Exercise 2.33.) The proof
that the factorization is unique is exactly the same
126
3. Integer Factorization and RSA
knows that n is composite; and if none of them is a witness for n, then Bob
suspects, but does not know for certain, that n is prime.
Unfortunately, intruding on this idyllic scene are barbaric numbers such
as 561. The
148
3. Integer Factorization and RSA
The function L(X) = e (ln X)(ln ln X) and other similar functions appear
prominently in the theory of factorization due to their close relationship to
the distribution of smooth numbers. It is thus important to underst
138
3. Integer Factorization and RSA
Example 3.34. We factor N = 203299. If we make a list of N + b2 for values
of b = 1, 2, 3, . . ., say up to b = 100, we do not find any square values. So next
we try listing the values of 3N + b2 and we find
3 203299 +
146
3. Integer Factorization and RSA
3.7
Smooth numbers, sieves, and building
relations for factorization
In this section we describe the two fastest known methods for doing hard
factorization problems, i.e., factoring numbers of the form N = pq, where p
92
2. Discrete Logarithms and DieHellman
q
2
3
5
e
1
2
4
e
g (p1)/q
11250
5029
5448
h(p1)/q
11250
10724
6909
e
!
e "x
e
Solve g (p1)/q
= h(p1)/q for x
1
4
511
Notice that the first problem is trivial, while the third one is the problem that
we solved in E
164
3. Integer Factorization and RSA
These in turn give linear relations for the discrete logarithms of 2, 3, and 5 to
the base g. For example, the first one says that
12708 = 3 logg (2) + 4 logg (3) + logg (5).
To ease notation, we let
x2 = logg (2),
x3
124
3. Integer Factorization and RSA
c1 me1 (mod N )
and
c2 me2 (mod N ),
she can take a solution to the equation
e1 u + e2 v = gcd(e1 , e2 )
and use it to compute
cu1 cv2 (me1 )u (me2 )v me1 u+e2 v mgcd(e1 ,e2 )
(mod N ).
If it happens that gcd(e1 , e2 )
104
2. Discrete Logarithms and DieHellman
Further, in a certain abstract sense it doesnt matter which irreducible polynomial we choose: we always get the same field. However, in a practical sense
it does make a dierence, because practical computations in
106
Exercises
2.5. Let p be an odd prime and let g be a primitive root modulo p. Prove that a
has a square root modulo p if and only if its discrete logarithm logg (a) modulo p is
even.
Section 2.3. DieHellman key exchange
2.6. Alice and Bob agree to use
Mount Saint Vincent University
BIOL 1153 Introductory Biology II
Dr. Jessica Boyd
Quiz I Study Guide
Chapter
Section
Pages
26 Origin of Life
26.1 Deep Time
26.2 Origins of Life
26.3 Evidence for Early Life
26.4 Earths Changing System
26.5 Ever- Changing L
134
3. Integer Factorization and RSA
The exponent k is not equal to 0, so it is quite unlikely that ak will be
congruent to 1 modulo q. Thus with very high probability, i.e., for most choices
of a, we find that
p divides aL 1
and
q does not divide aL 1.
B
Lab 5: Vertebrate
Animals. The Bony
Fish
Shirann Smith
Lab Section LA Tuesday am
February 11, 2016
2.What are the four features that all chordate animals have?
All chordate animals have scales, jaws, a bony skeleton, and gills.
3. Name four features that
BIOL 1153 Introductory Biology II Land Plant Innovations
Trait
Green Algae
Divisions or Phyla
Name _
Bryophytes Lycophytes
Chlorophytes Liverworts Club mosses
Charophytes Mosses
Hornworts
How trait benefits life on land
Dominant
Generation?
The Sporophyte
Lab 7
Flowers, fruits and seeds
Evolution of Angiosperms
Sticky resin on female pine cones
(gymnosperms) helped pollen grains
stick
Insects found this resin tasty, and
would fly from tree to tree while
inadvertently carrying pollen with
them.
Plants th
Mount Saint Vincent University
BIOL 1153 Introductory Biology II
Dr. Jessica Boyd
Quiz III Study Guide
Chapter
Section
Page
27 Viruses
27.1
27.2
27.4
27.5
The Nature of Viruses
Bacteriophages: Bacterial viruses
Other Viral Diseases
Prions and Viroids: Sub
Chapter 34 The Protostomes
Fate of the blastopore
Protostomes
first mouth
Deuterostomes
second mouth
Platyzo
a
Lophotrochozoa
Spiralia
Platyzo
a
Lophotrochozoa
Spiralia
No tissues
Tissues
Platyzo
a
Lophotrochozoa
Spiralia
Radial
symmetry
No tissues
Bilate
Mount Saint Vincent University
BIOL 1153 Introductory Biology II
Dr. Jessica Boyd
Quiz II Study Guide
Chapter
Section
29 Protists
29.1 Eukaryotic Origins and Endosymbiosis
29.2 Overview of Protists
29.3 Feeding Groove in Excavata
29.4 Secondary Endosymbio
Mount Saint Vincent University
BIOL 1153 Introductory Biology II
Quiz IV Study Guide
Dr. Jessica Boyd
Quiz Date April 3, 2017
Chapter
Section
Page
42 Animal Body
and Principles
of Regulation
42.1 Organization of the Vertebrate Body
42.2 Epithelial Tissue
LAB 3: FUNGI
Smith
Lab Section: Tuesday LA
February 4, 2016
Exercise 3.1 Phylum Zygomycota
1. Yes, some of the moulds on the bread do have hyphae modified as
sporangiophores.
2. The pigment is concentrated in the sporangiophore.
3. This is so that the win