1 Day 7 Number Theory 1 Number Theory In elementary number theory, integers are studied without use of techniques from other mathematical fields. Questions of divisibility, use of the Euclidean algorithm to compute greatest common divisors, factorization of integers into prime numbers, i ti ti f f t b d investigation of perfect numbers and congruences belong here. Several important discoveries of this field are Fermat's little theorem, Euler's theorem, the Chinese remainder theorem and the law of quadratic reciprocity. The properties of multiplicative functions such as the Möbius function, Euler's φ function, integer sequences, factorials and Fibonacci numbers all also fall into this area. 2 Divides Let and be integers, 0 divides an integer with n d d d n q n dq ≠ ↔∃ = if divides we write | if does not divide we write | d n nd d n n d 3
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1
Day 7
Number Theory
1
Number TheoryIn elementary number theory, integers are studied without use of techniques from other mathematical fields. Questions of divisibility, use of the Euclidean algorithm to compute greatest common divisors, factorization of integers into prime numbers, i ti ti f f t b dinvestigation of perfect numbers and congruences belong here. Several important discoveries of this field are Fermat's little theorem, Euler's theorem, the Chinese remainder theorem and the law of quadratic reciprocity. The properties of multiplicative functions such as the Möbius function, Euler's φ function, integer sequences, factorials and Fibonacci numbers all also fall into this area.
2
Divides
Let and be integers, 0 divides an integer with
n d dd n q n dq
≠↔ ∃ =
if divides we write |
if does not divide we write |
d n n d
d n n d
3
2
Theorem
( ) ( ) ( )m d n d m n d∧ → +| | |
( )
1 1
2 2
1 2 1 2
Proof: with with
then
d m q m q dd n q n q d
m n q d q d q q d
→∃ =→∃ =
+ = + = +
||
4
( ) ( ) ( )m d n d m n d∧ → −| | |
Theorem
( )
1 1
2 2
1 2 1 2
Proof: with with
then
d m q m q dd n q n q d
m n q d q d q q d
→∃ =→ ∃ =
− = − = −
||
5
( ) ( )m d n d mn d∧ →| | |
Theorem
6
( )
1 1
2 2
1 2 1 2
Proof: with with
then
d m q m q dd n q n q d
mn q dq d q q d d
→∃ =→∃ =
= =
||
3
Review Modular Arithmetic
x mod n means “remainder after dividing x by n”
x ≡ y (mod n) means “x and y have the same remainder mod n”
In C++ x%n means x mod n
7
Prime Number
A prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors. It follows that these two divisors are 1 and the prime
Euclid
number itself. There exists an infinitude of prime numbers, as demonstrated by Euclid in about 300 B.C.
The fundamental theorem of arithmetic states that every positive integer larger than 1 can be written as a product of one or more primes in a unique way, i.e. unique except for the order. Primes are thus the "basic building blocks" of the natural numbers.
8
Prime Number
An integer p>1 is prime iff it is not the product of two integers greater than 1:
p > 1 ∧ ¬∃ a, b ∈ N: a > 1, b > 1, a b = p
The only positive factors of a prime p are 1 and p itself
Some primes: 2, 3, 5, 7, 11, 13, 17, ...
9
4
Eratosthenes of CyreneA versatile scholar, Eratosthenes of Cyrene lived approximately 275-195 BC
He was the first to estimate accurately the diameter of the earth
Born in Cyrene (in modern day Libya)
276 BC - 194 BC
Born in Cyrene (in modern-day Libya), but worked and died in Alexandria, capital of Ptolemaic Egypt
Never married, he was reputedly known for his haughty character
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Sieve of EratosthenesPrime Numbers
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Euclid’s Proof Infinite Number of Primes
Assume there are a finite number n of primes, the largest being pn
C id th b th t i th d t f th lConsider the number that is the product of these, plus one: N = p1 ... pn + 1
By construction, N is not divisible by any of the pi’s
Hence it is either prime itself, or divisible by another prime greater than pn contradicting the assumption
12
5
Prime Number Algorithm
The sieve of Atkin is a fast, modern algorithm for finding all prime numbers up to a specified integer
It is an optimized version of the ancient sieve ofIt is an optimized version of the ancient sieve of Eratosthenes: the Atkin sieve does some preliminary work and then marks off multiples of primes squared, rather than multiples of primes
13
Prime? - Test for a Divisor
is_prime(n){
for 2 to nd ⎢ ⎥= ⎣ ⎦( )
( )
if mod 0 return d return}
n d ==
14
TI Test if prime
15
6
Prime Counting Function π(n)
( ) is the number of primes less than or equal to x xπ
( )lnnnn
π =
Counts the number of prime numbers less than or equal to some real number n
Gauss and Legendre approximated
Th th li t d f l t f
16
There are other more complicated formulas, most of which are beyond our course level, for example:
Where O is our “Big – O”
( ) ( )1
00 1
lim logln ln
k n
kk
dt dtn O n nt t
π−
→+
⎛ ⎞= + +⎜ ⎟
⎝ ⎠∫ ∫
Prime Counting Function
( ) is the number of primes less than or equal to x xπ ( )lnnnn
π =
17
Composite Number
A composite number is a positive integer which has a positive divisor other than one or itself
By definition, every integer greater than one is either a prime number or a composite number
The numbers zero and one are considered to be neither prime nor composite
For example, the integer 15 is a composite number because it can be factored as 3 · 5 and 315 can be written as 32 · 5 · 7
18
7
TI - Prime Factorization
19
Prime Factorization Algorithm
A prime factorization algorithm is any algorithm by which an integer is "decomposed" into a product of factors that are prime numbers
The fundamental theorem of arithmetic guarantees that this decomposition is unique
20
Recursive Factoring AlgorithmGiven a number n
if n is prime, this is the factorization, so stop here
if n is composite, divide n by the first prime p1
If it divides cleanly, recurse with the value n/p1Add p1 to the list of factors obtained for n/p1 to get a factorization for n
If it does not divide cleanly, divide n by the next prime p2, and so on
We need to test only primes from 2 to n 21
8
Least Common Multiple
The least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b
If there is no such positive integer, e.g., if a = 0 or b = 0, then lcm(a, b) is defined to be zero
22
TI Example
23
Greatest Common Divisor
The greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder
24
9
GCD pseudocode
gcd(n,m)Func
If n<m thenswap(n,m)
endifhil 0
swap(n,m)Func
n tm n
25
while m≠0n mod m rm nr m
endwhilereturn a
end
m nt m
end
GCD pseudocode
gcd(n,m)Func
Recursive Function
26
FuncIf m=0 then
return nelse
return(gcd(m, n mod m))endif
end
TI Example
27
10
Theorem
( ) ( ), gcd ,lcm a b a b a b⋅ = ⋅
Proof:Suppose factors of and area b
1 2 1 2
1 2 1 2
1 2
1 2 1 2
where are the common factorsand and are the factors unique to and
n n
n n
n
n n
a p p p r r rb q q q r r r
r r rp p p q q q
a b
= ⋅ ⋅ ⋅ ⋅= ⋅ ⋅ ⋅
⋅⋅ ⋅
28
( )( )
1 2 1 2 1 2
1 2
Then , =
and gcd ,n n n
n
lcm a b p p p q q q r r r
a b r r r
⋅ ⋅ ⋅ ⋅ ⋅
= ⋅
Theorem
( ) ( ), gcd ,lcm a b a b a b⋅ = ⋅
( )
( ) ( )( ) ( )
( ) ( )
1 2
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
d gcd ,Then
, gcd ,
n
n n n n
n n n n
a bp p p r r r q q q r r r
p p p q q q r r r r r r
lcm a b a b
⋅
= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
= ⋅
i
29
Relatively Prime
and are relatively prime or coprime if gcd( , ) 1a b
a b =
A set of integers {a1 , a2 ,…} is (pairwise) relatively prime if all pairs (ai , aj ), for i ≠ j, are relatively prime
30
11
Example
Neither 21 nor 10 is prime21=3·7 and 10=2·5
21 and 10
They have no common factors > 1They have no common factors > 1and their gcd = 1
They are coprime
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TI Test if coprime
32
programmer humor
There are 10 types of people in the world
Those that understand binary numbersand those that do not.
Binary Watch33
12
Sexagesimal Number System
The sexagesimal (base-sixty) is a numeral system with sixty as the base. It originated with the ancient Sumerians in the 2000s BC, and was transmitted to the Babylonians
Sexagesimal is seldom used for general computation or logic, but we retain some residual use of their system, in degrees of angle, and in the number of seconds in a minute, and minutes in an hour
34
Babylonian Symbols
35
Babylonian Multiplication Tablet
36table for 9
13
Decimal System
Possibly originated in China
This tablet was discovered in 1898 in Xiao dun in the An-yang district of Henan province dating from the 14th century BC
37
Arabic Numerals
Various Arabic SystemsToday
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Arabic NumeralsFibonacci, an Italian mathematician who had lived in North Africa, introduced the Arabic numeral system to Europe and promoted it with his book Liber Abaci, which was published in 1202
In the Muslim World until modern times the Arabic numeral system was used only by mathematicians
Muslim scientists used the Babylonian numeral system, and merchants used a numeral system similar to the Greek numeral system and the Hebrew numeral system
Therefore, it was not until Fibonacci that the Arabic numeral system was used by a large population
39
14
Computer Number Systems
40
Binary Number System
The binary system was used in Egypt 3,000 - 2,050 BC
The ancient Indian mathematician Pingala िपङ्गलpresented the first known description of a binary numeral system in the 3rd century BC
The modern binary number system was fully documented by Gottfried Leibniz in the 17th century in his article Explication de l'Arithmétique Binaire. Leibniz's system used 0 and 1, like the modern binary numeral system
41
Eye of Horus
ancient Egyptian symbol of protection and royal power from deities, in this case from Horus or Ra
42
The Eye Of Horus defined an Egyptian Old Kingdom rounded off number
A bit (binary digit) refers to a digit in the binary numeral system, which consists of base 2 digits.
Bits can be represented in many forms.
A bit of storage is like a light switch; it can be either (1) ff (0) A i l bit i ton (1) or off (0). A single bit is a one or a zero, a true
or a false, a "flag" which is "on" or "off“.
A byte is a collection of bits, originally variable in size but now almost always eight bits. Eight-bit bytes, also known as octets, can represent 256 values (28 values, 0–255). A four-bit quantity is known as a nibble, and can represent 16 values (24 values, 0–15).
53
Quaternary BaseThis is a base-4 system using the symbols 0, 1, 2, and 3
North and Central American natives used this base to represent the four cardinal directions
Many or all of the Chumashanused a base 4 counting system
54
used a base 4 counting system
The Chumashan are Native Americans inhabiting central and southern coastal regions of California
19
Quinary Number SystemQuinary (base-5) is a numeral system with five as the base. This originates from the five fingers on either hand: the most primitive numeral system
In the twentieth century, only the y yEast African Luo tribe of Kenya and the Yoruba of Nigeria were still using a base-five system
Women of Luo tribe drying fish55
Octal Number System
The octal numeral system is the base-8 number system, and uses the digits 0 to 7
The Yuki in California and the Pamean in Mexico have octalPamean in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves
Yuki Tribe 1858 56
Binary to Octal
Octal numerals can be made from binary numerals by grouping consecutive digits into groups of three (starting from the right)
Example:
The binary representation for decimal 74 is 1001010, which groups into 1 001 010
Nonary BaseThe Nenets language once used a base-9 system (nonary), but has since shifted to decimal under the influence of Russian
The word yúq originally meant 9, but took the value 10 on account of Russian influence; so in current Nenets the word for 9 is xasu-yúq, lit. 'Nenets yúq', whereas 10 is simply yúq, but in Eastern dialects also lúca-yúq, lit. 'Russian yúq’
59
Duodecimal Number System The word duodecimal comes from the Latin words for two and ten
The Egyptians were fond of counting in base twelve
The base-12 number system composed of the digits 0, 1 2 3 4 5 6 7 8 9 A B1, 2, 3, 4, 5, 6, 7, 8, 9, A, B
Merchants use a duodecimal system when they count by the dozen or gross (12 dozen or 144)
Base-12 systems (duodecimal or dozenal) have been popular because multiplication and division are easier than in base-10, with addition just as easy
This base is responsible for our 12 and 24 hour clock60
21
Duodecimal Number System Base-12 originated from the number of joints in the four fingers of a hand, using the thumb as a pointer
1
2
3
4
5
6 7
8
9
A
10B
61
A
A number of people in the Nigerian Middle Belt use a duodecimal system
Tridecimal BaseUses 13 different digits for representing numbers
Suitable digits for base 13 could be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, and T
According to Douglas Adams
62
g gand his Hitchhiker's Guide to the Galaxy, 42 is the ultimate answer to life, the universe and everything
With base 13, 6 x 9 = 42
Vigesimal Number System
The Maya civilization and other civilizations of Pre-Columbian Mesoamerica used base-20 (vigesimal), possibly originating from the number of a person's fingers and toes
Evidence of base 20 countingEvidence of base-20 countingsystems is also found in the languages of central and western Africa
63
22
Dual Base Number System
Many ancient counting systems use 5 as a primary base, almost surely coming from the number of fingers on a person's hand
Often these system are supplemented with a secondary base, sometimes ten, sometimes twentysecondary base, sometimes ten, sometimes twenty
In some African languages the word for 5 is the same as "hand" or "fist" (Dyola language of Guinea-Bissau, Banda language of Central Africa) 64
Hexadecimal Number SystemBase 16, hexadecimal, or simply hex, is a numeral system with a radix or base of 16, usually written using the symbols 0–9 and A–F or a–f
The current hexadecimal system was first introduced to the computing world in 1963 by IBM. An earlier
i i th di it 0 9 d i t d dversion, using the digits 0–9 and u–z, was introduced in 1956 by the Bendix Corporation
Hexadecimal is primarily used in computing as the most common form of expressing a human-readable string representation of a byte. All 256 possible values of a byte can be represented using 2 digits in hexadecimal notation
65
TI Hexadecimal
66
23
ASCII
ASCII (American Standard Code for Information Interchange) is a character encoding based on the English alphabet using one byte per symbol
2 1 2 3 2 6 2 13 2 261 0 1 01R R R R R` 1 0 1 01R R R R R= = = = =
10 226 11010=
74
TI Solution
75
26
Decimal Octal
0 1928 2 8 19 8 155
R=2 R=3 R=3 ↓ ↓ ↓
10 8155 233=
76
TI-89 Decimal to Octal
To convert to octal using the TI first convert to binary, separate into groups of three, then convert the groups into octal.
10 011 0112 3 3
77
Base b to Decimal
base_b_to_dec ( , , ){ dec_val=0 power=1
f 1 t 0
c n b
iInput: , ,Output: dec val
c n b
i
for 1 to 0 dec_val = dec_val+c power power = power } return dec_val}
i
b
=i
i
Output: dec_val
78
27
Decimal to Base b
( )
dec_to_base_b ( , , , ) { 1 While 0 {
1
m b c nn
mn n
= −
>
= +Input: ,Output:
m bc n 1
mod }}
n
n nc m bm m b
= +=
= ⎢ ⎥⎣ ⎦
Output: ,c n
79
Remarks
To convert from base a to base b
1)base a base 102)base 10 base b
We use both algorithms together
80
TI-89
Base Converter 1.2
"Base Converter" is a tool to convert any number (between 0 and 32767) from any base
baseconv.zip
81
to any other base (between 2 and 36)
This program and others similar can be found at www.ticalc.org
28
Modular Exponentiation
Modular exponentiation is a type of exponentiation performed over a modulus. It is particularly useful in computer science, especially in the field of cryptology
Doing a "modular exponentiation" means calculating theDoing a modular exponentiation means calculating the remainder when dividing by a positive integer m (called the modulus) a positive integer b (called the base) raised to the eth power (e is called the exponent)
The extended Euclidean algorithm is an extension to the Euclidean algorithm for finding the greatest common divisor (GCD) of a and b: it also finds the integers x and y in Bézout's identity
The extended Euclidean algorithm is particularly useful when a and b are coprime, since x is the multiplicative inverse of a modulo b
( )gcd ,ax by a b+ =
88
Review Euclidean AlgorithmThe Euclidean algorithm is a way to find the gcd of two positive integers, a and b
Example gcd(210,45)
210/45 = 4 with remainder 30 so 210 = 4 · 45 + 30210/45 = 4 with remainder 30, so 210 = 4 · 45 + 30
45/30 = 1 with remainder 15, so 45 = 1 · 30 + 15
30/15 = 2 with remainder 0, so 30 = 2 · 15 + 0
gcd(210,45) = 15
89
TI Solution
90
31
Review Euclidean Algorithm
Input : a, b Output: gcd
1 If a < b exchange a and b1. If a < b, exchange a and b. 2. Divide a by b and get the remainder, r If r = 0,
report b as the GCD of a and b. 3. Replace a by b and replace b by r. Return to the
65 / 40 1 with 25 25 65 1 40R= = → = − i40 / 25 1 with 15 15 40 1 25R= = → = − i
25 /15 1 with 10 10 25 1 15R= = → = − i15 /10 1 with 5 5 15 1 10R= = → = − i
10 / 5 2 with 0 5 10 2 5R= = → = − i
Therefore gcd(65, 40) 5=
93
32
Same Example - Back Substitution
( )( )
5 15 10 15 25 15 2 15 25
2 40 25 25 2 40 3 25
= −
= − − = −
= − − = −
i
i i( )( ) 2 40 3 65 40 5 40 3 65= − − = −i i i i
( )gcd(65, 40) 3 65 5 40= − +i i
94
recursive pseudocode
function extended_gcd(a, b) {
if a mod b = 0
return {0, 1}
elseelse
temp := extended_gcd(b, a mod b)
x := first(temp)
y := last(temp)
return {y, x-y*(a div b)
} 95
Example GCD( 69974450 , 6898073 )
69974450 mod 6898073
106898073 69974450
68980730993720
= 993720
6898073 mod 993720 = 935753
993720 mod 935753 = 57967
935753 mod 57967 = 8281
57967 mod 8281 = 0
GCD( 69974450 , 6898073 ) = 828196
33
Finding Inverses mod n
We want to find a -1 mod n
If gcd( a, n ) = 1, can use extended Euclidean to findx, y such that x · a + y · n = 1y y
Then x · a = 1 – y · n ≡ 1 mod n x ≡ a -1 mod n
Example: 5 · 3 – 2 · 7 = 1 5 ≡ 3-1 mod 7
97
TI Solution
modinv() is a function available at ticalc.org 98
Example
1 23 25 7 821 23 25mod82= ⋅ − ⋅= ⋅
Find the multiplicative inverse of 25 modulo 82
82 3 35 725 3 7 4
= ⋅ += ⋅ +
7 1 4 34 1 3 11 3
= ⋅ += ⋅ +=
23 is the inverse of 25mod82
99
34
TI Solution
Example Find the multiplicative inverse of 25 modulo 82
100
Pseudo code - Inverse mod n
s = m, t = x; /* set up for gcd(x,m)
a = 0, b = 1; /* 0*x = s and 1*x = t
while(t) {
q = s/t r = s%t; /* quotient and remainderq = s/t, r = s%t; / quotient and remainder
s = t, t = r; /* push back
temp = (a-b*q) % m; a = b, b = temp; /* push back
}
return a
101
CryptologyCryptology is an term for cryptography and cryptanalysis
In cryptography, a cipher (or cypher) is an algorithm for performing encryption and decryption — a series f ll d fi d t th t bof well-defined steps that can be
followed as a procedure
Enigma
102
An alternative term is encipherment
35
Cryptology
The original information is called plaintext
The encrypted form is called ciphertext
Encryption is the process of obscuring information to make it unreadable without special knowledgespecial knowledge, sometimes referred to as scrambling
3TL - 176 - 12 21 14 (In pencil; LUN)DHHAO FWQQM EIHBF BMHTT YFBHK YYXJK IXKDF RTSHB HLUEJ MFLACZRJDL CJZVK HFBYL GFSEW NRSGS KHLFW JKLLZ TFMWD QDQQV JUTJSVPRDE MUVPM BPBXX USOPG IVHFC ISGPY IYKST VQUIO CAVCW AKEQQEFRVM XSLQC FPFTF SPIIU ENLUW O =1 ABT GEN ST D H NR. 2050/38 G KDOS +
104
PlainText
Auf Befehl des Obersten Befehlshabers sind im Falle,
(z.Zt =) zur Zeit unwahrscheinlichen, Franzoesischen Angriffs
die Westbefestigungen jeder zahlenmaessigen Ueberlegenheit zum trotz zu halten. Fuehrung und Truppe muessen von dieser Ehrenpflicht durchdrungen sein.
Dem gemaess behalte ich mir die Ermaechtigung zurDem gemaess behalte ich mir die Ermaechtigung zur Aufgabe der Befestigungen oder auch von Teilen ausdruecklich persoenlich vor.
Aenderungen der Anweisung OKH/Gen/St/D/H Erste Abt Nr. 3321/38
G/KDos vom Juli 1938 bleibt vorbehalten.
Der Oberbefehlshaber des Heeres. 105
36
TranslationThe Commander-in-Chief orders as follows:
In the case of French attacks on the western fortifications, although unlikely at this moment, those fortifications must be held at all costs, even against numerically superior forces.
Commanders and troops must be imbued with the honor of this duty.
In accordance with orders I emphasize that I alone haveIn accordance with orders, I emphasize that I alone have the right to authorize the fortifications to be abandoned in whole or part.
I reserve the right to make changes to the order OKH/Gen/St/D/H
1. Abt. Nr. 3321/38 GKDos of July 1938.
The Commander-in-Chief of the Army.106
RSA
In cryptology, RSA is an algorithm for public-key encryption
It was the first algorithm known to be suitable for signing as well as encryption, and one of the first great advances in public key cryptographyadvances in public key cryptography
RSA involves a public and private key
The public key can be known to everyone and is used for encrypting messages
Messages encrypted with the public key can only be decrypted using the private key
107
RSA
The keys for the RSA algorithm are generated using modulo math with two prime numbers
Keys are typically 1024–2048 bits long
Secret key cryptography, also known as symmetric t h i l t k f b thcryptography uses a single secret key for both
encryption and decryption
It was developed by Ron Rivest, Adi Shamir, and Leonard Adleman in 1977
108
37
RSAAn analogy for public key encryption is that of a locked mailbox with a mail slot
The mail slot is exposed and accessible to the public; its location (the street address) is in essence the public keyy
Anyone knowing the street address can go to the door and drop a written message through the slot
However, only the person who possesses the key can open the mailbox and read the message
109
Rabin
The Rabin cryptosystem is an asymmetric cryptographic technique, whose security, like that of RSA, is related to the difficulty of factorization
Both RSA and Rabin use a trapdoor function, a function that is easy to compute in one direction yet believed tothat is easy to compute in one direction, yet believed to be difficult to compute in the opposite direction (finding its inverse) without special information, called the "trapdoor”
110
RSA
( )( )
RSA uses product of primes
public key , mod
private key , mod
e
d
n pq
e n C m n
d n M c n
=
=
=( )
RSA Chip
111
38
RSA Key Generation
( ) ( )
We select two different primes, say 5, 7Then we compute the following
351 1 24
p q
n p qp qϕ
= =
= =
= − − =
ii
5 we select so and , are coprime5 = inverse of modulo
e e e n ed e
ϕϕ
= <=
( )( )
From thse we havethe public key ,
the private key ,
e n
d n112
RSA CodingWe write our message “Discrete”
We find the number for each letter (call it m)A B C D E F … 1 2 3 4 5 6 …
W h l k h d d
Letter D i s c r e t e4 9 19 3 18 5 20 5
mod 09 04 24 33 23 10 20 10e
mm n
We then look up the code me mod nFor example for “D” 45 mod 35 = 9
D 4 9
113
RSA DecodingOur message is then 0904243323102010Break up into 09 04 24 33 23 10 20 10Take the first number, call it 09m′ =
5We then calculate mod , so 9 mod35 4dm m n′= =
09 04 24 33 23 10 20 10mod 4 9 19 3 18 5 20 5
Letter D i s c r e t e
d
mm n
′′
Letter number 4 corresponds to letter DWe took 9 4 D→ →