1. Prime Numbers: Theory and their
application in Public Key
Cryptography
Ashutosh Tripathi
10UCS011

2. A prime number is any natural number
greater than 1, which cannot be factored
as the product of two smaller numbers.
Examples: 2,3,5,7,11,13,17,…,991,…,99991,
…,9999991,…

3. The Convergence
There are 8 primes in first 20 counting
numbers
Only 4 in next 20
Does this sequence ultimately converge to
0?
The answer dates itself back to ~300 BC

4. Euclid’s Theorem
There are infinitely many primes
The proof to this is one of the earliest
appearance of Reductio-ad-absurdum
(proof by contradiction)
Every natural number greater than1 can be
uniquely factored into one or more primes
Fundamental Theorem of Arithmatic

5. The Prime Number Theorem
Gauss noticed, as Euclid did, that the prime
numbers begin to dwindle out as we get
higher and higher up the number ladder…
After lots of calculating and trial and error,
Gauss showed that:
Density of Primes ≈ 1/log n
More Precisely,
limlim ππ(x)(x) = 1= 1
xx →→ ∝∝
x/ln xx/ln x

6. Few Interesting Results On
Primes
The Largest Prime is 257,885,161
− 1
courtesy, GIMPS project
The Twin Prime Conjecture
Quartet for the End of Time. Olivier Messiaen
Vinogradov Theorem: All odd numbers
(sufficiently large can be expressed as sum of
three primes) Odd and Even Goldbach
conjectures
All large primes have 1,3,7,9 as their last digit

7. Randomness in Primes
Given n, there is no way we could tell the nth
prime, unlike the squares or even Fibonacci seq.
Although they are deterministic, but they seem to
appear randomly on the number line
“God may not play dice with the universe, but
something strange is going on with the prime
numbers” –Paul Erdos
This randomness aids in many modern day
cryptographic techniques which rely solely on the
“difficulties underlying prime factorization”

8. Fermat’s Little Theorem
If p is a prime then for every 0<a<p
ap-1
≡ 1 (mod p)
Proof?
All a.i(mod p) are distinct and range between 1
and p-1 (inclusive)
Fermat’s Test: for any N pick a<N randomly
if aN-1
≡ 1 (mod N) implies N is prime
Carmichael numbers: numbers that are not prime
yet fool the Fermat's Test. Ex., 561= 3.11.17

9. Private Key Schemes
Two parties secretly agree on some secret code
Each message therefrom is encoded by first
processing it with the secret code (XOR op.)
On the receiver's End, the same computation is
repeated to recover the original code
x ≡ secret code
g ≡ message to be encoded
Sender’s End e(g) = (g o x)
Receiver's End d(e(g)) = ((g o x) o x) = g (voila!)

10. The RSA Algorithm
by Rivest, Shamir & Adleman of MIT in
1977
best known & widely used public-key
scheme uses large integers (e.g., 1024 bits)
security due to cost of factoring large
numbers
Owes its reliability to prime factorization

11. The RSA Algorithm (… contd)
Unlike the previous protocol, the RSA scheme is
an example of public-key cryptography: anybody
can send a message to anybody else using publicly
available information, rather like addresses or
phone numbers. Each person has a public key
known to the whole world and a secret key known
only to him- or herself. When Alice wants to send
message x to Bob, she encodes it using his public
key. He decrypts it using his secret key, to retrieve
x . Eve is welcome to see as many encrypted
messages for Bob as she likes, but she will not be
able to decode them, under certain simple
assumptions.

12. The RSA Algorithm (… contd)
Property Pick any two primes p and q and let N =
p.q . For any e relatively prime to (p − 1)(q − 1) :
1. The mapping x→ xe
(mod N) is a bijection on {0, 1,
2, ... N-1}
2. Moreover the inverse mapping is easily realised,
let d be the inverse of e(modulo (p-1)(q-1))
then for all x e {0,1,…, N-1}
(xe
)d
= x mod N.

13. The RSA Algorithm (… contd)
The first property tells us that the mapping x 7→
xe mod N is a reasonable way to encode messages
x ; no information is lost. So, if Bob publishes (N,
e) as his public key, everyone else can use it to
send him encrypted messages. The second property
then tells us how decryption can be achieved. Bob
should retain the value d as his secret key, with
which he can decode all messages that come to him
by simply raising them to the d th power modulo N
.

14. Queries??
ashu0x00@gmail.com