The document discusses the fundamental theorem of arithmetic and its application to the RSA algorithm. It states that every composite number can be uniquely factored into a product of prime numbers. It then explains the key steps of the RSA algorithm, which relies on selecting two large prime numbers and using their relationship to encrypt and decrypt messages. Finally, it discusses how the fundamental theorem relates to concepts like divisibility, greatest common divisors, and representing numbers as products of prime factors.

1. The RSA Algorithm
Fundamental Theorem of Arithmetic
Presented By;
Antony P Saiji
S7 CSA ,Roll no:16
VJCET

2. Fundamental Theorem of Arithmetic
• Every Composite Number can be expressed (Factorized) as a product of
Prime Numbers, And these factorization is unique apart from the order in
which the Factor occur.

3. The RSA Algorithm
1 Select p,q
2 Calculate n= p xq
Ø(n)=(p-1) x(q-1)
3 Assume e : gcd(e, Ø(n))=1
4 Assume d : d ≡ e−1 (mod Ø (n))
or { d x e mod Ø (n) =1 }
5 Public Key = {e,n}
6 Private Key = {d,n}
p, q both large prime , p ≠ q
Primity Test –(Fermat Theorem)
Q(n) – Euler’s totient Function
1<e<Ø (n); e, Ø(n) Relatively Prime
• Encryption: Plaintext=M<n
Ciphertext , C =(Me)mod n
• Decryption:
Plaintext , M=(Ce)mod n

4. The RSA Algorithm

5. • An important requirement in a number of cryptographic algorithms is the
ability to choose a large prime number. An area of ongoing research is
the development of efficient algorithms for determining if a randomly
chosen large integer is a prime number.
• Its like Prime Numbers are the basic building block of a Number.

6. • Any integer a > 1 can be factored in a unique way as
where p1 < p2 < ... < pt are prime numbers and where each is a positive
integer.
This is known as thefundamental theorem of arithmetic
• 91 = 7 x 13
• 3600 = 24 x 32 x 52
• 11011 = 7 x 112 x 13

7. • If P is the set of all prime numbers, then any positive integer a can be
written uniquely in the following form:
• The right-hand side is the product over all possible prime numbers p; for any
particular value of a, most of the exponents ap will be 0.
• The value of any given positive integer can be specified by simply listing all
the nonzero exponents in the foregoing formulation.
• The integer 12 is represented by {a2 = 2, a3 = 1}.
• The integer 18 is represented by {a2 = 1, a3 = 2}.
• The integer 91 is represented by {a7 = 2, a13 = 1}.

8. • Multiplication of two numbers is equivalent to adding the corresponding
exponents.
• Define k = ab We know that the integer k can be expressed as the product
of powers of primes: . It follows that kp = ap + bp for all p e P.
• k = ab = 12 x 18 = (22 x 3) x (2 x 32) = 216
• k2 = 2 + 1 = 3; k3 = 1 + 2 = 3
• 216 = (23) x (33) = 8 x 27

9. What does it mean, in terms of the prime
factors of a and b, to say that a divides b?
• Any integer of the form can be divided only by an integer that is of a lesser or equal
power of the same prime number, with j <=n. Thus, we can say the following:
If a|b, then a_p b_p then for all p.
• a = 12; b = 36; 12|36
• 12 = 22 x 3 ; 36 = 22 x 32
• a2 = 2 = b2
• a3 = (1<= 2) = b3
• Thus, the inequality (a_p) <=(b_p) is satisfied for all prime numbers.

10. • It is easy to determine the Greatest Common Divisor of two positive
integers if we express each integer as the product of primes.
• 300 = (22 )x (31) x (52)
• 18 = (21) x (32)
• gcd(18,300) = (21) x (31) x (50) = 6
• The following relationship always holds:
If k = gcd(a,b) then kp = min(ap, bp) for all p