“Use of Number Theory in Cryptography”

 “Use of Number Theory in Cryptography”

With the growing quantity of digital data stored and communicated by electronic data processing systems, organizations in both the public and commercial sectors have felt the need to protect information from unwanted intrusion. Thus, there was a recent surge of interest by Mathematicians and computer scientists in Cryptography made from two Greek words “Krypto” meaning hidden and “graphien ” meaning to write. In this language codes are called Ciphers and the information to be concealed is called Plain text. The secret form of message is Cipher text. For encryption and decryption, we make use of the concept “Congruence Modulo n” in different context from Number Theory. The process of converting from Plaintext to Cipher text is said encrypting whereas the reverse process of changing from ciphertext back to plaintext is called decrypting or deciphering. The Caesar cipher can be described easily by using congruence theory. If P is the digital equivalent of plaintext letter and C is the digital equivalent of the corresponding Ciphertext letter, then C≡P+3(mod26) .

                   To recover the plaintext, the procedure is simply reversed by means of the congruence C≡C-3(mod26). Here, 26 is used as we are having 26 alphabets in all and here, we assign a numerical script to represent each of these. To illustrate Hill’s Cipher we use the
C₁≡aP₁+bP₂(mod26)
C₂≡cP₁+dP₂(mod26)

Where  must be selected so that gcd(ad-bc,26)=1 . The public key system most widely used was proposed in 1978 by Ronald Rivest, Adi Shamir and Leonard Adleman and is called RSA Cryptosystem. The RSA Cryptosystem uses computations in Z_n , where n is the product of two distinct odd primes p and q.  For such integer n, we have Φ(n) = (p − 1)(q − 1). Here, we make use of prime numbers, their properties to get the enciphering modulus. Also, Euler’s Phi function is used to obtain n. In this case the decryption process is carried out by using the Euclidean Algorithm to obtain the integer 1 <j<Φ (n) satisfying kj ≡ 1(modΦ(n)) . The security of the RSA Cryptosystem is based on the belief that the encryption function e_k(x)=x^bmod n is a one-way function so it will be computationally infeasible for an opponent to decrypt a Ciphertext. Thus, to compute the decryption exponent a in d_k(y)=y^amod n    Extended Euclidean Algorithm is used.   Some basic concepts of primitive roots and quadratic residues can be used in ElGamal Cryptosystem. Therefore, from above we can say that the basic concepts of Number Theory can be applied to develop more effective methods to secure information in many areas.

Dr. Alka Kumari

Assistant Professor, Department of Mathematics

Patna Women’s College (Autonomous)

Mail-ID: alka.math@patnawomenscollege.in