A Project Report On
Implementation of RSA Algorithm
by Md Saif H2016124031 Mounesh H2016124024
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI (RAJASTHAN) Hyderabad Campus (NOV 2016)
ABSTRACT This paper focuses on asymmetric Public key cryptographic method called RSA. RSA is one of the first practical public-key cryptosystems and is widely used for secure data transmission. In such a cryptosystem, the encryption key is public and differs from the decryption key which is kept secret. In RSA, this asymmetry is based on the practical difficulty of factoring the product of two large prime numbers, the factoring problem. RSA is made of the initial letters of the surnames of Ron Rivest, Adi Shamir, and Leonard Adleman, who first publicly described the algorithm in 1977. A user of RSA creates and then publishes a public key based on two large prime numbers, along with an auxiliary value. The prime numbers must be kept secret. Anyone can use the public key to encrypt a message, but with currently published methods, if the public key is large enough, only someone with knowledge of the prime numbers can feasibly decode the message. Hence second half of the paper discusses on security enhancement of RSA where two ciphers are generated which makes it difficult to decrypt the, message as to decrypt the message attacker now need, not only the private key but also random numbers k and s from which new ciphers are calculated.
CONTENTS
Cryptography Basic RSA Modified RSA Implementation of Modified RSA Flow chart showing Modified RSA Encryption Flow chart showing Modified RSA Decryption Simulation Results Discussion on Results References Appendix
Cryptography:[1] Is the practice and study of techniques for secure communication in the presence of third parties called adversaries. More generally, cryptography is about constructing and analyzing protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, and electrical engineering. Applications of cryptography include ATM cards, computer passwords, and electronic commerce. Modern cryptography is heavily based on mathematical theory and computer science practice; cryptographic algorithms are designed around computational hardness assumptions, making such algorithms hard to break in practice by any adversary (Attacker) The modern field of cryptography can be mainly divided into two cryptographic procedures. 1) Symmetric Cryptography 2) Asymmetric Cryptography [public key algorithm] Symmetric-key cryptography: Symmetric-key cryptography refers to encryption methods in which both the sender and receiver share the same key (or, less commonly, in which
their keys are different, but related in an easily computable way). This was the only kind of encryption publicly known until June 1976. Symmetric key ciphers are implemented as either Block ciphers Stream ciphers A block cipher enciphers input in blocks of plaintext. A stream cipher enciphers input as individual characters. The Data Encryption Standard (DES) and the Advanced Encryption Standard (AES) are block cipher designs that have been designated cryptography standards by the US government. DES remains quite popular; it is used across a wide range of applications, from ATM encryption to e-mail privacy and secure remote access. Asymmetric Cryptography [Public key cryptography]
Is any cryptographic system that uses pairs of keys: public keys which may be disseminated widely, and private keys which are known only to the owner. This accomplishes two functions: authentication, which is when the public key is used to verify that a holder of the paired private key sent the message, and encryption, whereby only the holder of the paired private key can decrypt the message encrypted with the public key. In a public key encryption system, any person can encrypt a message using the public key of the receiver, but such a message can be decrypted only with the receiver's private key. For this to work it must be computationally easy for a user to generate a public and private key-pair to be used for encryption and decryption. The strength of a public key cryptography system relies on the degree of difficulty (computational impracticality) for a properly generated private key to be determined from its corresponding public key. Security then depends only on keeping the private key private, and the public key may be published without compromising security. Public key cryptography systems often rely on cryptographic algorithms based on mathematical problems that currently admit no efficient solution—particularly those inherent in certain integer factorization, discrete logarithm etc. unlike symmetric key algorithms, asymmetric do not require a secure channel for the initial exchange of one (or more) secret keys between the parties. Because of the computational complexity of asymmetric encryption, it is usually used only for small blocks of data, typically the transfer of a symmetric encryption key. This symmetric key is then used to encrypt the rest of the potentially long message sequence. The symmetric encryption/decryption is based on simpler algorithms and is much faster. Public key cryptography finds application in, among others, the information technology security discipline, information security.
Information security (IS) is concerned with all aspects of protecting electronic information assets against security threats storage. RSA:[2] At present, the best known and most widely used public key system is RSA. It is considered secure when sufficiently long keys are used. The security of RSA depends on the difficulty of factoring large integers. The RSA algorithm involves four steps: 1. 2. 3. 4.
Key generation Key distribution Encryption Decryption.
RSA involves a public key and a private key. The public key can be known by everyone and is used for encrypting messages. The intention is that messages encrypted with the public key can only be decrypted in a reasonable amount of time using the private key. 1)Key generation: a) Select any two prime numbers p, q, where p and q should be large. Typical values for these primes are 512 to 4096bits. b) Compute 𝒏 = 𝒑∗𝒒 c) Compute totient function φ(n). Totient function counts the positive integers up to a given integer n that are relatively prime n. 𝝋(𝒏) = (𝒑 − 𝟏) ∗ (𝒒 − 𝟏)
d) Then public Key e is selected as follows 𝟏 < 𝒆 < 𝝋(𝒏) 𝑎𝑛𝑑
𝒈𝒄𝒅(𝝋(𝒏), 𝒆) = 𝟏
e) Private key is 𝒅 is calculated (𝒅 ∗ 𝒆)𝒎𝒐𝒅 𝝋(𝒏) = 𝟏 2) Key Distribution: Prior to the encryption and decryption, user needs generate public and private key as mentioned above. Public key is known to everyone. Anybody who wants to send the message uses the public key for encryption. Private key is unique and is known only to intended receiver. Private key is used to decrypt the cipher and therefore should not be shared with anyone. 3)Encryption: To encrypt the message public key is required. The length of message signal that can be encrypted is bounded by the constraint 𝟏<𝒎<𝒏 In order to increase the length of message to be encrypted large values of prime number p and q should be chosen. The encryption of message 𝒎 is 𝒄 = 𝒎^𝒆 (𝒎𝒐𝒅 𝒏) 4)Decryption: To decrypt the message private key 𝒅 is required. The decryption of cipher 𝒄 will be 𝒎 = 𝒄𝒆 (𝒎𝒐𝒅 𝒏)
RSA explained with Example: Key generation: Choose two prime numbers 𝒑 = 𝟏𝟎𝟏 𝒂𝒏𝒅 𝒒 = 𝟏𝟎𝟑 Compute the product 𝒏 = 𝒑 ∗ 𝒒 = 𝟏𝟎𝟏 ∗ 𝟏𝟎𝟑 = 𝟏𝟎𝟒𝟎𝟑 Compute totient function 𝝋(𝒏) = (𝒑 − 𝟏) ∗ (𝒒 − 𝟏) = 𝟏𝟎𝟎 ∗ 𝟏𝟎𝟐 = 𝟏𝟎𝟐𝟎𝟎 Choosing Public key Choose a number e such that 𝟏 < 𝒆 < 𝝋(𝒏) 𝒂𝒏𝒅 𝒆 𝒂𝒏𝒅 𝒏 𝒂𝒓𝒆 𝒄𝒐𝒑𝒓𝒊𝒎𝒆. 𝑳𝒆𝒕 𝒆 = 𝟕 Compute a value for private key 𝒅 such that it satisfies (𝒅 ∗ 𝒆) % 𝝋(𝒏) = 𝟏. [(𝟑 ∗ 𝟕) % 𝟐𝟎 = 𝟏] One of the solutions satisfying above condition is 𝒅 = 𝟖𝟕𝟒𝟑 Key Distribution: Public key is (𝒆, 𝒏) => (𝟕, 𝟏𝟎𝟒𝟎𝟑)
known to everyone
Private key is (𝒅, 𝒏) => (𝟖𝟕𝟒𝟑, 𝟏𝟎𝟒𝟎𝟑)
intended receiver
Encryption: 𝒄 = 𝒎^𝒆 (𝒎𝒐𝒅 𝒏) If we want encrypt letter h. ASCII value of h is 104. It is encrypted as follows 𝒄 = 𝟏𝟎𝟒𝟕 (𝒎𝒐𝒅 𝟏𝟎𝟒𝟎𝟑) 𝑐 comes out as
𝒄 = 𝟖𝟓𝟓𝟎
Decryption: 𝒎 = 𝒄𝒅 (𝒎𝒐𝒅 𝒏) 𝒎 = 𝟖𝟓𝟓𝟎𝟖𝟕𝟒𝟑 (𝒎𝒐𝒅 𝟏𝟎𝟒𝟎𝟑) 𝒎 = 𝟏𝟎𝟒
Advantages and Disadvantages of RSA Advantages: RSA relies on the fact that it is easy to multiply two large prime numbers together but extremely hard [time consuming] to factor them back to result. for ex: the product of 3391 and 23279 can be calculated easily its 78939089 but it’s extremely difficult to factorize the same number to get back factors 3391 23279
Disadvantages: 1. If factorization of n [p*q] is possible then whole RSA algorithm is compromised. 2. Encryption and Decryption takes a long time for large p and q. Hence we have to bring some modifications which increases the security of existing algorithm but not at the cost of time
POSSIBLE SOLUTIONS For Enhancing the security 1. Use of fake modulus i.e. instead of sending n , Fake modulus Fn is sent over the channel. 2. Instead of using Two prime to calculate n, we can use three prime there by making factorization more complex.
For Enhancing the Speed 1. Offline key generation 2. Montgomery Multiplication 3. Improving Modular Exponentiation 4. Many more mathematical improvements using concepts of number theory.
PROPOSED DESIGN RSA algorithm can be improved in two areas as mentioned 1)Security 2)Time We aim to increase the security of Basic RSA algorithm by introducing randomness in encryption process In proposed scheme we will be sending two ciphers One cipher is calculated using random number The other cipher will contain the information about the message text 𝐶1 : from random number selected 𝐶2 : from the message text Attacker has to break two ciphers to retrieve the message
IMPLIMENATAION Generation of keys As compared to basic RSA where 2 keys are generated we will generating three keys thus increasing the security Generation of public key e Choose two large prime numbers p and q find n and totient function φ(n) 𝒏=𝒑∗𝒒 𝝋(𝒏) = (𝒑 − 𝟏) ∗ (𝒒 − 𝟏) Next we compute "e " as 𝒈𝒄𝒅[𝒆, 𝝋(𝒏) ] = 𝟏 𝒘𝒉𝒆𝒓𝒆 𝟏 < 𝒆 < 𝝋(𝒏) Generation of Private key d Private key d is calculated using (𝒅 ∗ 𝒆) % 𝝋(𝒏) = 𝟏 Generation of random key k [ Modified part] we compute “k " as 𝒈𝒄𝒅[𝒌, 𝒏 ] = 𝟏
𝒘𝒉𝒆𝒓𝒆 𝟏 < 𝒌 < < 𝒏
Encryption: In our modified RSA Encryption is two step process 1) In first step we calculate first cipher 𝒄𝟏 using 𝒌 and public key 𝒆 𝒄𝟏 = 𝒌𝒆 𝒎𝒐𝒅 𝒏 2) In second step we encrypt our message using 𝒌 and public key 𝒆 𝒄𝟐 = (𝒌. 𝒎𝒆 )𝒎𝒐𝒅 𝒏 Final cipher that will sent to receiver is 𝒄 = [𝒄𝟏 𝒄𝟐 ] Decryption: In our modified RSA decryption is three step process 1) First cipher 𝒄𝟏 is decrypted to get random number 𝒌 using private key 𝒅 𝒌 = 𝒄𝒅𝟏 (𝒎𝒐𝒅 𝒏) 2) In second step we calculate unique integer 𝒔 using random number 𝒌 (𝒔. 𝒌) 𝒎𝒐𝒅 𝒏 = 𝟏
such that 𝟏 < 𝒔 < 𝒏
3) In third step we decrypt cipher 𝒄𝟐 using 𝒔 𝒎 = (𝒄𝟐 . 𝒔) 𝒎𝒐𝒅 𝒏
Flow Chart Modified RSA Encryption Input any two large prime numbers
Whether Prime
n=p*q ∅(𝑛) = (p-1)*(q-1)
Input any number
Input any number
e in the range 1
K in the range 1
Whether e and ∅(𝑛) are coprime
Whether k and ∅(𝑛) are coprime
Plain text Calculate private key d
Cipher 𝑐2
Cipher 𝑐1
Flow Chart Modified RSA Decryption
Receive 𝑐1, 𝑐2 𝑓𝑟𝑜𝑚 𝑠𝑒𝑛𝑑𝑒𝑟
Input number s
Calculate 𝑘 from 𝑐1
(𝑠 ∗ 𝑘)𝑚𝑜𝑑 𝑛 = 1
yes
From s and 𝑐2 decrypt message
Plain Text
No
SIMULATED RESULTS Basic RSA results
Modified RSA results Example 1:
Modified RSA results Example 2:
Modified RSA results Example 3:
Modified RSA results Example 4:
Discussion on results Basic version RSA lacks randomization component and if attacker succeeds in factorizing prime numbers. Then whole system is compromised. As we have seen from the results we have increased security by multilevel encryption and decryption. To decrypt the message attacker, need, not only private key but also random numbers k and s
REFERENCES [1]Ranjan Bose “ Information theory, coding and cryptography”, 2014, pp. 283-294. [2]Xin Zhou , Xiaofei Tang, “Research and Implementation of RSA Algorithm for Encryption and Decryption ,” The 6th International Forum on Strategic Technology, China,2011 [3]Alaa Hussein Al-Hamami, Ibrahem Abdallah Aldariseh , “Enhanced Method for RSA Cryptosystem Algorithm” Advanced Computer Science Applications and Technologies (ACSAT), 2012 International Conference, Jordan, 2012 [4]“Enhancing Security Features in RSA Cryptosystem,” IEEE Symposium on Computers & Informatics,2012 [5]M. Bahadori, M. R. Mali, O. Sarbishei, M. Atarodi and M. Sharifkhani “A novel approach for secure and fast generation of RSA public and private keys on SmartCard” NEWCAS Conference (NEWCAS), 2010 8th IEEE International, 2010, pp. 265-268. [6]H. Ge and S. R. Tate “Efficient Authenticated Key-Exchange for Devices with a Trusted Manager” Information Technology: New Generations, 2006 (ITNG2006). Third International Conference on, 2006, pp.198-203
Appendix MODIFIED RSA: Main Program clc; clearvars; p=input('enter the value of prime no p\n'); q=input('enter the value of prime no q\n'); [phi,n,d,e,k]=INIT_MOD(p,q); disp( sprintf( 'The value of n is n= %d\n', n ) ); disp( sprintf( 'The value of Totient function is phi(n)= %d\n', phi ) ); disp ( sprintf( 'The value public key is e= %d\n', e ) ); disp( sprintf( 'The value private key is d= %d\n', d ) ); disp( sprintf( 'The value random number in range (i
Function for Calclation of phi,n,d,e,k function[phi,n,d,e,k]=INIT(p,q) n=p*q; phi=(p-1)*(q-1); e=1; i=2; while i>1 e=e+1; i=gcd(phi,e); end
d=1; j=2; while j>1 if rem(d*e,phi)~=1 d=d+1; j=rem(d*e,phi); end end k=5; i=2; while i>1 k=k+5; i=gcd(n,k); end end
Function for Encryption function[en]=encryption_MOD_1(z,c,e,n) [e]=dtob(e); k = 65535; c = c; cf = 1; cf=mod(z*c*cf,n); for i=k-1:-1:1 c = mod(c*c,n); j=k-i+1; if e(j)==1 cf=mod(c*cf,n); end end en=cf;
Function for text to ASCII conversion function[c]=text2ascii(m) x=length(m); c=0; for j=1:x for i=0:122 if strcmp(m(j),char(i)) c(j)=i; end end end
Function for decimal to binary function[a]=dtob(d) i=1; a=zeros(1,65535); while d >= 2 r=rem(d,2); if r==1 a(i)=1; else a(i)=0; end i=i+1; d=floor(d/2); end if d == 2 a(i) = 0; else a(i) = 1; end
Function for Decryption function[en]=decryption_MOD(c,e,n) [e]=dtob(e); k = 65535; c = c; cf = 1; cf=mod(c*cf,n); for i=k-1:-1:1 c = mod(c*c,n); j=k-i+1; if e(j)==1 cf=mod(c*cf,n); end end en=cf;
Function for generation of random number k function[d]=random_MOD(e,n) d=1; j=2; while j>1 if rem(d*e,n)~=1 d=d+1; j=rem(d*e,n); end end end
Function for calculation of s function[s]=random_MOD(k,n) s=1; j=2; while j>1 if rem(s*k,phi)~=1 d=d+1; j=rem(s*k,phi); end end
