Application Of Mealy Machine And Recurrence Relations In Cryptography

Transcription

1 Applicatin Of Mealy Machine And Recurrence Relatins In Cryptgraphy P. A. Jytirmie 1, A. Chandra Sekhar 2, S. Uma Devi 3 1 Department f Engineering Mathematics, Andhra University, Visakhapatnam, IDIA 2 Department f Mathematics, GIT, GITAM University, Visakhapatnam, IDIA 3 Department f Engineering Mathematics, Andhra University, Visakhapatnam, IDIA 1. ABSTRACT Cryptgraphy is the study f techniques fr ensuring the secrecy and authenticatin f the infrmatin. Public key encryptin schemes are secure nly if the authenticity f the public key is ensured. The imprtance f security f data is ever expanding with increasing impact f internet as means f cmmunicatin and e-cmmerce. It is essential t prtect the infrmatin frm hackers and eavesdrppers. Secret sharing schemes are ideal fr string infrmatin that is highly sensitive. The mtivatin fr secret sharing is secure key management.in this paper, with the help f finite state machine (Mealy machine) and recurrence matrices a secret sharing scheme is develped fr secure cmmunicatin which is designed fr encryptin and als maintains secrecy f the message. Key wrds: Mealy Machine, Recurrence matrix, Fermat s and Mersenne s sequence. 1. ITRODUCTIO There is a scpe fr a wide range f applicatin f autmatn thery in the field f cryptlgy. In autmata thery, a branch f theretical cmputer science, a deterministic finite autmatn (DFA) als knwn as deterministic finite state machine is a finite state machine that accepts/rejects finite strings f symbls and nly prduces a unique cmputatin (r run) f the autmatn fr each input string. 'Deterministic' refers t the uniqueness f the cmputatin. The finite autmatn is a mathematical mdel f a system with discrete inputs and utputs. The system can be ne f a finite number f internal cnfiguratins r states [2][9][4]. The finite autmatn is a mathematical mdel r a system, with discrete inputs and utputs. When the finite autmata is mdified t allw zer, ne, r mre transitins frm a state n the same input symbl then it is called a nndeterministic finite autmata. Fr deterministic autmata the utcme is a state, i.e., an element f Q. Fr nndeterministic autmata the utcme is a subset f Q, where Q is a finite nnempty set f states. Autmata thery is the study f abstract cmputing devices r machines. It is a behavir mdel cmpsed f a finite number f states, transitin between thse states and actins in which ne can inspect the way lgic runs when certain cnditins are met. Recently finite state machines are used in cryptgraphy, nt nly t encrypt the message, but als t maintain secrecy f the message. In this paper, new secret sharing scheme is prpsed using finite state machines. Secret sharing schemes were discvered independently by Blakley and Shamir. The mtivatin fr secret sharing is t have secure key management. In sme situatins, there is usually ne secret key that prvides access t many imprtant files, if such a key is lst then all the imprtant files becme inaccessible. The basic idea in secret sharing is t divide the secret key int pieces and distribute the pieces t different persns s that certain subsets f the persns can get tgether t recver the key.[7][1] In Mealy Machine every finite state machine has a fixed utput. Mathematically Mealy machine is a six-tuple machine and is defined as : M= ( Q,,,, ', q0 ) Q : A nnempty finite set f state in Mealy machine : A nnempty finite set f inputs. : A nnempty finite set f utputs. 1286

2 : It is a transitin functin which takes tw arguments ne is input state and anther is input symbl. The utput f this functin is a single state. ' : Is a mapping functin which maps Q x t, giving the utput assciated The sequence 0, 1, 3, 7, 15, 31,... is the Mersenne sequence, and 2, 3, 5, 9, 17, 33,... is the Fermat sequence. These are just pwers f 2 plus r minus 1 with each transitin. q 0 : Is the initial state in Q Mealy machine can als be represented by transitin table, as well as transitin diagram. w, we cnsider a Mealy machine [4][8][9]. 2. Prpsed Algrithm Encryptin Step 1 Represent plain text with P Step 2 Divide the plain text int n number f texts i.e int square matrices. Step 3 Define a Finite state machine thrugh public channel. Step 4 Define input. Step 5 Get utput thrugh finite state machine. Fig 1 Mealy machine..in the abve diagram 0/1 represent input/utput. Recurrence Matrix Recurrence matrix is a matrix whse elements are taken frm a recurrence relatin [1]. The recurrence matrix in this paper is defined as R n = 1 C n+1 C n C n+1 1 C n+2 C n C n+2 1 R n is a symmetric matrix. where n 0 and C n s are either taken frm Fermat s sequence r Mersenne s sequence. Step 6 Define recurrence matrix and chse recurrence relatin. Step 7 Define the value f n f recurrence relatin. Step 8 Define cipher text at each stage fr all the plain texts. Step 9 Send the cipher text t the respective receivers. Decryptin The message is decrypted using the inverse peratin and key t get the riginal message. 1287

3 5 Implementatin 3 Perfrmance analysis Mathematical analysis Algrithm prpsed, is a simple applicatin f additin f tw matrices. But the recurrence matrix and elements f recurrence matrix are different at each stage depending n the input and utput. It is very difficult t break the cipher text withut prper key, defined peratin and the chsen finite state machine. The key is defined as the sum f all the elements f this plain text. We assign 1 t letter a, 2 t letter b and s n and 26 t the letter z. We assign 27 t full stp and 0 t space. Let us encrypt the Birds are singing. This sentence has eighteen characters which include space and full stp. Step 1 P= BIRDS ARE SIGIG Step 2 S. 4. Security analysis Extracting, the riginal infrmatin frm the Cipher text is difficult due t the selectin f the recurrence matrix, secret key and chsen finite state machines. Brute frce n key is als difficult because f the key size. ame f the 1 Cipher text 2 Knwn plain text 3 Chsen plain text 4 Adaptive chsen plain text 5 Chsen cipher text 6 Adaptive chsen cipher text Table 1 Security analysis Pssibility f the It is difficult t crack the cipher text. t crack Cipher text t crack Cipher text Remarks Because f the chsen finite state machine and the key. Because f the chsen finite state machines and key. Because f the chsen finite state machine and key. Because f the chsen finite state machines and different individual keys. Because f the chsen finite state machine, key and the recurrence matrix. Because f the chsen finite state machine, key and chsen recurrence matrix at each stage. As per the algrithm, we cnstruct tw plain texts A= And B= B I R D S A R E = S I G I G = w, we apply the remaining algrithm t plain texts A and B. Step 3 Mealy machine is publicized thrugh public channel. Step 4 All the elements f bth the plain texts are added and cnverted int binary frm. This is the input. The sum f all the elements f plain text A is equal t 76 = ( ) 2 And B is equal t 106 = ( ) 2 Step 5 The utput f the abve key is fund with the help f Mealy machine. Step 6 The recurrence matrix is defined as R n = 1 C n+1 C n C n+1 1 C n+2 C n C n

4 S l. In Pu t The elements f recurrence matrix are taken frm Out put / Tra nsiti n a recurrence relatin. Step 7 Let C i+1 be the cipher text at C i +i th state and is defined as C i+1 = C i +R n Key values Where R n depends n the input. Cipher text S l. In Pu t Out put / Tra nsiti n Key values Cipher text Receiver 2 Send cipher text And cipher text t receiver 1 t receiver 2. Fermat s sequence when input=0 R n = Mersenne s sequence when input = 1 Cncluding Remarks Step 8 Calculate cipher text at each state. Then the cipher text at each state is as fllws: Algrithm prpsed, is based n finite state machine and peratins n matrices. Secrecy is maintained at fur levels 1. The secret key. 2. The chsen finite state machine 3. The different peratins Receiver 1 4. The recurrence matrix. 1289

5 The btained cipher text becmes quite difficult t break r t extract the riginal infrmatin even when the algrithm is knwn. References [1] B.Krishna Gandhi, A.Chandra Sekhar, S.Srilakshmi Cryptgraphic scheme fr digital signals using finite state machine Internatinal jurnal f cmputer applicatins (September 2011). [2] Adesh K.Pandey. Reprint An intrductin t autmata thery and frmal languages S.K.Kararia & sns. ew Delhi. [3] A.Menezed, P.Van Orscht - Hand bk f Applied and S.Vanstne Cryptgraphy e-bk. [4] Jhn E.Hpcrft, Rajeev Mtwain, Jeffrey D.Ulman Intrductin t autmata thery, language, and cmputatin Vanstne3 rd impressin, 2007 CRC Press., Drling Kindersley (India) Pvt. Ltd. [5] [6] ELGamal. A public key Cryptsystem and a signature scheme based n discrete lgarithms. In Advances in Cryptlgy (CRYPTO 1984), Springer. [7] W.Diffi and M.E.Helman ew directins in Cryptgraphy. IEEE Transactins n Infrmatin thery, 22, , [8] A Curse in umber Thery and Cryptgraphy by eal Kblitz. [9] Thery f Cmputatins by Mishra and Chandrashekharan. 1290