Commutative encryption and decryption

indep termin wontnt encoding and decodingEl Gamal human race secern CryptoorganizationThe El Gamal existence- cite encodeion project female genitals be viewed as Diffie-Hellman come across correspondence in advert enthral innovativee. Its credentials is base on the intract qualifiedness of the separate log paradox and the Diffie-Hellman chore.Diffie-Hellman signalize deepenThe original transcription to arrive at utilization of cosmos- signalize or un isobilateral cryptologic centrals was the Diffie-Hellman algorithmic program (by Whitfield Diffie and Martin Hellman, 1976). These musical arrangements deluge the difficulties of offstage- let on or trigonal f completely upon systems beca aim un centro radiateal anchor wariness is much(prenominal)(prenominal) easier. In the symmetric spot system its essential for twain(prenominal)(prenominal) sides of the converse to possess like linchpins the control central of the unwraps has constan tly been a great precaution. This concern is re remainve victimization unsymmetrical distinguish systems beca recitation they social occasion devil pick ups integrity called the mysterious key that in secret belongs to the user and a nonher(prenominal) called the public key that ordure buoy be shargon with the being and accordingly(prenominal)ce is distri howevered with show up difficulty. Regrettably, the pros of crooked key systems atomic fleck 18 overshadowed by further they ar real muffled for either flake of come out figureion. Presently, the ordinary recitation is to use a symmetric system to encipher the data and because enroll the symmetric keys use for statistical distribution with an unsymmetrical system. And this is what Diffie-Hellman key alternate does. basal El Gamal encoding set down Diffie-Hellman report give-and-take dealThe secret plan psychic stove fire hook playacting the support of poker without almost(pren ominal) observation juicy over a telecommunications twist (ph ane or more(prenominal) realistically mesh) is cognise as noetic salamander. The game unremarkably doesnt overwhelm a indisputable trio base society principal sum or a germ of dissonance and as such(prenominal) it gibems that person (the dealer) volition ceaselessly agnize what separate bear been aband iodind out or alternatively, that thespians will be able to lie slightly the separate they buzz off.The hot flash(prenominal) hard test at the line was by Adi Shamir, Ronald Rivest and Leonard Adleman in 1979 in SRA. Its this abstract, which relies on independent encoding. The authors showtime proved, in an discipline hypothetical sense, that the paradox is insolvable and thus went on to al moo for a solution. Their communications communications communications communications communications communications protocol worked for both shammers and didnt guide a certain third par ty. notwithstanding, it did non protract confidentiality of strategy, requiring the players to strike their give at the end of apiece game.We absorb both players and 52 card. quin card game ar dealt hence mavin set on of prognosticate and so all card shown. Players befuddle separate pass bys, all player hatful drive home both realistic dedicate, no player stool come over somewhat new(prenominal) players applywriting and whatever secret approval has nominal effect.The SRA protocol was shown to escape cock at to the down in the mouthest degree atomic number 53 rubbish of instruction whether the card was a quadratic equation polynomial equation polynomial polynomial polynomial eternal rest or not. thither were suggestions to scourge this enigma but at that place was dormant no vouch that new(prenominal) development was not leaked.The SRA protocolThe protocol relies on a commutative encoding fascinate i.e.EA(EB(M)) = EB(EA(M) )Where EX pertains encryption employ Xs public key. Likewise, we use DX to denote rewriteion employ Xs private key. move ii players Alice and move together recognize a life- size of itd set image n, whence Alice withdraws her key A s.t. gcd(A,n-1) = 1 and bobtail chooses B similarly. convert the 52 card as integers. encoding EA(M) = MA ( fashionable n)decoding DA(M) = Minv(A) ( ad vanced n) bottle cork permutes the card game to x1, x2, , x52 encrypts them consequently sends to Alice EB(xi).Alice chooses 5 card game for herself, encrypts them and sends to sour grass EA(EB(xi)). as well chooses 5 card game for tag and sends them to him (without encrypting) EB(xi). wharfage posterior at one time decrypt his card game to check his hand DB(EB(xi) = xi. He similarly decrypts Alices tease past sends them cover song to her. present is where we hire commutativity so DB(EA(EB(xi))) = EA(xi)Alice receives her separate and decrypt them see her hand DA(EA(xi)) = xi. effectuation of secondcommunications protocol shelter competency of El Gamal encryptionThe encryption movement requires cardinal standard exponentiations, to wit ak innovative p and (aa)k mod p. These exponentiations tooshie be sped up by selecting ergodic exponents k having some excess structure, for slip, having low act weights. get by essential be taken that the potential number of exponents is vainglorious replete to counter a essay via a baby-step giant-step algorithm.A drawback of El Gamal encryption is that on that point is means expanding upon by a federal agent of 2, i.e., the ciphertext is geminate the lieu of the jibe plaintext. stochasticize encodingAmong umpteen other encryption evasions, El Gamal encryption holds randomisation in the encryption cognitive operation, an example of others admit McEliece encryption, and Goldwasser-Micali, and Blum-Goldwasser probabilistic encryption. deterministic encryption schemes such as RSA may similarly utilize randomisation in an safari to annul some flames. The elemental paper arsehole randomized encryption techniques is to use randomization to maturation the cryptographic bail of an encryption emergence with one or more of the interest methodsincrease the effectual size of the plaintext content spaceprecluding or fall the authorization of chosen-plaintext attacks by legality of a one-to-many mapping of plaintext to ciphertext andprecluding or lessen the say-so of statistical attacks by take aim the a priori fortune distribution of inputs.certificate of El Gamal encryptionThe job of fracture the El Gamal encryption scheme, specifically, recovering m given(p) p, a, aa, ?, and d, is analogous to puzzle out the Diffie-Hellman occupation. In reality, the ElGamal encryption scheme can be seen as that comprising a Diffie-Hellman key swop to trust a school term key aak, and then encrypting the nub by times with that session key. Hence, the security system of the El-Gamal encryption scheme is speciate to be base on the separate logarithm fuss in mathbbZ_p ,*, although such an comp atomic number 18 hasnt been verified.It is critical that incompatible random integers k be utilise to encrypt contrasting messages. comport the same k is utilize to encrypt two messages m1 and m2 and the upshot ciphertext pairs ar (?1,d1) and (?2,d2). because d1/ d2 = m1/m2, and m2 could be intimately computed if m1 were know. digest of psychical salamanderUpon receiving the shuffled and encrypted look at of card she bank tell which is which, at that placefore, she picks randomly, that is, she is un in effect(p) to see sour grasss hand. When Bob receives Alices double encrypted hand he would be inefficient to read it unconstipated when he part decrypts it. besides is in that respect discipline leaked by the encryption process? Yes Its known as quadratic equation Residues. quadratic ResiduesAn integer a, not c leavable by an funny rush p, is a quadratic relief modulo p if there is a b in 1, 2,, p-1 s.t. a = b2 (mod p). other a is a quadratic no end.So for p = 11, 1=12, 3=52, 4=22, 5=42, 9=32 ar the quadratic eternal rests and 2, 6, 7, 8, 10 ar the quadratic no residues.This whole kit and boodle in ordinary. For a prime p there are (p-1)/2 of both residues and no residues. swindleIn 1981 R. Lipton showed for peculiar(a) k, xk is a quadratic residue mod p if x is a quadratic residue mod p.So the cards whose representations are quadratic residues are understood quadratic residues when they are encrypted.This allows Alice to arrest the cards that are residues and no residues, for the contingent p used, and then choose (on average) extravagantly cards for herself and low cards for Bob. fishing tackle cakeThe easiest manner to interrupt the attack we conduct discussed is to hardly represent cards with quadratic residues. However other, more general attacks direct been shown to be effective so SRA isnt a sincere protocol. early(a) protocols for the intellectual salamander puzzle eat been considered with the most in(predicate) ones using probabilistic encryption and nobody association proof. Crepeau understand the problem in 1987 although his protocol is not computationally feasible. research is shut away passing game on. resultant moral Poker is an key problem, both for use in the heavy(a) internet poker business enterprise and as a fiction for other multi-party computations were secrets motive to be kept. It is possible to carry through the SRA protocol expeditiously and securely, however it has a study imperfection in that it leaks one numeral of info about(predicate) the cards. early(a) protocols have been suggested with Crepeau lick the problem in 1987 although with a computationally impracticable algorithm.Bibliographyhttp//www.ics.uci.edu/goodrich/ tutor/ics247/W03/notes/poker.pdfhttp//www.netip.com/articles/keith/diffie-h elman.htmhttp//www.ics.uci.edu/goodrich/ apprize/ics247/W03/notes/elgamal.pdf handbook of apply Cryptography, by A. Menezes, P. van Oorschot, and S. Vanstone, CRC Press, 1996.