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Privacy Homomorphisms for E-Gambling and Mental Poker Jordi Castella-Roca,` Vanesa Daza, Josep Domingo-Ferrer, Senior Member, IEEE, and Francesc Sebe´

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Privacy Homomorphisms for E-Gambling and Mental Poker Jordi Castella-Roca,` Vanesa Daza, Josep Domingo-Ferrer, Senior Member, IEEE, and Francesc Sebe´ Privacy Homomorphisms for E-Gambling and Mental Poker Jordi Castella-Roca,` Vanesa Daza, Josep Domingo-Ferrer, Senior Member, IEEE, and Francesc Sebe´ Abstract— With the development of computer networks, situ- 3) The encrypted deck of cards is processed sequentially by ations where a set of players remotely play a game (e-gaming) all the players. During this process, the player receives have become usual. Often players play for money (e-gambling), the set C, permutes its elements and then re-masks (i.e. which requires standards of security similar to those in physical gambling. Cryptographic tools have been commonly used so far re-encrypts) them. The set resulting from the previous to provide security to e-gambling. Homomorphic encryption is step corresponds to the shuffled encrypted deck of cards. an example of such tools. The shuffled encrypted deck of cards is made up of 52 In this paper we review the mental poker protocols, where encrypted values such that none of the players knows which players are assumed to remotely play poker. We focus on the key advantage of using cryptosystems with homomorphic properties card is contained in each. Note that if Step 3 consisted of a (privacy homomorphisms) because they offer the possibility of single permutation, at least the player who performed Step 2 manipulating cards in encrypted form. would know the content of the final shuffled and encrypted Index Terms— Cryptography, privacy homomorphism, mental deck since she knows which cryptogram corresponds to each poker card. For a secure generation of the encrypted shuffled deck, it is necessary that, after permuting the encrypted deck, its elements are re-masked. Re-masking consists of performing I. INTRODUCTION some operation over each encrypted value so that its cleartext The growth of computer networks has allowed many activ- does not change but the re-masked message is not linkable to ities that used to require physical presence to become doable the same message before re-masking. over the network. One example is e-gambling. In this paper we In this paper we survey the homomorphic properties of the focus on mental poker, i.e. a poker game played among players cryptosystems used in current state-of-the-art mental poker that are not physically together but communicate through protocols. computer networks. The rest of the paper is distributed as follows. Section II A mental poker system needs to provide protocols to reviews the concept of privacy homomorphism and gives some generate a deck of cards, shuffle it and confidentially deal examples. Section III surveys the state of the art of TTP-free the cards to players. Naive approaches assume the existence mental poker protocols and how homomorphims play a key of a trusted third party (TTP), who performs some or all of role in them. Finally, Section IV is a conclusion. the aforementioned operations [1]. Since the assumption on all players trusting this central node II. PRIVACY HOMOMORPHISM is not always realistic, research on solutions not requiring Privacy homomorphisms (PH) were introduced by Rivest, a trusted party has become a hot topic [2]–[5]. To ensure Adleman and Dertouzos in [6]. A PH is an encryption function the players a fair play, standards of security similar to those which maps a set F of operations on plaintext to another set in physical gambling must be guaranteed. For instance, it is F of operations on ciphertext. Operations on encrypted data good for trust generation that all players participate in the are computed without knowledge of the decryption key. computation of the shuffled deck of cards. Current proposals Privacy homomorphisms are used as building blocks in in the literature are based on a paradigm consisting of the secure computation applications such as computing delegation, following steps: data delegation and e-gambling: the idea is to encrypt data at 1) The players generate the deck of 52 face-up cards D = a classified level, to process them at an unclassified level and {d1,...,d52}. All players agree on which card each di to decrypt the result at a classified level. corresponds to. Next we describe some examples of privacy homomor- 2) The elements of set D are encrypted using some cryp- phisms. tosystem. The result is the encrypted (or face-down) deck of cards C = {c1,...,c52}, ci = E(di). A. Homomorphic properties of ElGamal cryptosystem The ElGamal cryptosystem is a famous public key cryp- The authors are partly supported by the Catalan Government under grant ⊂ Z∗ 2005 SGR 00446, and by the Spanish Ministry of Science and Education tosystem described in [7]. It is defined over a group G p ∗ through project SEG2004-04352-C04-01 “PROPRIETAS”. of order q|p − 1, where p and q are large primes. Let g ∈ Zp The authors are with the Rovira i Virgili University of Tarragona, Dept. be a generator of G. of Computer Engineering and Maths, Av. Pa¨ısos Catalans, 26, E-43007 ∈ Z Tarragona, Catalonia (e-mail: {jordi.castella, vanesa.daza, josep.domingo, The secret key corresponds to some secret value x q x francesc.sebe}@urv.net) and the public key is h = g mod p. 1-4244-0133-X/06/$20.001-4244-0134-8/06/$20.00 ©© 22006006 IIEEEEEE 788 ∗ A message m ∈ Zp is encrypted by generating a random D. Domingo-Ferrer’s 2002 algebraic privacy homomorphism ∈ Z t t value t q and then computing E(m)=(c, c )=(mh ,g ). Another PH preserving the addition and multiplication is The security of ElGamal relies on the so-called Decisional presented in [9]. Public parameters are a positive integer δ>2 Diffie-Hellman (DDH) assumption, which states that the fol- and a large integer N. N should have many small divisors and t1 t2 t3 t1 t2 t1t2 lowing two ensembles (g, g ,g ,g ) and (g, g ,g ,g ) at the same time there should be many integers less than N are computationally indistinguishable, where t1,t2 and t3 are that can be inverted modulo N. The secret parameters are Z −1 chosen at random in q. y ∈ ZN such that there exists an inverse y mod N and a The ElGamal cryptosystem is a multiplicative privacy ho- small divisor N > 1 of N. The secret key is (y,N). momorphism. Let us assume two messages m1 and m2 en- The encryption of a message m ∈ ZN is as follows. First, crypted under the same public key h, i.e. E(m1)=(c1,c )= 1 m is randomly split into secret shares s1,...,sδ such that t1 t1 t2 t2 δ (m1h ,g ) and E(m2)=(c2,c2)=(m2h ,g ). m = sj mod N sj ∈ ZN j=1 and . The encrypted message By computing E(m1)E(m2)=(c1c2,c1c2), we obtain t t t t t +t t +t is: 1 2 1 2 1 2 1 2 1 2 1 2 δ (m m h h ,g g )=(m m h ,g ), which cor- E(m)=(s1 · y mod N,...,sδ · y mod N) responds to E(m1m2). Addition and multiplication are performed in a way similar as described in Section II-C. This privacy homomorphism is B. Homomorphic properties of Kurosawa et al. general public secure against ciphertext-only attacks. key cryptosystem Kurosawa et al. presented in [2] a public key cryptosystem III. MENTAL POKER PROTOCOLS BASED ON th based on the r residue problem. In this cryptosystem, the HOMOMORPHIC ENCRYPTION secret key is a pair of large prime numbers (p, q) and the Mental poker is played like ordinary poker but without public key parameters are (n, r, y), where n = pq and r and physical elements (like cards) nor verbal communication; y are elements satisfying some technical requirements detailed all exchanges between players must be accomplished using in [2]. messages [10]. Any player may try to cheat. ≤ Then, in order to encrypt a plaintext m, 0 m<r, A mental poker protocol must guarantee the fairness of a number t is chosen at random and the value E(m)= m r the game and, if a player tries to cheat, the protocol must y t mod n is computed thereafter. detect or avoid the cheating. In [11], Crepeau´ enumerated the This cryptosystem is an additive privacy homomorphism. requirements and properties that must be met by a mental That is, let m1 and m2 be two messages encrypted using poker protocol. m1 r the same public key (n, r, y), i.e. E(m1)=y t1 mod n Some mental poker protocols require the disclouse of the m2 r and E(m2)=y t2 mod n, (for some randomly cho- secret information at the end of the game to verify the game sen t1 and t2); then it holds that E(m1 + m2 mod r)= fairness. Nonetheless, the secret information also reveals the E(m1)E(m2)modn. strategy of players. The most advanced protocols use zero- m r m r Note that E(m1)E(m2)=y 1 t1 y 2 t2 = knowledge proofs to ensure the honesty of the game without m1+m2 r m1+m2 r y (t1t2) = y t3 = E(m1 + m2 mod r). revealing the strategies of players. A zero-knowledge proof is an interactive and probabilistic method for a player (prover) to efficiently prove to another C. Domingo-Ferrer’s 1996 algebraic privacy homomorphism player (verifier) that a statement is true, without conveying The PH proposed in [8] is additive and multiplicative. The any additional knowledge other than the statement. The wide public parameters are δ and n, where δ is a security parameter applicability of zero-knowledge proofs is due to the fact and n is the product of two sufficiently large primes p and q.
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