PEA: Polymorphic Encryption Algorithm based on quantum computation. Nikos Komninos* and Georgios Mantas

Transcription

1 Int. J. Systes, Control and Counications, Vol. 3, No., PEA: Polyorphic Encryption Algorith based on quantu coputation Nikos Koninos* and Georgios Mantas Algoriths and Security Group, Athens Inforation Technology, GR-9 Peania Attiki, Greece E-ail: E-ail: *Corresponding author Abstract: In this paper, a Polyorphic Encryption Algorith (PEA), based on basic quantu coputations, is proposed for the encryption of binary bits. PEA is a syetric key encryption algorith that applies different cobinations of quantu gates to encrypt binary bits. PEA is also polyorphic since the states of the shared secret key control the different cobinations of the ciphertext. It is shown that PEA achieves perfect secrecy and is resilient to eavesdropping and Trojan horse attacks. A security analysis of PEA is also described. Keywords: encryption algorith; polyorphis; quantu coputations; CNOT and SWAP quantu gates. Reference to this paper should be ade as follows: Koninos, N. and Mantas, G. () PEA: Polyorphic Encryption Algorith based on quantu coputation, Int. J. Systes, Control and Counications, Vol. 3, No., pp. 8. Biographical notes: Nikos Koninos is currently an Assistant Professor in Applied Cryptography and Network Security at Athens Inforation Technology (AIT). He has over 5 years of R&D experience in the acadeia and industry working on the evaluation, analysis and developent of practical secure counication systes. He has written over 5 journals and conference publications, patents, books and technical reports in the inforation security research area. He has been invited as international advisor and technical progra coittee, guest editor in international conferences and journals and speaker with honors in the field. He is also senior eber in IEEE. Georgios Mantas received his Diploa in Electrical and Coputer Engineering fro University of Patras, in 5. In 8, he obtained the Master of Science in Inforation Networking offered by Carnegie Mellon University in collaboration with Athens Inforation Technology (AIT). Since 8, he has been pursuing his PhD in the Departent of Electrical and Coputer Engineering at University of Patras. His research areas of interest include authentication, data integrity, key agreeent, routing security in wired and wireless networks and sart cards security. He has knowledge of various prograing languages and siulation tools. Copyright Inderscience Enterprises Ltd.

2 N. Koninos and G. Mantas Introduction Conventional cryptography consists of two types of cryptosystes; syetric key cryptosystes and asyetric key, or public key cryptosystes. Syetric key cryptosystes, as the nae iplies, use identical keys for encryption and decryption and both the sender and the receiver share the sae key(s). For this reason, such cryptosystes rely on the secrecy of the key. Otherwise, any alicious actors can deduce the plaintext fro the ciphertext. The ajor disadvantage of such cryptosystes is the need of frequent and reliable key distribution. On the contrary, cryptosystes based on asyetric key cryptography use different keys for ciphering and for deciphering. In ciphering, the key is announced publicly, but in deciphering, the key reains secret. The security of public key cryptosystes stes fro the fact that the key pair generation is based on coputational coplexity of certain difficult atheatical probles. However, the ain drawback of this cryptosyste is the coputational cost of generating the key pair (Stallings, 5; Menezes et al., 4). In the early 97s, a new type of Cryptography was proposed by Stephen Wiesner (Nielsen and Chuang, 4). He proposed the idea of Quantu Cryptography as he introduced the concept of quantu conjugate coding at the sae year. Quantu Cryptography is based on quantu coputation. In contrast to Conventional Cryptography that uses digital bits for encoding inforation, Quantu Cryptography uses quantu particles (i.e., photons) and their quantised properties (i.e., photon s polarisation) to do that. Each photon carries one bit of quantu inforation, called qubit. A qubit defines not only the two binary eigenstates and but also the superposition of the two. Furtherore, the security of Quantu Cryptography is based on fundaental quantu echanical principles. It uses quantu echanics in order to overcoe the key distribution proble which is the ost serious proble in the both types of conventional cryptography (Knill et al., ; Nielsen and Chuang, 4; Steane and Rieffel, ). Quantu Cryptography is considered as the basis for next generation cryptographic systes that ay be able to replace the public key cryptosystes (Gisin and Wolf, 999; Pike, ; Benbasat, 999; Li and Barnu, 4; Benenti and Casatti, 4). Conventional cryptosystes are susceptible to advanced technology, as new and ore powerful coputers (e.g., quantu coputers) ay be able to override the burden of coputationally coplex probles in the future. Thus, no atter how strong a cryptographic algorith is, soon or later it will be broken. On the contrary, Quantu Cryptography is independent of technological advance in the future as its security is provable inforation theoretically (Pike, ; Nabu and Kosaka, 3). During the last two decades, the Quantu Cryptography has focused on three fields concerned to key sharing between the transitter and the receiver. These fields are the following: quantu key distribution, quantu secret sharing (Nielsen and Chuang, 4) and quantu bit coitent (Zhou et al., 6). Furtherore, a new breakthrough appeared regarding the evolution of quantu encryption algoriths. The concept of this new field was first introduced by Boykin and Roychowdhury (Boykin and Roychowdhury, 3) in. Thus, Quantu Cryptography not only provides a secure key exchange between two parties, but also provides a special type of quantu encryption at transitter and recipient. Moreover, Quantu Cryptography solves the two ajor practical probles of one-tie pad encryption schee as it provides secure key distribution and is also able to

3 PEA: Polyorphic Encryption Algorith based on quantu coputation 3 generate sets of rando nubers at the two counicating parties. One tie pad is the only provably unconditionally secure cryptosyste, which can not be coproised even in the face of unliited tie and coputational power (Mullins, ; Stallings, 5; Menezes et al., 4). Therefore, the existing encryption algoriths for quantu inforation take the for of one-tie pad encryption ethod. Several quantu encryption algoriths for classical binary bits fro different aspects were proposed in Hirota et al. (5), Bechann-Pasquinucci and Gisin (3), Zhou and Zeng (4). A realisable quantu encryption algorith for qubits was also presented in Zhou and Zeng (5). Motivated by these and the challenge of quantu coputing field, we propose a PEA, which uses different cobinations of quantu CNOT and SWAP gates in order to encrypt essages without changing the key. The Controlled NOT gate (CNOT) is the quantu analogue of the XOR gate. It has two input qubits, the first is the controlled qubit and the second is the target qubit. If the controlled qubit is zero then the target qubit is intact. If the controlled qubit is one then the target qubit is flipped. Generally, the notation CNOT x, y, eans that the index x is the controlled qubit and the index y is the target qubit. The quantu SWAP gate has two input qubits and swaps the. The notation SWAP x, y, eans that this gate swaps the qubit x with the qubit y. The security of the encryption algorith is analysed fro several aspects and it is shown that the PEA can prevent quantu and classical attack. Following this introduction, the paper is organised as follows. Section describes the PEA, along with its encryption and decryption processes. Section 3, presents a security analysis of PEA and shows how resilient it is to quantu and classical attacks. Finally, we conclude our algorith in Section 4 and propose future work. Polyorphic Encryption Algorith The PEA is a syetric key algorith based on quantu coputation for encryption of classical essages. PEA requires four groups of keys to be exchanged between sender and recipient before the encryption takes place. The first group key is responsible for the choice of quantu ancilla bits. The second group key is responsible for the choice of the first level of encryption. The third group key is responsible for the second level of encryption and, finally, the fourth group key is responsible for leading the encrypted data to non-orthogonality. Hence, the pre-sharing of these keys is essential as they are used during encryption of PEA. The polyorphis of this algorith is based on the pre-shared keys, as they are the factors that designate the internal paths that the algorith should follow in order to give the encrypted output. Generally, the classical essage, which is encrypted by PEA, consists of bits and for each bit of the essage one quantu state, C, is created based on the current bit and the key. So, quantu states are going to be created. As illustrated in Figure, the input of PEA is C, which, according to the second group of key eleent k, can follow one of the two paths. If k is equal to, the quantu state will go through CNOT, gate. Whereas k is equal to, the quantu state will go through SWAP, gate. Next, the output of CNOT, gate, C (CNOT, ), will pass either by the path which has no gate or by the SWAP,3, according to the third group key eleent k 3. Likewise, the output of SWAP, gate, C (SWAP, ), will pass either by CNOT, gate or by CNOT 3,, according to the third group key eleent k 3.

4 4 N. Koninos and G. Mantas Figure PEA encryption algorith. Encryption The encryption process consists of the following phases: preparation; first and second level of encryption; and non-orthogonality phases. Phase : Preparation In Phase, the key k is used for the definition of the valid quantu states. Let us define the quantu ancilla state represented as kk. The k and the k are two key eleents of the ith key pair in k. When the classical key eleent pairs are,, and we have the corresponding quantu ancilla states,,,, respectively. For each bit, ( or ), of the classical essage, we have the corresponding quantu state. Now, the cobination of the quantu ancilla state with the corresponding quantu state of each bit, will give all the valid quantu states. This cobination is achieved by using the tensor product which generates the tensor product state kk. To siplify our calculations, we have assued an 8-bit classical essage as a block size for our algorith. Thus, we will have the following eight valid quantu states:,,,,,,,. These valid states can be represented by equation (): C = k k >, where, k, k {,}. () Phase : First level of encryption In Phase, the second key, k, is responsible for the first level of encryption. The input to the first level of encryption is one of the eight valid values defined by equation (). According to the values of k, there is a different path that the input can follow.

5 PEA: Polyorphic Encryption Algorith based on quantu coputation 5 When k is equal to, the input will pass by the CNOT, gate and when k is equal to, the input will pass by the SWAP, gate. The index, of CNOT gate defines the second qubit as control and the first qubit as target. Thus, the CNOT, gate inverts the first qubit when the second qubit is equal to. Likewise, the index, of SWAP gate defines that the first qubit will be swapped with the second qubit. We have selected the CNOT, and SWAP, gates to create diffusion to the data bit found in the first qubit of the quantu states, kk, of Phase. The output C that results fro CNOT, and SWAP, quantu gates (see Figure ), denoted as C (CNOT, ) and C (SWAP, ) respectively, is represented by equation (): C (CNOT ) = t k k, where, k, k {, } and t = k,, C(SWAP ) = k, k, where, k, k {,}., () Based on the second group of key eleent k, all possible cobinations of C (CNOT, ) and C (SWAP, ) states are shown in Table : Table All possible C outputs C k C (CNOT, ) k C (SWAP, ) Phase 3: Second level of encryption In Phase 3, the third key, k 3, is responsible for the second level of encryption. In the second level of encryption, its inputs are the outputs of Phase. The outputs, C (CNOT, ) and C (SWAP, ), are the inputs to the second level of encryption and according to the values of k 3, there is a different path that each input can follow. C (CNOT, ) will pass either fro the path which has no gate, for k 3 equal to, or fro the SWAP,3 gate, for k 3 equal to. When C (CNOT, ) follows the path which has no gate, the output will be the sae as the input. The index,3 of SWAP gate defines that the first qubit will be swapped with the third qubit. Siilar to Phase, we have selected the SWAP,3 gate to create diffusion to the inverted data bit which is found in the first qubit in the quantu states, tkk, of Phase. The output C 3 that results fro the path without quantu gate and the path with SWAP,3 quantu gate (see Figure ), denoted as C 3 (NO GATE) and C 3 (SWAP,3 ) respectively, is represented by equation (3): C (NO GATE) = t k k, where, k, k {, } and t = k, 3 C (SWAP ) = k, k, t, where, k, k {, } and t = k. 3,3 (3)

6 6 N. Koninos and G. Mantas Based on the third group of key, k 3, all possible cobinations of C 3 (NO GATE) and C 3 (SWAP,3 ) states are shown in Table : Table All possible C 3 outputs with C (CNOT, ) input C (CNOT, ) k 3 C 3 (NO GATE) k 3 C 3 (SWAP,3 ) C (SWAP, ) will pass either fro the CNOT, gate, when k 3 is equal to or by CNOT 3, when k 3 is equal to. The index, of CNOT gate defines the first qubit as control and the second qubit as target. Thus, the CNOT, gate inverts the second qubit when the first qubit is equal to. Likewise, the index 3, of CNOT gate defines the third qubit as control and the second qubit as target. Thus, the CNOT 3, gate inverts the second qubit when the third qubit is equal to. We have selected the CNOT, and CNOT 3, gates to create diffusion to the data bit which is found in the second qubit in the quantu states, kk, of Phase. The output C 3 that results fro CNOT, and CNOT 3, quantu gates (see Figure ), denoted as C 3 (CNOT, ) and C 3 (CNOT 3, ) respectively, is represented by equation (4): C (CNOT ) = k t, k, where, k, k {, } and t = k 3, C (CNOT ) = k t, k, where, k, k {,} and t = k. 3 3, (4) Based on the third group of key eleent k 3, all possible cobinations of C 3 (CNOT, ) and C 3 (CNOT 3, ) states are shown in Table 3: Table 3 All possible C 3 outputs with C (SWAP, ) input C (SWAP, ) k 3 C 3 (CNOT, ) k 3 C 3 (CNOT 3, )

7 PEA: Polyorphic Encryption Algorith based on quantu coputation 7 Phase 4: Non-orthogonality In Phase 4, the fourth group key k 4 is responsible for leading the encrypted data to non-orthogonality. The encrypted data derived fro the second level of encryption are states which are orthogonal. However, orthogonality is a property which should be avoided so that the encrypted data are securely transferred through the counication channel. Orthogonality perits states to be distinguished and thus, Phase 4 akes the outputs states non-orthogonal. Non-orthogonality is a preferable property for PEA. Non-orthogonality is achieved by using the fourth key, k 4. According to the key, k 4, we change the second qubit in the following approach: If the key eleent is or, then we do not change the second qubit of state C 3. Thus, the second qubit reains in the state or. However, when the key eleent is or, then we perfor the following coputations on the second qubit: when the key eleent has value equal to, we apply Hadaard (H) gate (Nielsen and Chuang, 4) to the second qubit and the resulting output state is + if the input state is, or if the input state is. If the key eleent is Nielsen and Chuang (4), we apply ZH gate [] to the second qubit and the resulting output state is if the input state is, or + if the input state is. We select the second qubit to be passed by H or ZH gates because in the second qubit there is the quantu representation of a data bit after Phase 3, and particularly into two outputs out of four. The Non-orthogonality process is shown in Figure. For each output C 3 (NO GATE), C 3 (SWAP,3 ), C 3 (CNOT, ), and C 3 (CNOT 3, ), non-orthogonality is also illustrated in Figure. Figure Non-orthogonality process To prove that the utilisation of Hadaard gates and ZH gates offers non-orthogonality, we need to follow the next steps:

8 8 N. Koninos and G. Mantas Firstly, it is known that the Hadaard gate is given by equation (5) (Nielsen and Chuang, 4): ^ H =. It is also known that the atrix representations of the and the are given by equation (6) (Nielsen and Chuang, 4): >= and >=. Fro equations (5) and (6) we can calculate the following: ^ H >= ( ) = = >+ > + ^ H >= ( ) = = > >. Furtherore, the ZH gate is given by equation (): and ^ ZH = >= and >=. Fro equations (8) and (9) we can calculate the following: ^ ZH >= ( ) = = > > > ^ ZH >= ( ). = = >+ > +> For input C 3 (NO GATE), the output C 4 results fro not changing the second qubit of the input C 3 (NO GATE) or applying H/ZH quantu gates to the second qubit of the input C 3 (NO GATE). These results denoted as C 4 (NO CHANGE), C 4 (H gate) and C 4 (ZH gate) are represented by equation (): C (NO CHANGE) = k,, k, where k,, k {, } when k = or 4 4 C (H gate) =, ±, k, where, k, k {, } when k = 4 4 C(ZH gate) =,, k, where, k, k {,}whenk =. 4 4 Based on the fourth group of key eleent k 4, all possible cobinations of C 4 (NO CHANGE), C 4 (H gate) and C 4 (ZH gate) states are shown in Table 4: (5) (6) (7) (8) (9) () ()

9 PEA: Polyorphic Encryption Algorith based on quantu coputation 9 Table 4 All possible C 4 outputs with C 3 (NO GATE) input C 3 (NO GATE) k 4 C 4 (NO CHANGE) k 4 C 4 (H gate) k 4 C 4 (ZH gate) or + or + or + or + or + or + or + or + For input C 3 (SWAP,3 ), the output C 4 results fro not changing the second qubit of the input C 3 (SWAP,3 ), or fro applying H or ZH quantu gates to the second qubit of the input C 3 (SWAP,3 ). These results denoted as C 4 (NO CHANGE), C 4 (H gate) and C 4 (ZH gate), respectively, are represented by equation (): C (NO CHANGE) = k, k,, where, k, k {, } when k = or 4 4 C (H gate) = k ±,, where, k, k {, } when k = 4 4 C (ZH gate) = k,,, where, k, k {,} when k =. 4 4 () Based on the fourth group of key eleent k 4, all possible cobinations of C 4 (NO CHANGE), C 4 (H gate) and C 4 (ZH gate) states are shown in Table 5: Table 5 All possible C 4 outputs with C 3 (SWAP,3 ) input C 3 (SWAP,3 ) k 4 C 4 (NO CHANGE) k 4 C 4 (H gate) k 4 C 4 (ZH gate) or + or + or + or + or + or + or + or + For input C 3 (CNOT, ), the output C 4 results fro not changing the second qubit of the input C 3 (CNOT, ), or fro applying H/ZH quantu gates to the second qubit of the input C 3 (CNOT, ). These results, denoted as C 4 (NO CHANGE), C 4 (H gate) and C 4 (ZH gate) are represented by equation (3):

10 N. Koninos and G. Mantas C (NO CHANGE) = k,, k, where, k, k {, } when k = or 4 4 C (H gate) = k, ±, k, where, k, k {, } when k = 4 4 C (ZH gate) = k,, k, where, k, k {, } when k =. 4 4 Based on the fourth group of the key eleent k 4, all possible cobinations of C 4 (NO CHANGE), C 4 (H gate) and C 4 (ZH gate) states are shown in Table 6: (3) Table 6 All possible C 4 outputs with C 3 (CNOT, ) input C 3 (CNOT, ) k 4 C 4 (NO CHANGE) k 4 C 4 (H gate) k 4 C 4 (ZH gate) or + or + or + or + or + or + or + or + For input C 3 (CNOT 3, ), the output C 4 results fro not changing the second qubit of the input C 3 (CNOT 3, ), or fro applying H/ZH quantu gates to the second qubit of the input C 3 (CNOT 3, ). These results denoted as C 4 (NO CHANGE), C 4 (H gate) and C 4 (ZH gate) are represented by equation (4): C (NO CHANGE) = k,, k, where, k, k {, } when k = or 4 4 C (H gate) = k, ±, k, where, k, k {, } when k = 4 4 C (ZH gate) = k,, k, where, k, k {, } when k =. 4 4 (4) Based on the fourth group of key eleent k 4, all possible cobinations of C 4 (NO CHANGE), C 4 (H gate) and C 4 (ZH gate) states are shown in Table 7: Table 7 All possible C 4 outputs with C 3 (CNOT 3, ) input C 3 (CNOT 3, ) k 4 C 4 (NO CHANGE) K 4 C 4 (H gate) k 4 C 4 (ZH gate) or + or + or + or + or + or + or + or +

11 PEA: Polyorphic Encryption Algorith based on quantu coputation. Decryption All gates (CNOT gates, SWAP gates, H gates, ZH gates) used in the encryption and non-orthogonality processes are unitary operators. Hence, the decryption is the inverse process of the encryption using the pre-shared four group keys. The decryption process takes place at the recipient side. Thus, the decryption process consists of the following inverse phases: Phase : Inverse process of Phase 4 of encryption process In Phase, the fourth group key k 4 is responsible for the first level of decryption. The recipient receives the non-orthogonal states (ciphertext) fro the sender and applies the sae process as the one applied in Phase 4 during encryption. When the key eleent is or we do not change the second qubit of state C 4 and thus, the second qubit reains in state or. However, when the key eleent is or, then we perfor the following coputations on the second qubit of the ciphertext: If the key eleent has value equal to, we apply H gate to the second qubit and the output state is when the input state is +, or when the input state is. If the key eleent is, we apply ZH gate to the second qubit and the resulting output state is when the input state is, or when the input state is +. Therefore, the output of this phase is the state C 3 of the encryption process. Phase : Inverse process of Phase 3 of encryption process In Phase, the third and second group keys k 3 and k are responsible for the second level of decryption. The inputs of Phase are the outputs of the first level of decryption. Here the recipient applies the sae process with the process applied in Phase 3 during the encryption process. For exaple, when k is equal to, state C 3 passes either by the path which has no gate for k 3 equal to, or by SWAP,3, for k 3 equal to. When k is equal to state C 3 passes either by the CNOT, gate for k 3 equal to, or by CNOT 3, for k 3 equal to. Hence, the output of this phase is the state C of the encryption process. Phase 3: Inverse process of Phase of encryption process In Phase 3, the second group of key eleent k is responsible for the third level of decryption. The inputs of Phase 3 are the outputs of the second level of decryption. As before, the receiver applies the sae process as the one applied in Phase during encryption. For exaple, when k is equal to, state C passes by the CNOT, gate, but when k is equal to, state C passes by the SWAP, gate. Hence the output of this phase is the state C of the encryption process and the first qubit of each quantu state corresponds to the initial data bit. Phase 4: Acquisition of plaintext In Phase 4, we obtain all the first qubits of each state that correspond to the original bits and acquire the bit sequence of the original plaintext. 3 Security analysis of PEA To evaluate PEA we have conducted a security analysis to exaine its resilience to the Trojan horse and eavesdropping attacks and to study its hoogeneity and perfect secrecy.

12 N. Koninos and G. Mantas 3. Hoogeneity and perfect secrecy It is essential for an encryption algorith to produce hoogeneous ciphertexts and, ideally, have perfect secrecy, i.e., the ciphertext gives no inforation about the plaintext (Menezes et al., 4). To prove that our algorith is hoogeneous and achieves perfect secrecy, it is enough to show that the density atrix of n ciphertexts states related to the n bits classical essage is the identity atrix. To prove it, let ψ wz be the linear cobination of all possible states with equal probability in the ciphertext set C w (w =,, 3), which corresponds to the zth bit of essage. The density atrices are calculated by equation (5): ψ z ψ z = I ψ z ψ z = I ψ 3z ψ 3z = I. (5) The density atrix of n ciphertext states ψ 3 for a essage which consists of n bits is given by equation (6): ψ 3 ψ 3 = ψ 3 ψ 3 ψ 3 ψ 3 ψ 33 ψ ψ 3n ψ 3n. (6) To calculate equation (6), firstly, it is required to calculate the outer product of each eleent ψ 3i : ψ 3 ψ 3, ψ 3 ψ 3, ψ 33 ψ 33, ψ 34 ψ 34, ψ 35 ψ 35, ψ 36 ψ 36, ψ 37 ψ 37, ψ 38 ψ 38. Then, it is required to calculate the tensor products of the above outer products. We know that the eleents ψ 3i represented as the following:,,,,,,,, can be also represented as vectors. By using vectors we can ake siple calculations and prove that the ciphertext is hoogenous. For exaple, the vector representation of the first states is given by equation (7): = =. = = Likewise, we can construct the rest of the states,,,,,, and as vectors. According to Figure, when the output is given by C 3 (NO GATE) we have the following possible states: (7) Table 8 C 3 (NO GATE) output

13 PEA: Polyorphic Encryption Algorith based on quantu coputation 3 Fro Table 8, we have: ψ 3 =, ψ 3 =, ψ 33 =, ψ 34 =, ψ 35 =, ψ 36 =, ψ 37 =, ψ 38 =. Next, the outer products can be calculated by equation (8): T ψ ψ = = ee = E 3 3 T where e = [], E = Diag{,,,,,,, }. (8) Likewise, we can calculate all the outer products ψ 3 ψ 3, ψ 33 ψ 33, ψ 34 ψ 34, ψ 35 ψ 35, ψ 36 ψ 36, ψ 37 ψ 37, and ψ 38 ψ 38. According to Figure, when the output is given by C 3 (SWAP,3) we have the following possible states: Table 9 C 3 (SWAP,3) output Fro Table 9, we have: ψ 3 =, ψ 3 =, ψ 33 =, ψ 34 =, ψ 35 =, ψ 36 =, ψ 37 =, ψ 38 =. The outer product of ψ 3 ψ 3 is given by equation (9): T ψ ψ = = ee = E 3 3 T where e = [], E = Diag{,,,,,,, }. (9) Likewise, we can calculate ψ 3 ψ 3, ψ 33 ψ 33, ψ 34 ψ 34, ψ 35 ψ 35, ψ 36 ψ 36, ψ 37 ψ 37, and ψ 38 ψ 38. According to Figure, when the output is given by C 3 (CNOT, ) we have the following possible states: Table C 3 (CNOT, ) output Fro Table, we have: ψ 3 =, ψ 3 =, ψ 33 = ψ 34 =, ψ 35 =, ψ 36 =, ψ 37 =, ψ 38 =.

14 4 N. Koninos and G. Mantas The outer product of ψ 3 ψ 3 is given by equation (): T ψ ψ = = ee = E 3 3 T where e = [], E = Diag{,,,,,,, }. () Likewise, we can calculate ψ 3 ψ 3, ψ 33 ψ 33, ψ 34 ψ 34, ψ 35 ψ 35, ψ 36 ψ 36, ψ 37 ψ 37, and ψ 38 ψ 38. According to Figure, when the output is given by C 3 (CNOT 3, ) we have the following possible states: Table C 3 (CNOT 3, ) output Fro Table, we have: ψ 3 =, ψ 3 =, ψ 33 =, ψ 34 =, ψ 35 =, ψ 36 =, ψ 37 =, ψ 38 =. The outer product of ψ 3 ψ 3 is given by equation (): T ψ ψ = = ee = E 3 3 T where e = [], E = Diag{,,,,,,, }. () Likewise, we can calculate ψ 3 ψ 3, ψ 33 ψ 33, ψ 34 ψ 34, ψ 35 ψ 35, ψ 36 ψ 36, ψ 37 ψ 37, and ψ 38 ψ 38. We notice that the first output ( ) given by C 3 (CNOT, ) (see Table ) is the sae as the first output ( ) given by C 3 (CNOT 3, ) (see Table ). Consequently, the linear cobination ψ 3 corresponds to the state is the sae for the both cases. Thus, equations () and () are identical. According to the assuption that k, k, k 3, k 4, are uniforly distributed and based on the above atrices of the outer products, we can calculate the tensor products and obtain the following density atrix: ψ 3 ψ 3 = ψ 3 ψ 3 ψ 3 ψ 3 ψ 33 ψ ψ 3n ψ 3n = I. () Equation () deonstrates that the ciphertext is hoogeneous and has perfect secrecy, since the density atrix is equal to the identity atrix. 3. Trojan horse attacks tolerance The Trojan horse attacks are the ost serious threats to coputer networking security. However, the proposed algorith is tolerant to Trojan horse attacks. If a Trojan horse, denoted as T, has invaded in the sender or the recipient in order to identify the quantu states and, then the Trojan horse is going to take the following ciphertext state:

15 PEA: Polyorphic Encryption Algorith based on quantu coputation 5 4i T = ( ψ ( ) ? ????? ????? ?????????? ???. ) (3) In equation (3), we have used three sybols. The sybol is the inforation obtained by the Trojan horse when the quantu state is. The sybol is the inforation obtained by the Trojan horse when the quantu state is. Whereas, the sybol? is the inforation obtained by the Trojan horse when the quantu state is + or. It eans that when the state is + or due to non-orthogonality that is applied in the fourth phase of encryption process, the Trojan horse cannot decide which the valid state is. Moreover, according to the design of encryption algorith, the quantu representation of a data bit can be in the first qubit or in the second qubit or in the third qubit. Thus, the Trojan horse is able to take any inforation related to the plaintext as well as which qubit is related to the ancilla qubit. 3.3 Eavesdropping attacks tolerance According to the design of PEA, the ciphertext states are non-orthogonal, leading the ciphertext states to be indistinguishable to an eavesdropping attacker. The non-orthogonality is applied in the fourth phase of encryption process and we can prove it by calculating the ψ 4i ψ 4i. If the value of inner product ψ 4i ψ 4i is larger than, it eans that the ciphertext states are non-orthogonal (Nielsen and Chuang, 4). We know that ψ 4i (equation (3)) ust be represented as a vector in order to calculate the value of ψ 4i ψ 4i. The states,,,,,,, and have already been fored as vectors and it is necessary to also represent the states +,, +,, +, +,, and as vectors. For exaple, the vector representation of + is given by equation (4):

16 6 N. Koninos and G. Mantas + = + =. = = (4) Likewise, we can calculate, +,, +, +,, and. Hence, according to equations (7), (5) and their further calculations we have: ψ 4i = = Next, we can calculate the inner product ψ 4i ψ 4i as shown in equation (6): + + ψ 4i ψ4 i = (5) (6) and: = ( + ). 3

17 PEA: Polyorphic Encryption Algorith based on quantu coputation 7 ψ 4i ψ 4i = ( + ) >. (7) 3 By proving that the value of the inner product ψ 4i ψ4 i is larger than, it eans that the ciphertext states are non-orthogonal. 4 Conclusion To our knowledge not any good and feasible quantu encryption algoriths have yet been proposed. With the rapid progress of quantu inforation theory and technology, quantu inforation coes into real life quietly. When the quantu coputers becoe a reality soe day, it will be necessary, but not always possible, to transfer the existing encryption algoriths into quantu inforation. Based on the basic principle of quantu coputation, a quantu cryptographic algorith to encrypt the classical binary bits was proposed. The security of the encryption algorith was analysed in detail. It was shown that the proposed algorith can prevent quantu, as well as classical, attacks. PEA has several properties. First of all, no quantu state is pre-shared or stored, aking PEA possible and efficient in real applications. Second, it achieves perfect secrecy with the condition that the key is uniforly distributed. Third, both encryption and decryption are based on siple quantu coputation, and particularly on a cobination of the quantu CNOT and SWAP gates. Fourth, its ipleentation is feasible with the existing technology. At last, the sae algorith can be extended to encrypt quantu inforation. References Bechann-Pasquinucci, H. and Gisin, N. (3) Interediate states in quantu cryptography and Bell inequalities, Physical Review A, Vol. 67, pp Benbasat, A.Y. (999) A Survey of Current Optical Security Techniques, MIT Report 6.637, at Benenti, G. and Casatti, G. (4) Principles of Quantu Coputation, Vol. I: Basic Concepts, World Scientific Publishing, New Jersey. Boykin, P.O. and Roychowdhury, V. (3) Optial encryption of quantu bits, Physical Review A, Vol. 67, pp Gisin, N. and Wolf, S. (999) Quantu cryptography on noisy channels: quantu versus classical key-agreeent protocols, Phys. Rev. Lett., Vol. 83, pp Hirota, O., Soha, M., Fuse, M. and Kato, K. (5) Quantu strea cipher by the Yuen protocol: design and experient by an intensity-odulation schee, Physical Review A, Vol. 7, pp Knill, E., Laflae, R., Barnu, H.N., Dalvit, D.A., Dziaraga, J.J., Gubernatis, J.E., Gurvits, L., Ortiz, G., Viola, L. and Zurek, W.H. () Quantu inforation processing: a hands-on prier, Los Alaos Science, Vol., No. 7, pp. 37. Li, X.Y. and Barnu, H. (4) Quantu authentication using entangled states, International Journal of Foundations Coputer Science, Vol. 5, No. 4, pp Menezes, A.J., Vanstone, S.A. and Van Oorschot, P.C. (4) Handbook of Applied Cryptography, CRC Press, Inc., USA. Mullins, J. () Making Unbreakable Code, IEEE Spectru, May.

18 8 N. Koninos and G. Mantas Nabu, Y. and Kosaka, H. (3) Introduction of quantu cryptography and its developent, NEC Res. and Develop., Vol. 44, No. 3, pp Nielsen, M.A. and Chuang, I.L. (4) Quantu Coputation and Quantu Inforation, Cabridge Press, UK. Pike, R. () An Introduction to Quantu Coputation and Quantu Counication, Bell Labs, Lucent Technologies, pp. 35. Stallings, W. (5) Cryptography and Network Security Principles and Practices, Prentice-Hall, Noveber. Steane, A.M. and Rieffel, E.G. () Beyond bits: the future of quantu inforation processing, IEEE Coputer, Vol. 33, No., pp Zhou, N. and Zeng, G. (4) Progress on cryptography, The Kluwer International Series in Engineering and Coputer Science, Kluwer Acadeic Publishers, Boston, pp.95. Zhou, N. and Zeng, G. (5) A realizable quantu encryption algorith for qubits, Chinese Physics, Vol. 4, No., pp Zhou, N., Zeng, G., Nie, Y., Xiong, J. and Zhu, F. (6) A novel quantu block encryption algorith based on quantu coputation, Physica A: Statistical Mechanics and Its Applications, Vol. 36, No., April, pp