Quantum-Resistant Network for Classical Client Compatibility
ITC 2/50
Information Technology and Control
Vol. 50 / No. 2 / 2021 pp. 224-235
DOI 10.5755/j01.itc.50.2.27570
Quantum-Resistant Network for Classical Client Compatibility Received 2020/08/30 Accepted after revision 2021/03/19
http://dx.doi.org/10.5755/j01.itc.50.2.27570
HOW TO CITE: Lin, T.-Y., Fuh, C.-S. (2021). Quantum-Resistant Network for Classical Client Compatibility. Information Technology and Control, 50(2), 224-235. https://doi.org/10.5755/j01.itc.50.2.27570
Corresponding author: [email protected]
Te-Yuan Lin
Ph. D. Candidate of Department of Computer Science and Information Engineering, National Taiwan University;
No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan; phone: + 886 2 33664888; fax: +886 2-23628167;
e-mail: [email protected]
Chiou-Shann Fuh
Professor of Department of Computer Science and Information Engineering, National Taiwan University;
No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan; phone: + 886 2 33664888; fax: +886 2-23628167;
e-mail: [email protected]
Quantum computing is no longer a thing of the future. Shor’s algorithm proved that a quantum computer could traverse key of factoring problems in polynomial time. Because the time-complexity of the exhaustive key search for quantum computing has not reliably exceeded the reasonable expiry of crypto key validity, it is be- lieved that current cryptography systems built on top of computational security are not quantum-safe. Quan- tum key distribution fundamentally solves the problem of eavesdropping; nevertheless, it requires quantum preparatory work and quantum-network infrastructure, and these remain unrealistic with classical computers.
In transitioning to a mature quantum world, developing a quantum-resistant mechanism becomes a stringent problem. In this research, we innovatively tackled this challenge using a non-computational difficulty scheme with zero-knowledge proof in order to achieve repellency against quantum computing cryptanalysis attacks for universal classical clients.
KEYWORDS: Cloud Computing Security, Homomorphic Encryption, Quantum Computing, Zero-Knowledge.
1. Introduction
Computing can occur in any location and using a wide range of devices. The path to this accomplishment has passed through mainframe and personal computing,
and then to Internet computing. Computation oper- ates in an increasingly distributed manner; thus, data leakage threats have become ubiquitous. The current
approach to information security depends heavily on public key cryptography systems of computational se- curity, where the security commitment comes from the time required for exhaustive key search exceeding that required for cryptography key validity, and the attack- ing resource cost outweighs the value of the message itself. The safety keeps assured until Shor’s algorithm [26] formally proved that a quantum computer could traverse key of factoring problems in polynomial time.
As quantum computing power grows, computational complexity loses its intractability [16]. Eavesdroppers can be the Man-In-The-Middle (MITM), malicious communication counterparts, inadvertent/ intention- al cloud providers, and even governments. Although owning a private quantum computer is unrealistic, attackers can leverage quantum computing services from cloud providers with quantum supremacy to con- duct cryptanalysis attacks. Hence, it is imperative to identify different approaches to classical cryptography algorithms, which are known as Post-Quantum Cryp- tographies (PQC).
1.1. Paper Organization and Contributions This paper is organized as follows:
Section 1 gives a brief overview of quantum cryptog- raphy and reviews research related to quantum re- sistance. Section 2 provides a detailed analysis of the important methods used for the proposed scheme.
In Section 3, we further explain the methods in Sec- tion 2 and depict how we transform Zero-Knowledge Proof system concept to concrete proving/ verifying steps which suffice to support the idea of replacing public key exchange, to establish a quantum-resistant network. The algorithms comprising the scheme are given in Section 3, and the algorithm is summarized in Figure 4. The scheme extends the capability of tra- ditional homomorphic encryption for addressing the vulnerability that quantum computers may bring. We offer a different perspective as an alternative to plac- ing all bets on PKI improvements. We tested the real quantum computing provided by IBM Q experience of factorization and simulated the scheme algorithms in Section 4. The paper is summarized and concluded in Section 5.
Our main contributions are summarized below:
_ The proposed scheme is compatible with classical computers and networks, allowing them to be quantum-resistant without the need for augmented quantum preparations.
_ The novelty of the proposed scheme is how it achieves no public key-exchange during the entire protocol so that it greatly reduces the risk of quantum computing attacks.
_ We establish an initiative that combines ZKP and FHE methods to reduce possible breaches caused by third-party validation.
1.2. Quantum Cryptography
Similar to quantum computing from the principles of quantum mechanics, quantum cryptography takes advantage of qubit states with its un-trackable and no perfect cloning nature. Suppose there is an existing state of a quantum system A, denoted as |ψ⟩A, which we wish to clone without any prior knowledge. Then we take another independent quantum system, A’, of initial state with identical Hilbert Space [17], denoted as |e⟩A’. If we try to perform a measurement on A, the measurement immediately collapses the system into a certain eigenstate of the observable, totally corrupt- ing the information contained in the original qubit.
The alternative is to entangle the state of A and A’ as a pair; these two systems can be seen as a composite object described by the tensor product A⊗A’ of two vector spaces A and A’, and its composite state is |ψ⟩A
|e⟩A’. Next, to perform a unitary linear transformation (U) on quantum states to approximate the state |ψ⟩A, this cloning can be denoted as
U |ψ⟩A |e⟩A’ = |ψ⟩A |ψ⟩A’. (1) The imperfection that the theoretical bounds were derived on the fidelity of cloned quantum states has been proved. In classical communication channels, we use public key/ private key pair and a trusted third party to promise the cryptographic key distribution is assured, based on computational intractability. In a quantum communication channel, the proven limita- tion that guarantees no eavesdropper cannot create an exact copy of a quantum cryptography key is criti- cal. The feature can be useful in Quantum Key Distri- bution (QKD) protocol, by which a private key can be generated between two parties over a public channel.
QKD is provably secure [4] because the eavesdropper, Eve, cannot gain the exact information from the qu- bits transmitted from Alice to Bob without interfer- ing with their state. By calibrating the threshold of the error rate, it is easy for Alice and Bob to be aware of whether eavesdropping is happening. Once the error
rate is higher than pre-defined threshold, they can abandon the key and re-initialize the key-negotiation process until the low error rate is met. The shared key can then be used to encrypt and decrypt the classical information for communicating parties as we do now.
The first famous QKD scheme, invented by Charles Bennett and Gilles Brassard in 1984, is known as BB84 [3, 4].
QKD is rigorously secure (many researchers consid- er it unconditionally secure) [1, 12, 18], but it requires quantum network infrastructure and quantum devic- es at both sides to transmit quantum bits and generate their shared key. QKD has emerged from the labora- tory but remains in the preliminary implementation stage [8]. It is not widely adopted for most communi- cators due to noise interfering and performance and cost concerns [13, 19]. The aforementioned limits entail little compatibility with the current classical client running infrastructure.
1.3. Previous Quantum Resistance Research In 2012, the Computer Security Resource Center in National Institute of Standards and Technology (NIST) initiated a project called Post-Quantum Cryp- tography standards [22], for reviewing contemporary technologies to develop effective new algorithms for protecting electronic information from attack by the computers of both tomorrow and today. The goal is simple—keeping existing public key infrastructure intact in a future era of quantum computing. The most promising replacements of public key infrastructure are proposed and analyzed; these fall into a couple of large algorithm families: lattices-based, code-based, multivariate-based, hash-based, and isogenies-based.
Each has its own pros and cons. The comparison of each main family, today versus PQC effectiveness, is listed in the following table (see Table 1). Some other modernized implementations, such as Post-Quantum Yao and PQ-QT [6], still fall in the main family here.
The common and noteworthy merit of these algo- rithms is compatibility with classical clients, without the need for augmented quantum preparation, qubits measurement, or quantum channel transmission. In contrast to quantum augmentations and quantum channels, the cost of the above algorithms is quite ac- ceptable; however, except hash-based family, the se- curity promises of the remaining rely heavily on the mathematical difficulty barrier, i.e., computational
Table 1
Main algorithms of post-quantum cryptography [21]
Main Family
Proposed Signatures Key
Exchange
Size Efficient
Lattices-based 2.7 kb 1 kb Y
Code-based 192 kb 1 mb Y
Multivariate-
based Usually smaller
than public key 10-100kb Y
Hash-based 41 kb 1 kb Y
Other – Isogenies-
based 1228 kb 330 bytes N
security. In a hash-based algorithm, instead of en- crypting and decrypting messages directly, they are primarily used on digital signatures, so additional cryptographic complements would be necessary for the unhandled methods.
In this paper, we emphasize that a PQC scheme whose security commitment relies on none of three hard mathematical problems (i.e., the integer factorization problem, the discrete logarithm problem, and the el- liptic-curve discrete logarithm problem) finding a path in the isogeny graph of super-singular elliptic curves.
2. Methods Analysis
2.1. Proof System for Authentication
The essential disparity between quantum computing and classical computing is computing speed. With that in mind, instead of complexity confrontation, devising a scheme that is radically independent of the mathematical difficulty barrier may be the best way to avoid quantum attacking. The Zero-Knowledge Proof (ZKP) method is one of the problem-solving candi- dates.
A ZKP is a system to prove the authenticity of a statement without leaking extra information of statement. A famous metaphor, Ali Baba’s Cave [25], is quite a good example to show the ZKP philosophy intuitively. The cave layout is illustrated as in Fig- ure 1, with the entrance on one side (Point E) and the magic door blocking in the middle walkway at
the cave’s opposite side; only the person who knows the secret word of a magic spell can unlock the door.
The verifier (V) wants to know whether the prover (P) holds the secret word (w); P also wants to prove his awareness of w, but he does not want to reveal his knowledge (i.e., w) to V or to reveal his knowledge to other observers (O). To achieve this with ZKP, they can label the left and right paths as A and B and set some ground rules. First, V waits outside the cave at E as P goes in. P chooses either path A or B. Second, V is not allowed to see which path P takes. Then, V enters the cave and stands at point M, shouting the name of the path via which that V wants P to return, either A or B, chosen at random. Providing P really does know the magic word; this is easy: P opens the door using w, if necessary, and returns along the desired path. However, suppose P did not know the magic word, w, then P would only be able to return via the same path by which he had entered. Since V would choose A or B at random, P would have a 50 percent chance of guessing correctly. If they con- tinued the round many times, say 20 times in a row, P’s chance of successfully anticipating all of V’s re- quests would become extremely low (i.e., 1/220, about one in a million). Thus, if P repeatedly appears at the exit V requests, then V can conclude that it is very likely that P does know w. From start to finish, w is unknown to the V and the other observers, O, and this embodies zero-knowledge.
Figure 1
Ali Baba’s Cave – An example of the Zero-Knowledge Proof System
2.2. Homomorphic Encryption System for Confidentiality
An intuitive approach for preserving the privacy of user information in cloud-based services is to en- crypt everything before sending it to the cloud. This is secure, but the cloud service cannot operate on data to provide computing advantages before de- cryption. The dilemma is that we can never secure- ly transmit a conventional decrypting key. Thus, we wonder whether there exist encryption schemes that allow some computation to be performed directly on encrypted data (without first decrypting it). The birth of homomorphic encryption (HE) was exactly to this end, computation on ciphertexts without de- crypting sensitive data. This empowers the users to outsource their computing work to cloud service pro- viders securely or to chain different services (secure multi-party computation) together. A general HE scheme, ε, is primarily characterized by four operat- ing phases: KeyGenε, Encε, Decε, and Evalε, as shown in Figure 2.
Figure 2
The relations of four operating phases of homomorphic encryption
Let plaintext m ∈ M, ciphertext c ∈ C, and a key pair k ∈ K, where K is a secret key space for encryption and decryption. In KeyGenε, we design a specific algo- rithm to generate a key pair used for the next encryp- tion, decryption, and evaluation phases.
The output of KeyGenε can be denoted as k = (ke, kd). It is known as symmetric HE if ke = kd; asymmetric HE if ke ≠ kd. Here, we use the symmetric key and deem it the same as P’s secret key (a simplified w). To avoid some sophisticated attacks, the actual ke is a hashed mixture of a random oracle and an initial vector composed to-
gether as a dynamic key generating process. We will elaborate on the mechanism further in Section 3.
In phase Encε, the inputs are k and m, so the output, c, can be denoted as
c = Encε (ke, m). (2)
In phase Decε, the inputs are k and c, so the output, m, can be denoted as
m = Decε (kd, c). (3)
In the phase Evalε (a.k.a. re-encryption), it is correlat- ed with a set of operating functions Fε and the output can be denoted as
Evalε (ke, f, c1, c2, … ct,) = f (m1, m2, … mt) (4) for every Boolean function f ∈ Fε and arbitrary cipher- text c1, c2, … ct, where ci = Encε (ke, mi).
∀ f ∈ Fε, ∀ m ∈ M, ciphertext c ∈ C. (5) Thus, the correctness of the scheme is satisfied when
C←Evalε (ke, f, c) ↔ f (m1, m2, … mt) = Decε (kd, c). (6) In short, the operation Fε in ciphertext space, C, which is constructed by the encryption after the completion of f on its corresponding plaintext space, M, can be de- noted as
Fε (C) = Encε (f (M)), (7)
where f is usually a time-consuming/ compute-inten- sive task, with HE, since f (M) and C are homomor- phic, any Fε operation executing on C by the third-par- ty service provider is equivalent to its counterpart, Encε (f (M)).
As a result, decrypt the Fε (C), and we get f (M), so we are allowed to delegate Encε (f (M)) to any third-party service providers securely.
In the early version, Encε is classified as a Partially Homomorphic Encryption (PHE) algorithm of f, be- cause f can only operate addition or multiplication under encryption, but not both; then, an advanced version scheme comes in and is classified as Some- what Homomorphic Encryption (SHE) since f can operate both addition and multiplication with limited rounds. Finally, f can operate both addition and multi- plication with no infinite rounds, so the Encε scheme is deemed a Fully Homomorphic Encryption (FHE), which was implemented in 2009 based on PHE and SHE by Gentry [31].
FHE cryptosystems have better practical implica- tions in the outsourcing of private data computa- tions. An intuitive example is a line of assembling gold or diamonds into rings. A piece of gold (data) is locked inside a glovebox (encrypted by FHE) so that a worker (the cloud service provider) may transform it into a ring (the computed result). The ring is later taken out by the ring owner (data owner) when the glovebox is unlocked by the ring owner’s key (data owner’s secret key). This prevents the key from transmitting risk (hacking/ embezzled) and the pos- sibility of gold stolen (data stolen), shown as the fol- lowing figure 3.
Figure 3
Glovebox idea and FHE [20]
(1) (2) (3) (4) (5)
_ The eavesdropper, E: Eve, the eavesdropper who is spying on the classical channel between Alice and Bob. Eve can be an imposter and conducts MITM attacks. Furthermore, Eve can delegate her computing tasks to any cloud service provider, even attainable to the quantum grade.
_ The statement, x: Since Alice uses w to encrypt the data to be processed, the statement is that Alice knows the valid data owner's secret key w.
_ The proof, π: It is related to the parameters x and w, i.e., π ← Prove (x, w); without revealing w to V, P has to convince V that her knowledge of w to hold the inequality 1 ← Verify (x, π) true.
_ The private data set was pre-processed by a specific FHE algorithm, ε, before transmitting to the verifier, V. The encryption key, w, needs to be generated in the phase, KeyGenε, and used to encrypt the data set.
For the entire proving protocol, V can learn nothing about P’s knowledge of w.
3.1. Pseudo-code of Two ZKP-Cave Algorithms
3. Proposed Scheme
Based on the above-elaborated analysis on currently known methods, we can merge the two systems into a new scheme in a novel way.
Instead of relying on public key infrastructure, we transform the data privacy problem of outsourcing cloud computing into a zero-knowledge proof man- ner. The scheme mixes ZKP and FHE, acting as Ali Baba’s cave with an FHE glovebox in-between the classical client (Alice) and her cloud service provider (Bob). We name the scheme ZKP-Cave after this phi- losophy (as shown in Figure 4).
Figure 4
ZKP-Cave — a fully homomorphic encryption scheme that conforms with zero-knowledge proof and is compatible with classical parties
ZKP-Cave elements:
_ The prover, P: Alice, the cloud service consumer who has her secret key, w, to encrypt her data and plans to outsource her computing tasks to the cloud service provider, Bob. Before Bob accepts Alice’s tasks, Alice must prove she really is the secret key owner who uses the key to encrypt the data within the tasks.
_ The verifier, V: Bob, the cloud service provider who has sufficient computing and memory resources, even attainable to the quantum-grade. Bob verifies the consumer’s identity and ownership before processing the computing tasks.
Algorithm 1. Avoid the public key exchange risk via the proving algorithm
Initialized inputs:
rounds: the rounds of executing the dummy test task randNum : a random integer number generated from a Random class
1: private static void main(String args[]) { 2: int t = 0;
3: int operand1, operand2;
4: //Define a custom class array to store operand pairs
5: OperandPair[] operandSet = new OperandPair[t];
6: int[] operandType = new int[t];
7: int[] operationResultSet = new int[t];
8: while (t < rounds) {
9: operand1 = randNum.nextInt();
10: operand2 = randNum.nextInt();
11: operandType[t] = randomOperatorPickup();
12: operationResultSet[t] = calculateOperands(operand1, operand2, operandType[t]);
13: t ++;
14: }
15: //Storing each encrypted pair of operands
16: for (int i = 0; i <
operandSet.length; i++) { 17: operandSet[i] = new
OperandPair (Enc(operand1, w), Enc(operand2, w))
18: }
19: // Shuffle the element sequence of each set and submit them to cloud provers
20: submitShuffleSet(operandSet, operationResultSet);
21: 22: }
23: public static int
calculateOperands(int operand1, int operand2, String operator)
24: {
25: int operationFHE, operationResult;
26: // Each operational result must be processed by FHE with private key w
27: operationFHE =
EncFHE(String.valueOf(operand1) + operator.charAt(0) +
String.valueOf(operand2));
28: operationResult = (int) operationFHE.toString();
29: return operationResult;
30: }
31: public static String
randomOperatorPickup() { 32: char[] Operators= new
char[]{'+','*'};
33: if(randNum == 0 || randNum % 2
== 0) Operators [0] = '+';
34: else Operators [0] = '*';
35: return
String.valueOf(Operators[0]);
36: }
Output: Shuffled operandSet and
operationResultSet pair
Algorithm 2. The verifying algorithm (Processing the privacy data only after the prover passed verification) Initialized inputs:
confidence: the minimum level to accept P as true hitRateLevel: an adjustable threshold that considers reasonable network noise
1: private static void main(String args[]) { 2: int t = 0;
3: long operand = 0;
4: long operandResult = 0;
5: Long returningResult = 0;
6: int confidence = 0;
7: float hitRateLevel = 99.0;
8: 9: // Initialize the operand object of shuffleSet for verification
10: OperandPair op =
OperandPair.initInstance(shuffleSet);
11: 12: // Supply the encrypted operand pair object randomly and compare the result
13: while (t < rounds) {
14: operand = op.getRandomOperand();
15: operandResult = op.getRandomOp- erandResult();
16:
17: //Collect Prover’s returning answer
18: returningResult = op.send(operand);
19: if(returningResult ==
operandResult) { confidence ++};
20:
21: t ++;
22: };
23: 24: //Verify the result confidence
25: if(confidence / t >= hitRateLevel) { 26: //Proving process passed, the session
is secure
27: //The cloud service provider can proceed to execute FHE task
28: 29: } else { //Hacker detected, invalidate the session}
30: 31: }
Output: Pass or fail the verifying session
3.2. Incorporate with Improved Fully Homomorphic Symmetric Encryption
The version of Fully Homomorphic Encryption (FHE) has two main classes: asymmetric and sym- metric. In the early version of the asymmetry-based approach, the size of encrypted data proliferates rap- idly. It is inevitable for a bootstrapping framework, which accumulates noise of a large number of keys per computation [9]. Most as-is schemes are devoted to decreasing growing data size and processing over- head; the papers [5, 7, 20, 23, 29] present the improved version of FHE without Gentry’s bootstrapping pre- requisite.
However, almost all asymmetry-based approaches are based on the same assumption (i.e., the large inte- ger factorization remains a hard problem). Given the evolving maturity of quantum computing, the expo- sure of a public key can lead to the compromise of its private key used in the first place.
Our scheme sets the symmetric version FHE as the basis for encryption/ decryption-sensitive data to become immune to the above attacking problems in a quasi-quantum cloud computing world. Per the KeyGenε process discussed in the section of meth- ods analysis, caution taken with the use of symmet- ric FHE in our scenario should design to be resis- tant to chosen and known-plaintext attacks from on-premises network interceptors, although the threat has nothing to do with the initial proposition of quantum-power attacks from the cloud or inter- net eavesdroppers. Our solution to this type of attack is incorporating every time dynamic key generation and dynamic block encryption while executing ho- momorphic encryption. Likewise, the extra inversed process should be involved during the decryption process of symmetric FHE, referring to P’s secret key (i.e., w). Hariss et al. [9, 10] have proposed the detailed implementation of dynamic key generation and dynamic block encryption.
4. Test and Evaluation
To analyze as-is quantum attacking force and the pro- posed scheme capability, we conducted several tests and evaluations to evaluate its effectiveness.
4.1. Factorization Attack Test on IBM Quantum Cloud Computers
The implementations are done with Python, Jupy- ter Notebooks, and IBM QISKit [11, 24], which allow developers to explore IBM Q Experience [14, 15] – a real cloud-enabled platform of quantum processors.
In this work, the Shor Factorizing algorithm has been executed on IBM’s quantum computer, as shown in Figure 5.
Figure 5
Shor Factorization Test on IBM Q Experience, as of early 2020
In earlier research, the theoretical circuits of Shor’s algorithm use 2n+3 qubits for factoring [2], yet the circuits consume 4n+2 qubits for factoring with IBM Q Experience in practice, according to our observa- tion. We infer that the overhead comes from auxiliary quantum registers used in addition and multiplication of algorithm implementation. Furthermore, our test result shows that burst time in quantum processors consumes rapidly when the slow rise of small integer factorization. The trend has self-explained enough, even with IBM’s state-of-the-art 53-qubit quantum computer [21], still hard to deal with the length of 13-bit integer factorization in a flash time, not to say threaten 1024-bit or 2048-bit key length in which we found commonly used by cryptographic algorithms
today. However, should Moore's Law or even Neven’s Law [28] be applied to the number of quantum com- puting power qubits growth, in less than two decades, the factoring run chart result would be quite different.
4.2. Scheme Effectiveness Simulation
Referring back to the definitions of the ZKP-Cave el- ements in the section of the proposed scheme, we can simulate the interactions of proving steps and verify- ing steps.
ZKP-Cave Proving Steps:
_ A cloud service consumer P first generates a dummy test task that includes random plaintext numbers(operands). P chooses a small number of plaintext numbers and calculates them with a random Boolean operator, either multiplication or addition.
_ P encrypts the chosen plaintext operands and their Boolean operational result, respectively, with P’s secret key (i.e., w). P keeps these ciphertexts as a set of dummy test task results.
_ P repeats the above process for t rounds and hence gets t sets of dummy task results. Within the dummy task results, there are two kinds of groups;
one is the operand set, another is the operational result set. Every operand set has a corresponding operational result in the operational result sets.
_ Next, P submits ciphertexts of operand sets and operational result in scrambled order to V over a classical channel.
ZKP-Cave Verifying Steps:
A cloud service provider V receives and stores the ci- phertexts of operand and operational result sets from P for later processing and verification.
_ V verifies P, who claims to be authentic, by sending an encrypted operand pair that is randomly picked up out of the stored operand sets and asks for the correct encrypted operational result repeatedly.
Once V gets the response, regardless of whom, he immediately compares whether that matches anything in the operational result sets sent from the authentic P in the very beginning.
_ When an eavesdropper, E, tries to trick V into believing that E = P since E has no idea of which operand set (encrypted) is corresponding to which operational result set (encrypted), she can only respond the challenge by guessing one from the set pairs earlier sent by P. As the examining rounds Table 2
Dummy Test Tasks Simulations
Round 1 2 … t
Operand Set (in plaintext)
Operand 1 5 42022 … 77
Operator ( * or +) * + … +
Operational Result Set (in plaintext)
Operand 2 3 998 … 32
Operation Result 15 43020 … 109
Dummy Test Task Set 1 2 … t
Operand Set (in ciphertext)
Operand 1 (encrypted) 70994920…7074870821 62090366…7954462948 … 62312036 …7582581582 Operand 2 (encrypted) 70994920 ...7774870819 62090366…7954421924 … 62312036…4782581537
Operation Result Set (in ciphertext)
Operation Result (encrypted) 50402786...7687472399 12418073…08884872 … 12462407…5165163119
grow, V will find the hit rate abnormally decreased as the number of dummy test rounds increases.
From the scene, V knows there must be someone in the middle who is trying a hack engineering;
thus, V can discard the rest of the verifying process and invalidate any request by the session. The soundness can be achieved since the cheating P can always be rejected by the honest V.
_ After t rounds of operand sets have been traversed with a high hit rate, V would have a high confidence level that 1 ← Verify (x, π) holds true to accept as P; otherwise, V invalidates the session. The completeness can thereby be achieved.
The evaluation result is summarized in Table 2. With several rounds of dummy test execution generated by cloud service consumers, we can observe that the scope for a cloud service provider is merely execut- ing massive instructions on the operation result in ciphertext with pre-defined FHE evaluate algorithm of ZKP-Cave. Notice that the data-in-process always stays in a ciphered manner during the protocol. ZKP- Cave prevents malicious cloud service providers from spying on sensitive content. The scheme promises ze- ro-knowledge commitment against its secret knowl- edge processors and any other eavesdroppers.
The effectiveness of our scheme maintains a simu- lator to produce a scrambled and indistinguishable output to avoid linkage with any meaning during the interaction between the prover and the verifier. In the case of a malicious adversary/ service provider armed with quantum computing power, since there is no key-exchange during ZKP-Cave protocol, there is no PKI computational vulnerability that can be breached by quantum computation.
5. Discussions
When dealing with issues such as secure multi-party computation, the approach of simulation is most com- monly used. The critical key of simulation is to build an undistinguishable carrier to an adversary that in- teracts with all parties involved. In our scheme, we see the nature of quantum computing, i.e., the result is a measuring state with higher probability; in other words, there is always a chance of lower probability to get wrong answers. Our scheme promises data pri-
vacy based on FHE, and symmetric key encryption casts away the asymmetric keys and artfully builds an example of this carrier on top of the proving and verifying processes to reject possible quantum eaves- droppers who always have a slight chance to respond wrong answers. However, the challenge is, we now have very limited knowledge of the internal space of quantum adversaries about their all generic quantum states, actions, and communications behaviors. We need to carry on with modeling quantum adversary‘s attacking capability. The success of continuously proving quantum security heavily relies on the com- pleteness of modeling works [27, 30]. These questions remain open, and their model structure analysis will be covered in our future research interests.
6. Conclusions
In this paper, after introducing quantum comput- ing, quantum cryptography, and discussing various post-quantum cryptography schemes and limits, we propose an original scheme to include a zero-knowl- edge system and FHE symmetric encryption. The proposed method is an instance to combine the au- thentication of ZKP and the confidentiality of FHE in a communicating system to preserve data privacy in the scenario of immature quantum computation.
Quantum computers will eventually be helpful to solve innovation problems as well as giving rise to various trusted computing issues. Convergence of technology is one direction for achieving this, which deserves to long, hard contemplation among academ- ics.
We look forward to motivating more possibilities in the quantum resistance field and continual improve- ments for cloud computing security. Coupling differ- ent measures to a certain extent of secure diversity are our suggested way to sustain safety under uncer- tain quantum computing attacks.
Acknowledgement
We sincerely thank National Taiwan University-IBM Q Hub, who sponsored the state-of-the-art quantum computers and application interfaces to contribute this research as a great testing platform for simulat- ing the factorization with the Shor algorithm.
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