908 research outputs found
A New Upperbound for the Oblivious Transfer Capacity of Discrete Memoryless Channels
We derive a new upper bound on the string oblivious transfer capacity of
discrete memoryless channels. The main tool we use is the tension region of a
pair of random variables introduced in Prabhakaran and Prabhakaran (2014) where
it was used to derive upper bounds on rates of secure sampling in the source
model. In this paper, we consider secure computation of string oblivious
transfer in the channel model. Our bound is based on a monotonicity property of
the tension region in the channel model. We show that our bound strictly
improves upon the upper bound of Ahlswede and Csisz\'ar (2013).Comment: 7 pages, 3 figures, extended version of submission to IEEE
Information Theory Workshop, 201
Privacy-Preserving Quantum Two-Party Geometric Intersection
Privacy-preserving computational geometry is the research area on the
intersection of the domains of secure multi-party computation (SMC) and
computational geometry. As an important field, the privacy-preserving geometric
intersection (PGI) problem is when each of the multiple parties has a private
geometric graph and seeks to determine whether their graphs intersect or not
without revealing their private information. In this study, through
representing Alice's (Bob's) private geometric graph G_A (G_B) as the set of
numbered grids S_A (S_B), an efficient privacy-preserving quantum two-party
geometric intersection (PQGI) protocol is proposed. In the protocol, the oracle
operation O_A (O_B) is firstly utilized to encode the private elements of
S_A=(a_0, a_1, ..., a_(M-1)) (S_B=(b_0, b_1, ..., b_(N-1))) into the quantum
states, and then the oracle operation O_f is applied to obtain a new quantum
state which includes the XOR results between each element of S_A and S_B.
Finally, the quantum counting is introduced to get the amount (t) of the states
|a_i+b_j> equaling to |0>, and the intersection result can be obtained by
judging t>0 or not. Compared with classical PGI protocols, our proposed
protocol not only has higher security, but also holds lower communication
complexity
Computational Perspectives on Bell Inequalities and Many-body Quantum Correlations
The predictions of quantum mechanics cannot be resolved with a completely
classical view of the world. In particular, the statistics of space-like
separated measurements on entangled quantum systems violate a Bell inequality.
We put forward a computational perspective on a broad class of Bell tests that
study correlators, or the statistics of joint measurement outcomes. We
associate particular maps, or functions to particular theories. The violation
of a Bell inequality then implies the ability to perform some functions, or
computations that classical, or more generally, local hidden variable (LHV)
theories cannot.Comment: PhD thesis: 169 pages, 3 figures, and hopefully is of use to someon
On perfectly secure 2PC in the OT-hybrid model
A well known result by Kilian (ACM 1988) asserts that general secure two computation (2PC) with statistical security, can be based on OT. Specifically, in the client-server model, where only one party -- the client -- receives an output, Kilianโs result shows that given the ability to call an ideal oracle that computes OT, two parties can securely compute an arbitrary function of their inputs with unconditional security. Ishai et al. (EUROCRYPT 2011) further showed that this can be done efficiently for every two-party functionality in in a single round.
However, their results only achieve statistical security, namely, it is allowed to have some error in security. This leaves open the natural question as to which client-server functionalities can be computed with perfect security in the OT-hybrid model, and what is the round complexity of such computation. So far, only a handful of functionalities were known to have such protocols. In addition to the obvious theoretical appeal of the question towards better understanding secure computation, perfect, as opposed to statistical reductions, may be useful for designing secure multiparty protocols with high concrete efficiency, achieved by eliminating the dependence on a security parameter.
In this work, we identify a large class of client-server functionalities , where the server\u27s domain is larger than the client\u27s domain , that have a perfect reduction to OT. Furthermore, our reduction is 1-round using an oracle to secure evaluation of many parallel invocations of -bit-OT, as done by Ishai et al. (EUROCRYPT 2011). Interestingly, the set of functions that we are able to compute was previously identified by Asharov (TCC 2014) in the context of fairness in two-party computation, naming these functions full-dimensional. Our result also extends to randomized non-Boolean functions satisfying
Privacy-preserving proximity detection with secure multi-party computational geometry
Over the last years, Location-Based Services (LBSs) have become popular due to the global use of smartphones and improvement in Global Positioning System (GPS) and other positioning methods. Location-based services employ users' location to offer relevant information to users or provide them with useful recommendations. Meanwhile, with the development of social applications, location-based social networking services (LBSNS) have attracted millions of users because the geographic position of users can be used to enhance the services provided by those social applications. Proximity detection, as one type of location-based function, makes LBSNS more flexible and notifies mobile users when they are in proximity. Despite all the desirable features that such applications provide, disclosing the exact location of individuals to a centralized server and/or their social friends might put users at risk of falling their information in wrong hands, since locations may disclose sensitive information about people including political and religious affiliations, lifestyle, health status, etc. Consequently, users might be unwilling to participate in such applications.
To this end, private proximity detection schemes enable two parties to check whether they are in close proximity while keeping their exact locations secret. In particular, running a private proximity detection protocol between two parties only results in a boolean value to the querier. Besides, it guarantees that no other information can be leaked to the participants regarding the other party's location. However, most proposed private proximity detection protocols enable users to choose only a simple geometric range on the map, such as a circle or a rectangle, in order to test for proximity.
In this thesis, we take inspiration from the field of Computational Geometry and develop two privacy-preserving proximity detection protocols that allow a mobile user to specify an arbitrary complex polygon on the map and check whether his/her friends are located therein. We also analyzed the efficiency of our solutions in terms of computational and communication costs. Our evaluation shows that compared to the similar earlier work, the proposed solution increases the computational efficiency by up to 50%, and reduces the communication overhead by up to 90%. Therefore, we have achieved a significant reduction of computational and communication complexity
์ ๋ณด ๋ณดํธ ๊ธฐ๊ณ ํ์ต์ ์ํธํ ๊ธฐ๋ฐ ๊ธฐ์ : ๊ทผ์ฌ ๋ํ ์ํธ์ ๋ถํธ ๊ธฐ๋ฐ ์ํธ
ํ์๋
ผ๋ฌธ (๋ฐ์ฌ) -- ์์ธ๋ํ๊ต ๋ํ์ : ๊ณต๊ณผ๋ํ ์ ๊ธฐยท์ ๋ณด๊ณตํ๋ถ, 2021. 2. ๋
ธ์ข
์ .In this dissertation, three main contributions are given as; i) a protocol of privacy-preserving machine learning using network resources, ii) the development of approximate homomorphic encryption that achieves less error and high-precision bootstrapping algorithm without compromising performance and security, iii) the cryptanalysis and the modification of code-based cryptosystems: cryptanalysis on IKKR cryptosystem and modification of the pqsigRM, a digital signature scheme proposed to the post-quantum cryptography (PQC) standardization of National Institute of Standards and Technology (NIST).
The recent development of machine learning, cloud computing, and blockchain raises a new privacy problem; how can one outsource computation on confidential data? Moreover, as research on quantum computers shows success, the need for PQC is also emerging. Multi-party computation (MPC) is the cryptographic protocol that makes computation on data without revealing it. Since MPC is designed based on homomorphic encryption (HE) and PQC, research on designing efficient and safe HE and PQC is actively being conducted.
First, I propose a protocol for privacy-preserving machine learning (PPML) that replaces bootstrapping of homomorphic encryption with network resources. In general, the HE ciphertext has a limited depth of circuit that can be calculated, called the level of a ciphertext. We call bootstrapping restoring the level of ciphertext that has exhausted its level through a method such as homomorphic decryption. Bootstrapping of homomorphic encryption is, in general, very expensive in time and space. However, when deep computations like deep learning are performed, it is required to do bootstrapping. In this protocol, both the client's message and servers' intermediate values are kept secure, while the client's computation and communication complexity are light.
Second, I propose an improved bootstrapping algorithm for the CKKS scheme and a method to reduce the error by homomorphic operations in the CKKS scheme. The Cheon-Kim-Kim-Song (CKKS) scheme (Asiacrypt '17) is one of the highlighted fully homomorphic encryption (FHE) schemes as it is efficient to deal with encrypted real numbers, which are the usual data type for many applications such as machine learning. However, the precision drop due to the error growth is a drawback of the CKKS scheme for data processing. I propose a method to achieve high-precision approximate FHE using the following two methods .First, I apply the signal-to-noise ratio (SNR) concept and propose methods to maximize SNR by reordering homomorphic operations in the CKKS scheme. For that, the error variance is minimized instead of the upper bound of error when we deal with the encrypted data. Second, from the same perspective of minimizing error variance, I propose a new method to find the approximate polynomials for the CKKS scheme. The approximation method is especially applied to the CKKS scheme's bootstrapping, where we achieve bootstrapping with smaller error variance compared to the prior arts. In addition to the above variance-minimizing method, I cast the problem of finding an approximate polynomial for a modulus reduction into an L2-norm minimization problem. As a result, I find an approximate polynomial for the modulus reduction without using the sine function, which is the upper bound for the polynomial approximation of the modulus reduction. By using the proposed method, the constraint of q = O(m^{3/2}) is relaxed as O(m), and thus the level loss in bootstrapping can be reduced. The performance improvement by the proposed methods is verified by implementation over HE libraries, that is, HEAAN and SEAL. The implementation shows that by reordering homomorphic operations and using the proposed polynomial approximation, the reliability of the CKKS scheme is improved. Therefore, the quality of services of various applications using the proposed CKKS scheme, such as PPML, can be improved without compromising performance and security.
Finally, I propose an improved code-based signature scheme and cryptanalysis of code-based cryptosystems. A novel code-based signature scheme with small parameters and an attack algorithm on recent code-based cryptosystems are presented in this dissertation. This scheme is based on a modified Reed-Muller (RM) code, which reduces the signing complexity and key size compared with existing code-based signature schemes. The proposed scheme has the advantage of the pqsigRM decoder and uses public codes that are more difficult to distinguish from random codes. I use (U, U+V) -codes with the high-dimensional hull to overcome the disadvantages of code-based schemes. The proposed a decoder which efficiently samples from coset elements with small Hamming weight for any given syndrome. The proposed signature scheme resists various known attacks on RM code-based cryptography. For 128 bits of classical security, the signature size is 4096 bits, and the public key size is less than 1 MB. Recently, Ivanov, Kabatiansky, Krouk, and Rumenko (IKKR) proposed three new variants of the McEliece cryptosystem (CBCrypto 2020, affiliated with Eurocrypt 2020). This dissertation shows that one of the IKKR cryptosystems is equal to the McEliece cryptosystem. Furthermore, a polynomial-time attack algorithm for the other two IKKR cryptosystems is proposed. The proposed attack algorithm utilizes the linearity of IKKR cryptosystems. Also, an implementation of the IKKR cryptosystems and the proposed attack is given. The proposed attack algorithm finds the plaintext within 0.2 sec, which is faster than the elapsed time for legitimate decryption.๋ณธ ๋
ผ๋ฌธ์ ํฌ๊ฒ ๋ค์์ ์ธ ๊ฐ์ง์ ๊ธฐ์ฌ๋ฅผ ํฌํจํ๋ค. i) ๋คํธ์ํฌ๋ฅผ ํ์ฉํด์ ์ ๋ณด ๋ณดํธ ๋ฅ๋ฌ๋์ ๊ฐ์ ํ๋ ํ๋กํ ์ฝ ii) ๊ทผ์ฌ ๋ํ ์ํธ์์ ๋ณด์์ฑ๊ณผ ์ฑ๋ฅ์ ์ํด ์์ด ์๋ฌ๋ฅผ ๋ฎ์ถ๊ณ ๋์ ์ ํ๋๋ก ๋ถํธ์คํธ๋ํ ํ๋ ๋ฐฉ๋ฒ iii) IKKR ์ํธ ์์คํ
๊ณผ pqsigRM ๋ฑ ๋ถํธ ๊ธฐ๋ฐ ์ํธ๋ฅผ ๊ณต๊ฒฉํ๋ ๋ฐฉ๋ฒ๊ณผ ํจ์จ์ ์ธ ๋ถํธ ๊ธฐ๋ฐ ์ ์ ์๋ช
์์คํ
.
๊ทผ๋์ ๊ธฐ๊ณํ์ต๊ณผ ๋ธ๋ก์ฒด์ธ ๊ธฐ์ ์ ๋ฐ์ ์ผ๋ก ์ธํด์ ๊ธฐ๋ฐ ๋ฐ์ดํฐ์ ๋ํ ์ฐ์ฐ์ ์ด๋ป๊ฒ ์ธ์ฃผํ ์ ์๋๋์ ๋ํ ์๋ก์ด ๋ณด์ ๋ฌธ์ ๊ฐ ๋๋๋๊ณ ์๋ค. ๋ํ, ์์ ์ปดํจํฐ์ ๊ดํ ์ฐ๊ตฌ๊ฐ ์ฑ๊ณต์ ๊ฑฐ๋ญํ๋ฉด์, ์ด๋ฅผ ์ด์ฉํ ๊ณต๊ฒฉ์ ์ ํญํ๋ ํฌ์คํธ ์์ ์ํธ์ ํ์์ฑ ๋ํ ์ปค์ง๊ณ ์๋ค. ๋ค์๊ฐ ์ปดํจํ
์ ๋ฐ์ดํฐ๋ฅผ ๊ณต๊ฐํ์ง ์๊ณ ๋ฐ์ดํฐ์ ๋ํ ์ฐ์ฐ์ ์ํํ ์ ์๋๋ก ํ๋ ์ํธํ์ ํ๋กํ ์ฝ์ ์ด์นญ์ด๋ค. ๋ค์๊ฐ ์ปดํจํ
์ ๋ํ ์ํธ์ ํฌ์คํธ ์์ ์ํธ์ ๊ธฐ๋ฐํ๊ณ ์์ผ๋ฏ๋ก, ํจ์จ์ ์ธ ๋ํ ์ํธ์ ํฌ์คํธ ์์ ์ํธ์ ๊ดํ ์ฐ๊ตฌ๊ฐ ํ๋ฐํ๊ฒ ์ํ๋๊ณ ์๋ค.
๋ํ ์ํธ๋ ์ํธํ๋ ๋ฐ์ดํฐ์ ๋ํ ์ฐ์ฐ์ด ๊ฐ๋ฅํ ํน์ํ ์ํธํ ์๊ณ ๋ฆฌ์ฆ์ด๋ค. ์ผ๋ฐ์ ์ผ๋ก ๋ํ ์ํธ์ ์ํธ๋ฌธ์ ๋ํด์ ์ํ ๊ฐ๋ฅํ ์ฐ์ฐ์ ๊น์ด๊ฐ ์ ํด์ ธ ์์ผ๋ฉฐ, ์ด๋ฅผ ์ํธ๋ฌธ์ ๋ ๋ฒจ์ด๋ผ๊ณ ์นญํ๋ค. ๋ ๋ฒจ์ ๋ชจ๋ ์๋นํ ์ํธ๋ฌธ์ ๋ ๋ฒจ์ ๋ค์ ๋ณต์ํ๋ ๊ณผ์ ์ ๋ถํธ์คํธ๋ํ (bootstrapping)์ด๋ผ๊ณ ์นญํ๋ค. ์ผ๋ฐ์ ์ผ๋ก ๋ถํธ์คํธ๋ํ์ ๋งค์ฐ ์ค๋ ๊ฑธ๋ฆฌ๋ ์ฐ์ฐ์ด๋ฉฐ ์๊ฐ ๋ฐ ๊ณต๊ฐ ๋ณต์ก๋๊ฐ ํฌ๋ค. ๊ทธ๋ฌ๋, ๋ฅ๋ฌ๋๊ณผ ๊ฐ์ด ๊น์ด๊ฐ ํฐ ์ฐ์ฐ์ ์ํํ๋ ๊ฒฝ์ฐ ๋ถํธ์คํธ๋ํ์ด ํ์์ ์ด๋ค. ๋ณธ ๋
ผ๋ฌธ์์๋ ์ ๋ณด ๋ณดํธ ๊ธฐ๊ณํ์ต์ ์ํ ์๋ก์ด ํ๋กํ ์ฝ์ ์ ์ํ๋ค. ์ด ํ๋กํ ์ฝ์์๋ ์
๋ ฅ ๋ฉ์์ง์ ๋๋ถ์ด ์ ๊ฒฝ๋ง์ ์ค๊ฐ๊ฐ๋ค ๋ํ ์์ ํ๊ฒ ๋ณดํธ๋๋ค. ๊ทธ๋ฌ๋ ์ฌ์ ํ ์ฌ์ฉ์์ ํต์ ๋ฐ ์ฐ์ฐ ๋ณต์ก๋๋ ๋ฎ๊ฒ ์ ์ง๋๋ค.
Cheon, Kim, Kim ๊ทธ๋ฆฌ๊ณ Song (CKKS)๊ฐ ์ ์ํ ์ํธ ์์คํ
(Asiacrypt 17)์ ๊ธฐ๊ณํ์ต ๋ฑ์์ ๊ฐ์ฅ ๋๋ฆฌ ์ฐ์ด๋ ๋ฐ์ดํฐ์ธ ์ค์๋ฅผ ํจ์จ์ ์ผ๋ก ๋ค๋ฃฐ ์ ์์ผ๋ฏ๋ก ๊ฐ์ฅ ์ด๋ง๋ฐ๋ ์์ ๋ํ ์ํธ ์์คํ
์ด๋ค. ๊ทธ๋ฌ๋, ์ค๋ฅ์ ์ฆํญ๊ณผ ์ ํ๊ฐ CKKS ์ํธ ์์คํ
์ ๊ฐ์ฅ ํฐ ๋จ์ ์ด๋ค. ์ด ๋
ผ๋ฌธ์์๋ ์๋์ ๊ธฐ์ ์ ํ์ฉํ์ฌ CKKS ์ํธ ์์คํ
์ ์ค๋ฅ๋ฅผ ์ค์ด๋ ๋ฐฉ๋ฒ์ ์ ์ํ๋ฉฐ, ์ด๋ ๊ทผ์ฌ ๋ํ ์ํธ์ ์ผ๋ฐํํ์ฌ ์ ์ฉํ ์ ์๋ค. ์ฒซ์งธ, ์ ํธ ๋๋น ์ก์ ๋น (signal-to-noise ratio, SNR)์ ๊ฐ๋
์ ๋์
ํ์ฌ, SNR๋ฅผ ์ต๋ํํ๋๋ก ์ฐ์ฐ์ ์์๋ฅผ ์ฌ์กฐ์ ํ๋ค. ๊ทธ๋ฌ๊ธฐ ์ํด์๋, ์ค๋ฅ์ ์ต๋์น ๋์ ๋ถ์ฐ์ด ์ต์ํ๋์ด์ผ ํ๋ฉฐ, ์ด๋ฅผ ๊ด๋ฆฌํด์ผ ํ๋ค. ๋์งธ, ์ค๋ฅ์ ๋ถ์ฐ์ ์ต์ํํ๋ค๋ ๊ฐ์ ๊ด์ ์์ ์๋ก์ด ๋คํญ์ ๊ทผ์ฌ ๋ฐฉ๋ฒ์ ์ ์ํ๋ค. ์ด ๊ทผ์ฌ ๋ฐฉ๋ฒ์ ํนํ, CKKS ์ํธ ์์คํ
์ ๋ถํธ์คํธ๋ํ์ ์ ์ฉ๋์์ผ๋ฉฐ, ์ข
๋ ๊ธฐ์ ๋ณด๋ค ๋ ๋ฎ์ ์ค๋ฅ๋ฅผ ๋ฌ์ฑํ๋ค. ์์ ๋ฐฉ๋ฒ์ ๋ํ์ฌ, ๊ทผ์ฌ ๋คํญ์์ ๊ตฌํ๋ ๋ฌธ์ ๋ฅผ L2-norm ์ต์ํ ๋ฌธ์ ๋ก ์นํํ๋ ๋ฐฉ๋ฒ์ ์ ์ํ๋ค. ์ด๋ฅผ ํตํด์ ์ฌ์ธ ํจ์์ ๋์
์์ด ๊ทผ์ฌ ๋คํญ์์ ๊ตฌํ๋ ๋ฐฉ๋ฒ์ ์ ์ํ๋ค. ์ ์๋ ๋ฐฉ๋ฒ์ ์ฌ์ฉํ๋ฉด, q=O(m^{3/2})๋ผ๋ ์ ์ฝ์ q=O(m)์ผ๋ก ์ค์ผ ์ ์์ผ๋ฉฐ, ๋ถํธ์คํธ๋ํ์ ํ์ํ ๋ ๋ฒจ ์๋ชจ๋ฅผ ์ค์ผ ์ ์๋ค. ์ฑ๋ฅ ํฅ์์ HEAAN๊ณผ SEAL ๋ฑ์ ๋ํ ์ํธ ๋ผ์ด๋ธ๋ฌ๋ฆฌ๋ฅผ ํ์ฉํ ๊ตฌํ์ ํตํด ์ฆ๋ช
ํ์ผ๋ฉฐ, ๊ตฌํ์ ํตํด์ ์ฐ์ฐ ์ฌ์ ๋ ฌ๊ณผ ์๋ก์ด ๋ถํธ์คํธ๋ํ์ด CKKS ์ํธ ์์คํ
์ ์ฑ๋ฅ์ ํฅ์ํจ์ ํ์ธํ๋ค. ๋ฐ๋ผ์, ๋ณด์์ฑ๊ณผ ์ฑ๋ฅ์ ํํ ์์ด ๊ทผ์ฌ ๋ํ ์ํธ๋ฅผ ์ฌ์ฉํ๋ ์๋น์ค์ ์ง์ ํฅ์ํ ์ ์๋ค.
์์ ์ปดํจํฐ๋ฅผ ํ์ฉํ์ฌ ์ ํต์ ์ธ ๊ณต๊ฐํค ์ํธ๋ฅผ ๊ณต๊ฒฉํ๋ ํจ์จ์ ์ธ ์๊ณ ๋ฆฌ์ฆ์ด ๊ณต๊ฐ๋๋ฉด์, ํฌ์คํธ ์์ ์ํธ์ ๋ํ ํ์์ฑ์ด ์ฆ๋ํ๋ค. ๋ถํธ ๊ธฐ๋ฐ ์ํธ๋ ํฌ์คํธ ์์ ์ํธ๋ก์จ ๋๋ฆฌ ์ฐ๊ตฌ๋์๋ค. ์์ ํค ํฌ๊ธฐ๋ฅผ ๊ฐ๋ ์๋ก์ด ๋ถํธ ๊ธฐ๋ฐ ์ ์ ์๋ช
์์คํ
๊ณผ ๋ถํธ ๊ธฐ๋ฐ ์ํธ๋ฅผ ๊ณต๊ฒฉํ๋ ๋ฐฉ๋ฒ์ด ๋
ผ๋ฌธ์ ์ ์๋์ด ์๋ค. pqsigRM์ด๋ผ ๋ช
๋ช
ํ ์ ์ ์๋ช
์์คํ
์ด ๊ทธ๊ฒ์ด๋ค.
์ด ์ ์ ์๋ช
์์คํ
์ ์์ ๋ Reed-Muller (RM) ๋ถํธ๋ฅผ ํ์ฉํ๋ฉฐ, ์๋ช
์ ๋ณต์ก๋์ ํค ํฌ๊ธฐ๋ฅผ ์ข
๋ ๊ธฐ์ ๋ณด๋ค ๋ง์ด ์ค์ธ๋ค. pqsigRM์ hull์ ์ฐจ์์ด ํฐ (U, U+V) ๋ถํธ์ ์ด์ ๋ณตํธํ๋ฅผ ์ด์ฉํ์ฌ, ์๋ช
์์ ํฐ ์ด๋์ด ์๋ค. ์ด ๋ณตํธํ ์๊ณ ๋ฆฌ์ฆ์ ์ฃผ์ด์ง ๋ชจ๋ ์ฝ์
(coset)์ ์์์ ๋ํ์ฌ ์์ ํค๋ฐ ๋ฌด๊ฒ๋ฅผ ๊ฐ๋ ์์๋ฅผ ๋ฐํํ๋ค. ๋ํ, ์์ ๋ RM ๋ถํธ๋ฅผ ์ด์ฉํ์ฌ, ์๋ ค์ง ๋ชจ๋ ๊ณต๊ฒฉ์ ์ ํญํ๋ค. 128๋นํธ ์์ ์ฑ์ ๋ํด์ ์๋ช
์ ํฌ๊ธฐ๋ 4096 ๋นํธ์ด๊ณ , ๊ณต๊ฐ ํค์ ํฌ๊ธฐ๋ 1MB๋ณด๋ค ์๋ค. ์ต๊ทผ, Ivanov, Kabatiansky, Krouk, ๊ทธ๋ฆฌ๊ณ Rumenko (IKKR)๊ฐ McEliece ์ํธ ์์คํ
์ ์ธ ๊ฐ์ง ๋ณํ์ ๋ฐํํ๋ค (CBCrypto 2020, Eurocrypt 2020์ ํจ๊ป ์งํ). ๋ณธ ๋
ผ๋ฌธ์์๋ IKKR ์ํธ ์์คํ
์ค ํ๋๊ฐ McEliece ์ํธ ์์คํ
๊ณผ ๋์น์์ ์ฆ๋ช
ํ๋ค. ๋ํ ๋๋จธ์ง IKKR ์ํธ ์์คํ
์ ๋ํ ๋คํญ ์๊ฐ ๊ณต๊ฒฉ์ ์ ์ํ๋ค. ์ ์ํ๋ ๊ณต๊ฒฉ์ IKKR ์ํธ ์์คํ
์ ์ ํ์ฑ์ ํ์ฉํ๋ค. ๋ํ, ์ด ๋
ผ๋ฌธ์ ์ ์ํ ๊ณต๊ฒฉ์ ๊ตฌํ์ ํฌํจํ๋ฉฐ, ์ ์๋ ๊ณต๊ฒฉ์ 0.2์ด ์ด๋ด์ ๋ฉ์์ง๋ฅผ ๋ณต์ํ๊ณ , ์ด๋ ์ ์์ ์ธ ๋ณตํธํ๋ณด๋ค ๋น ๋ฅธ ์๋์ด๋ค.Contents
Abstract i
Contents iv
List of Tables ix
List of Figures xi
1 Introduction 1
1.1 Homomorphic Encryption and Privacy-Preserving Machine Learning 4
1.2 High-Precision CKKS Scheme and Its Bootstrapping 5
1.2.1 Near-Optimal Bootstrapping of the CKKS Scheme Using Least Squares Method 6
1.2.2 Variance-Minimizing and Optimal Bootstrapping of the CKKS Scheme 8
1.3 Efficient Code-Based Signature Scheme and Cryptanalysis of the Ivanov-Kabatiansky-Krouk-Rumenko Cryptosystems 10
1.3.1 Modified pqsigRM: An Efficient Code-Based Signature Scheme 11
1.3.2 Ivanov-Kabatiansky-Krouk-Rumenko Cryptosystems and Its Equality 13
1.4 Organization of the Dissertation 14
2 Preliminaries 15
2.1 Basic Notation 15
2.2 Privacy-Preserving Machine Learning and Security Terms 16
2.2.1 Privacy-Preserving Machine Learning and Security Terms 16
2.2.2 Privacy-Preserving Machine Learning 17
2.3 The CKKS Scheme and Its Bootstrapping 18
2.3.1 The CKKS Scheme 18
2.3.2 CKKS Scheme in RNS 22
2.3.3 Bootstrapping of the CKKS Scheme 24
2.3.4 Statistical Characteristics of Modulus Reduction and Failure Probability of Bootstrapping of the CKKS Scheme 26
2.4 Approximate Polynomial and Signal-to-Noise Perspective for Approximate Homomorphic Encryption 27
2.4.1 Chebyshev Polynomials 27
2.4.2 Signal-to-Noise Perspective of the CKKS Scheme 28
2.5 Preliminary for Code-Based Cryptography 29
2.5.1 The McEliece Cryptosystem 29
2.5.2 CFS Signature Scheme 30
2.5.3 ReedMuller Codes and Recursive Decoding 31
2.5.4 IKKR Cryptosystems 33
3 Privacy-Preserving Machine Learning via FHEWithout Bootstrapping 37
3.1 Introduction 37
3.2 Information Theoretic Secrecy and HE for Privacy-Preserving Machine Learning 38
3.2.1 The Failure Probability of Ordinary CKKS Bootstrapping 39
3.3 Comparison With Existing Methods 43
3.3.1 Comparison With the Hybrid Method 43
3.3.2 Comparison With FHE Method 44
3.4 Comparison for Evaluating Neural Network 45
4 High-Precision Approximate Homomorphic Encryption and Its Bootstrapping by Error Variance Minimization and Convex Optimization 50
4.1 Introduction 50
4.2 Optimization of Error Variance in the Encrypted Data 51
4.2.1 Tagged Information for Ciphertext 52
4.2.2 WorstCase Assumption 53
4.2.3 Error in Homomorphic Operations of the CKKS Scheme 54
4.2.4 Reordering Homomorphic Operations 59
4.3 Near-Optimal Polynomial for Modulus Reduction 66
4.3.1 Approximate Polynomial Using L2-Norm optimization 66
4.3.2 Efficient Homomorphic Evaluation of the Approximate Polynomial 70
4.4 Optimal Approximate Polynomial and Bootstrapping of the CKKS Scheme 73
4.4.1 Polynomial Basis Error and Polynomial Evaluation in the CKKS Scheme 73
4.4.2 Variance-Minimizing Polynomial Approximation 74
4.4.3 Optimal Approximate Polynomial for Bootstrapping and Magnitude of Its Coefficients 75
4.4.4 Reducing Complexity and Error Using Odd Function 79
4.4.5 Generalization of Weight Constants and Numerical Method 80
4.5 Comparison and Implementation 84
4.6 Reduction of Level Loss in Bootstrapping 89
4.7 Implementation of the Proposed Method and Performance Comparison 92
4.7.1 Error Variance Minimization 92
4.7.2 Weight Constant and Minimum Error Variance 93
4.7.3 Comparison of the Proposed MethodWith the Previous Methods 96
5 Efficient Code-Based Signature Scheme and Cryptanalysis of Code-Based Cryptosystems 104
5.1 Introduction 104
5.2 Modified ReedMuller Codes and Proposed Signature Scheme 105
5.2.1 Partial Permutation of Generator Matrix and Modified ReedMuller Codes 105
5.2.2 Decoding of Modified ReedMuller Codes 108
5.2.3 Proposed Signature Scheme 110
5.3 Security Analysis of Modified pqsigRM 111
5.3.1 Decoding One Out of Many 112
5.3.2 Security Against Key Substitution Attacks 114
5.3.3 EUFCMA Security 114
5.4 Indistinguishability of the Public Code and Signature 120
5.4.1 Modifications of Public Code 121
5.4.2 Public Code Indistinguishability 124
5.4.3 Signature Leaks 126
5.5 Parameter Selection 126
5.5.1 Parameter Sets 126
5.5.2 Statistical Analysis for Determining Number of Partial Permutations 128
5.6 Equivalence of the Prototype IKKR and the McEliece Cryptosystems 131
5.7 Cryptanalysis of the IKKR Cryptosystems 133
5.7.1 Linearity of Two Variants of IKKR Cryptosystems 133
5.7.2 The Attack Algorithm 134
5.7.3 Implementation 135
6 Conclusion 139
6.1 Privacy-Preserving Machine Learning Without Bootstrapping 139
6.2 Variance-Minimization in the CKKS Scheme 140
6.3 L2-Norm Minimization for the Bootstrapping of the CKKS Scheme 141
6.4 Modified pqsigRM: RM Code-Based Signature Scheme 142
6.5 Cryptanalysis of the IKKR Cryptosystem 143
Abstract (In Korean) 155
Acknowlegement 158Docto
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