Leakage-resilient Cryptography Research Articles

Protecting Distributed Primitives Against Leakage: Equivocal Secret Sharing and more

Leakage-resilient cryptography aims to protect cryptographic primitives from so-called “side channel attacks” that exploit their physical implementation to learn their input or secret state. Starting from the works of Ishai, Sahai and Wagner (CRYPTO‘03) and Micali and Reyzin (TCC‘04), most works on leakage-resilient cryptography either focus on protecting general computations, such as circuits or multiparty computation protocols, or on specific non-interactive primitives such as storage, encryption, and signatures. This work focuses on leakage resilience for the middle ground, namely for distributed and interactive cryptographic primitives. Our main technical contribution is designing the first secret sharing scheme that is equivocal, resists adaptive probing of a constant fraction of bits from each share, while incurs only a constant blowup in share size. Equivocation is a strong leakage-resilience guarantee, recently introduced by Hazay et al. (ITC, 2021). Our construction is obtained via a general compiler which we introduce, that transforms any secret sharing scheme into an equivocal scheme against adaptive leakage. An attractive feature of our compiler is that it respects additive reconstruction; namely, if the original scheme has additive reconstruction, then the transformed scheme has linear reconstruction. We extend our compiler to a general paradigm for protecting distributed primitives against leakage and show its applicability to various primitives, including secret sharing, verifiable secret sharing, function secret sharing, distributed encryption and signatures, and distributed zero-knowledge proofs. For each of these primitives, our paradigm transforms any construction of the primitive into a scheme that resists adaptive party corruptions, as well as adaptive probing leakage of a constant fraction of bits in each share when the share is stored in memory (but not when it is used in computations). Moreover, the transformation incurs only a constant blowup in the share size and respects additive reconstruction—an important feature for several of these primitives, such as function secret sharing and distributed encryption.

Leakage-Resilient Hybrid Signcryption in Heterogeneous Public-key Systems

Signcryption integrates both signature and encryption schemes into single scheme to ensure both content unforgeability (authentication) and message confidentiality while reducing computational complexity. Typically, both signers (senders) and decrypters (receivers) in a signcryption scheme belong to the same public-key systems. When signers and decrypters in a signcryption scheme belong to heterogeneous public-key systems, this scheme is called a hybrid signcryption scheme which provides more elastic usage than typical signcryption schemes. In recent years, a new kind of attack, named side-channel attack, allows adversaries to learn a portion of the secret keys used in cryptographic algorithms. To resist such an attack, leakage-resilient cryptography has been widely discussed and studied while a large number of leakage-resilient schemes have been proposed. Also, numerous hybrid signcryption schemes under heterogeneous public-key systems were proposed, but none of them possesses leakage-resilient property. In this paper, we propose the first hybrid signcryption scheme with leakage resilience, called leakage-resilient hybrid signcryption scheme, in heterogeneous public-key systems (LR-HSC-HPKS). Security proofs are demonstrated to show that the proposed scheme provides both authentication and confidentiality against two types of adversaries in heterogeneous public-key systems.

On Provable White-Box Security in the Strong Incompressibility Model

Incompressibility is a popular security notion for white-box cryptography and captures that a large encryption program cannot be compressed without losing functionality. Fouque, Karpman, Kirchner and Minaud (FKKM) defined strong incompressibility, where a compressed program should not even help to distinguish encryptions of two messages of equal length. Equivalently, the notion can be phrased as indistinguishability under chosen-plaintext attacks and key-leakage (LK-IND-CPA), where the leakage rate is high.In this paper, we show that LK-IND-CPA security with superlogarithmic-length leakage, and thus strong incompressibility, cannot be proven under standard (i.e. single-stage) assumptions, if the encryption scheme is key-fixing, i.e. a polynomial number of message-ciphertext pairs uniquely determine the key with high probability. Our impossibility result refutes a claim by FKKM that their big-key generation mechanism achieves strong incompressibility when combined with any PRG or any conventional encryption scheme, since the claim is not true for encryption schemes which are key-fixing (or for PRGs which are injective). In particular, we prove that the cipher block chaining (CBC) block cipher mode is key-fixing when modelling the cipher as a truly random permutation for each key. Subsequent to and inspired by our work, FKKM prove that their original big-key generation mechanism can be combined with a random oracle into an LK-IND-CPA-secure encryption scheme, circumventing the impossibility result by the use of an idealised model.Along the way, our work also helps clarifying the relations between incompressible white-box cryptography, big-key symmetric encryption, and general leakage resilient cryptography, and their limitations.

Leakage-Resilient Anonymous Multireceiver Certificateless Encryption Resistant to Side-Channel Attacks

Anonymous multireceiver encryption (AMRE) allows a data provider to transmit a ciphertext to a designated group of receivers. Any receiver of the designated group can get the plaintext by decrypting this ciphertext while the receiver's identity is concealed to other receivers. In the past, several anonymous multireceiver certificateless encryption (AMR-CLE) schemes based on the certificateless public-key cryptography (CL-PKC) were proposed to remove both the certificate management problem of AMRE schemes based on the conventional public-key cryptography and the key escrow problem of AMR-IBE schemes based on the ID-based public-key cryptography. Recently, side-channel attacks endanger the existing cryptographic schemes, AMRE, AMR-IBE, AMR-CLE, etc. To resist to such attacks, leakage-resilient cryptography is a novel way. However, until now, there exists no multireceiver encryption or AMRE scheme resistant to side-channel attacks. In this article, we present the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$first$</tex-math></inline-formula> leakage-resilient anonymous multireceiver certificateless encryption (LR-AMR-CLE) scheme. In our scheme, attackers are granted to repeatedly extract fractional content of each secret key participated in our scheme and retains the original security of AMR-CLE schemes.

Fully Continuous Leakage-Resilient Certificate-Based Signcryption Scheme for Mobile Communications

Due to the popularity of mobile communication, many computing devices are exposed to remote environments without physical protection so that these devices easily suffer from leakage attacks (e.g., side-channel attacks). Under such leakage attacks, when a computing device performs some cryptographic algorithm, an adversary may acquire partial bits of secret keys participated in this cryptographic algorithm. To resist leakage attacks, researchers offer leakage-resilient cryptography as a solution. A signcryption scheme combines signing and encrypting processes to simultaneously provide both authentication and confidentiality, which is an important cryptographic primitive. Indeed, many leakage-resilient signcryption schemes under various public key system (PKS) settings were proposed. Unfortunately, these schemes still have two shortcomings, namely, bounded leakage resilience and conditionally continuous leakage resilience. In this paper, a “fully” continuous leakage-resilient certificate-based signcryption (FCLR-CBSC) scheme is proposed. Security analysis is formally proved to show that our scheme possesses both authentication and confidentiality against two types of adversaries in the certificate-based PKS setting. Performance analysis and simulation experience show that our scheme is suited to run on both a PC and a mobile device.

Exploring Integrity of AEADs with Faults: Definitions and Constructions

Implementation-based attacks are major concerns for modern cryptography. For symmetric-key cryptography, a significant amount of exploration has taken place in this regard for primitives such as block ciphers. Concerning symmetric-key operating modes, such as Authenticated Encryption with Associated Data (AEAD), the stateof-the-art mainly addresses the passive Side-Channel Attacks (SCA) in the form of leakage resilient cryptography. So far, only a handful of work address Fault Attacks (FA) in the context of AEADs concerning the fundamental properties – integrity and confidentiality. In this paper, we address this gap by exploring mode-level issues arising due to FAs. We emphasize that FAs can be fatal even in cases where the adversary does not aim to extract the long-term secret, but rather tries to violate the basic security requirements (integrity and confidentiality). Notably, we show novel integrity attack examples on state-of-the-art AEAD constructions and even on a prior fault-resilient AEAD construction called SIV$. On the constructive side, we first present new security notions of fault-resilience, for PRF (frPRF), MAC (frMAC) and AEAD (frAE), the latter can be seen as an improved version of the notion introduced by Fischlin and Gunther at CT-RSA’20. Then, we propose new constructions to turn a frPRF into a fault-resilient MAC frMAC (hash-then-frPRF) and into a fault-resilient AEAD frAE (MAC-then-Encrypt-then-MAC or MEM).

The Mother of All Leakages: How to Simulate Noisy Leakages via Bounded Leakage (Almost) for Free

We show that the most common flavors of noisy leakage can be simulated in the information-theoretic setting using a single query of bounded leakage, up to a small statistical simulation error and a slight loss in the leakage parameter. The latter holds true in particular for one of the most used noisy-leakage models, where the noisiness is measured using the conditional average min-entropy (Naor and Segev, CRYPTO’09 and SICOMP’12). Our reductions between noisy and bounded leakage are achieved in two steps. First, we put forward a new leakage model (dubbed the dense leakage model) and prove that dense leakage can be simulated in the information-theoretic setting using a single query of bounded leakage, up to small statistical distance. Second, we show that the most common noisy-leakage models fall within the class of dense leakage, with good parameters. Third, we prove lower bounds on the amount of bounded leakage required for simulation with sub-constant error, showing that our reductions are nearly optimal. In particular, our results imply that useful general simulation of noisy leakage based on statistical distance and mutual information is impossible. We also provide a complete picture of the relationships between different noisy-leakage models. Our result finds applications to leakage-resilient cryptography, where we are often able to lift security in the presence of bounded leakage to security in the presence of noisy leakage, both in the information-theoretic and in the computational setting. Remarkably, this lifting procedure makes only black-box use of the underlying schemes. Additionally, we show how to use lower bounds in communication complexity to prove that bounded-collusion protocols (Kumar, Meka, and Sahai, FOCS’19) for certain functions do not only require long transcripts, but also necessarily need to reveal enough information about the inputs.

Indistinguishable Leakage-Resilient Circuit Compiler

Abstract Transforming any circuit into a circuit with the same functionality but resilience against well-defined classes of leakage is an essential subject of cryptography. Whereas, according to the relationship between $VBB$ and simulated leakage-resilient circuit compiler, the compiling needs a specific mechanism to guarantee that its input is free of any leakage. Unfortunately, it makes it tough to implement this strategy in the context of actual leaks. In this work, we formally propose the indistinguishable bounded leakage-resilient circuit compiler $\bar{C}$ to solve the problem mentioned above. Specifically, it specifies that any polynomial-time adversary who derives the bound secret information from the circuit compiler can not distinguish an equivalent compiled circuit $\bar{C}^{\prime}$ with the same output. To achieve this goal, first, we utilize the leakage-resilient indistinguishable encoding as its input. Second, it takes advantage of the indistinguishability obfuscation $(IO)$ program to construct an obfuscated circuit that removes the random coins of the input encoding . Its total running time is the original circuit $C$ running time plus $poly(\kappa )$, and the size is independent of the original $C$. As this new compiler overcomes the restriction of the simulated-based leakage resilient circuit compiler,it will bring an independent interest result in leakage-resilient cryptography.

Review on Leakage Resilient Key Exchange Security Model

In leakage resilient cryptography, leakage resilient key exchange protocols are constructed to defend against leakage attacks. Then, the key exchange protocol is proved with leakage resilient security model to determine whether its security proof can provide the security properties it claimed or to find out any unexamined flaw during protocol building. It is an interesting work to review the meaningful security properties provided by these security models. This work review how a leakage resilient security model for a key exchange protocol has been evolved over years according to the increasing security requirement which covers a different range of attacks. The relationship on how an adversary capability in the leakage resilient security model can be related to real-world attack scenarios is studied. The analysis work for each leakage resilient security model here enables a better knowledge on how an adversary query addresses different leakage attacks setting, thereby understand the motive of design for a cryptographic primitive in the security model.

Leakage-Resilient Revocable Certificateless Encryption with an Outsourced Revocation Authority

To resolve both certificate management and key escrow problems, a certificateless public-key system (CLPKS) has been proposed. However, a CLPKS setting must provide a revocation mechanism to revoke compromised users. Thus, a revocable certificateless public-key system (RCLPKS) was presented to address the revocation issue and, in such a system, the key generation centre (KGC) is responsible to run this revocation functionality. Furthermore, a RCLPKS setting with an outsourced revocation authority (ORA), named RCLPKS-ORA setting, was proposed to employ the ORA to alleviate the KGC’s computational burden. Very recently it was noticed that adversaries may adopt side-channel attacks to threaten these existing conventional public-key systems (including CLPKS, RCLPKS and RCLPKS-ORA). Fortunately, leakage-resilient cryptography offers a solution to resist such attacks. In this article, the first leakage-resilient revocable certificateless encryption scheme with an ORA, termed LR-RCLE-ORA scheme, is proposed. The proposed scheme is formally shown to be semantically secure against three types of adversaries in the RCLPKS and RCLPKS-ORA settings while resisting side-channel attacks. In the proposed scheme, adversaries are allowed to continually extract partial ingredients of secret keys participated in various computational algorithms of the proposed scheme while retaining its security.

Leakage-Resilient Outsourced Revocable Certificateless Signature with a Cloud Revocation Server

Certificateless public-key system (CL-PKS) is a significant public-key cryptography and it solves both the key escrow and certificate management problems. Outsourced revocable certificateless public-key system (ORCL-PKS) with a cloud revocation server (CRS) not only provides a revocation mechanism, but also further outsources the revocation functionality to the CRS to reduce the computational burden of the key generation center (KGC). Recently, side-channel attacks have threatened some existing conventional cryptography (including CL-PKS). Indeed, adversaries can apply side-channel attacks to derive fractional constituents of private (or secret) keys to damage the security of these cryptographic protocols (or schemes). To withstand such attacks, leakage-resilient cryptography is an attractive approach. However, little research concerns with leakage-resilient certificateless cryptography. In this paper, the first leakage-resilient outsourced revocable certificateless signature (LR-ORCLS) scheme is presented. The proposed scheme allows adversaries to continually derive fractional constituents of private (or secret) keys and possesses overall unbounded leakage property. In the generic bilinear group (GBG) model, our scheme is shown to be existential unforgeable against adversaries. Finally, the comparisons between the proposed scheme and the previous revocable certificateless signature schemes are provided to demonstrate the merits of the proposed scheme.

New approach to practical leakage-resilient public-key cryptography

AbstractWe present a new approach to construct several leakage-resilient cryptographic primitives, including leakage-resilient public-key encryption (PKE) schemes, authenticated key exchange (AKE) protocols and low-latency key exchange (LLKE) protocols. To this end, we introduce a new primitive called leakage-resilient non-interactive key exchange (LR-NIKE) protocol. We introduce an appropriate security model for LR-NIKE protocols in the bounded memory leakage (BML) settings. We then show a secure construction of the LR-NIKE protocol in the BML setting that achieves an optimal leakage rate, i.e., 1 – o(1). Our construction of LR-NIKE requires a minimal use of a leak-free hardware component. We argue that the use of such a leak-free hardware component seems to be unavoidable in any construction of an LR-NIKE protocol, even in the BML setting. Finally, we show how to construct the aforementioned leakage-resilient primitives from such an LR-NIKE protocol as summarized below. All these primitives also achieve the same (optimal) leakage rate as the underlying LR-NIKE protocol. We show how to construct a leakage-resilient (LR) IND-CCA-2-secure PKE scheme in the BML model generically from a bounded LR-NIKE (BLR-NIKE) protocol. Our construction of LR-IND-CCA-2 secure PKE differs significantly from the state-of-the-art constructions of these primitives, which mainly use hash proof techniques to achieve leakage resilience. Moreover, our transformation preserves the leakage-rate of the underlying BLR-NIKE protocol. We introduce a new leakage model for AKE protocols, in the BML setting, and present a leakage-resilient AKE protocol construction from the LR-NIKE protocol. We introduce the first-ever leakage model for LLKE protocols in the BML setting and the first construction of such a leakage-resilient LLKE from the LR-NIKE protocol.

Practical continuous leakage-resilient CCA secure identity-based encryption

Leakage of private information including private keys of user has become a threat to the security of computing systems. It has become a common security requirement that a cryptographic scheme should withstand various leakage attacks. In the real life, an adversary can break the security of cryptography primitive by performing continuous leakage attacks. Although, some research on the leakage-resilient cryptography had been made, there are still some remaining issued in previous attempts. The identity-based encryption (IBE) constructions were designed in the bounded-leakage model, and might not be able to meet their claimed security under the continuous-leakage attacks. In the real applications, the leakage is unbounded. That is, a practical cryptography scheme should keep its original security in the continuous leakage setting. The previous continuous leakage-resilient IBE schemes either only achieve chosen-plaintext attacks security or the chosen-ciphertext attacks (CCA) security is proved in the selective identity model. Aiming to solve these problems, in this paper, we show how to construct the continuous leakage-resilient IBE scheme, and the scheme’s adaptive CCA security is proved in the standard model based on the hardness of decisional bilinear Diffie-Hellman exponent assumption. For any adversary, all elements in the ciphertext are random, and an adversary cannot obtain any leakage on the private key of user from the corresponding given ciphertext. Moreover, the leakage parameter of our proposal is independent of the plaintext space and has a constant size.

A Leakage-Resilient Certificateless Authenticated Key Exchange Protocol Withstanding Side-Channel Attacks

Certificateless public-key cryptography has conquered both the certificate management problem in the traditional public-key cryptography and the key escrow problem in the ID-based public-key cryptography. Certificateless authenticated key exchange (CLAKE) protocol is an important primitive of the certificateless public-key cryptography. A CLAKE protocol is employed to provide both mutual authentication and establishing a session key between two participators. Indeed, all conventional public-key cryptographies have encountered a new kind of attack, named “side-channel attacks”. Fortunately, leakage-resilient cryptography is a flexible approach to withstand such attacks. However, the design of leakage-resilient CLAKE (LR-CLAKE) protocols is not studied. In the article, by extending the well-known extended-Canetti–Krawczyk (eCK) model, we present the security notions (adversary model) of LR-CLAKE protocols, called continual-leakage-resilient eCK (CLReCK) model. The first LR-CLAKE protocol withstanding side-channel attacks is proposed. By employing the proof technique of the generic bilinear group (GBG) model, we formally prove the security of our protocol in the CLReCK model.

Leakage-Resilient Certificate-based Key Encapsulation Scheme Resistant to Continual Leakage

In the past, the security of most public-key encryption or key encapsulation schemes is shown in an ideal model, where private keys, secret keys and random values are assumed to be absolutely secure to adversaries. However, this ideal model is not practical due to side-channel attacks in the sense that adversaries could gain partial information of these secret values involved in decryption operations by perceiving energy consumption or execution timing. In such a case, these schemes under the ideal model could suffer from side-channel attacks. Recently, leakage-resilient cryptography resistant to side-channel attacks is an emerging research topic. Certificate-based encryption (CBE) or certificate-based key encapsulation (CB-KE) schemes are a class of important public-key encryption. However, little work addresses the design of leakage-resilient CBE (LR-CBE) or leakage-resilient CB-KE (LR-CB-KE) schemes. In this paper, we present the first LR-CB-KE scheme with overall unbounded leakage property which permits adversaries to continuously gain partial information of the system secret key of a trusted certificate authority (CA), the private keys and certificates of users, and random values. In the generic bilinear group model, formal security analysis is made to prove that the proposed LR-CB-KE scheme is secure against chosen ciphertext attacks.

CCA Secure Public Key Encryption against After-the-Fact Leakage without NIZK Proofs

In leakage resilient cryptography, there is a seemingly inherent restraint on the ability of the adversary that it cannot get access to the leakage oracle after the challenge. Recently, a series of works made a breakthrough to consider a postchallenge leakage. They presented achievable public key encryption (PKE) schemes which are semantically secure against after-the-fact leakage in the split-state model. This model puts a more acceptable constraint on adversary’s ability that the adversary cannot query the leakage of secret states as a whole but the functions of several parts separately instead of prechallenge query only. To obtain security against chosen ciphertext attack (CCA) for PKE schemes against after-the-fact leakage attack (AFL), existing works followed the paradigm of “double encryption” which needs noninteractive zero knowledge (NIZK) proofs in the encryption algorithm. We present an alternative way to achieve AFL-CCA security via lossy trapdoor functions (LTFs) without NIZK proofs. First, we formalize the definition of LTFs secure against AFL (AFLR-LTFs) and all-but-one variants (ABO). Then, we show how to realize this primitive in the split-state model. This primitive can be used to construct AFLR-CCA secure PKE scheme in the same way as the method of “CCA from LTFs” in traditional sense.

Novel updatable identity-based hash proof system and its applications

Alwen et al. in Eurocrypt 2010, showed that an identity-based hash proof system (IB-HPS), where IB-HPS generalizes the notion of hash proof system (HPS) to the identity-based setting, almost immediately yields an identity-based encryption (IBE) scheme which is secure against partial leakage of the target identity's decryption key. That is, an IBE scheme with bounded leakage resilience can be naturally created from an IB-HPS. However, in the real world, the leakage is unbounded, and any adversary can break the security of cryptography shceme by performing continuous leakage attacks. To further increase the practicability, a cryptography scheme must hold the claimed security in the continuous leakage setting. Dodis et al. in FOCS 2010, showed a generic method how to create a cryptography shceme with continuous leakage resilience from the bounded leakage-resilient cryptosystem by performing an additional key update algorithm while the public parameters keep unchanged. To construct a continuous leakage-resilient cryptography scheme, a new primitive, called it updatable identity-based hash proof system (U-IB-HPS), is proposed, which is an improved IB-HPS. In particular, the improved system has an additional key update algorithm, which can push some new randomness into the private key of user (or the master secret key), the updated results are random in the adversary's view, and the leakage of previous private key of user (or the master secret key) does not work for the updated results. However, the previous instantiations of U-IB-HPS cannot achieve the claimed security. To solve the above problems, in this paper, two instantiations of U-IB-HPS with better performance are created, and the security of proposed system is proved, in the standard model, based on the classic decisional bilinear Diffie-Hellman assumption. The corresponding IBE scheme created with our U-IB-HPS allows continuous leakage of multiple keys, i.e., continuous leakage of the master secret key and the private key of user. Additionally, our U-IB-HPS can also be employed as an underlying basic tool to build the generic construction of continuous leakage-amplified public-key encryption scheme, continuous leakage-resilient identity-based authenticated key exchange protocol, and continuous leakage-resilient public-key encryption scheme with keyword search, etc.

Review on Leakage Resilient Key Exchange Security Model

In leakage resilient cryptography, leakage resilient key exchange protocols are constructed to defend against leakage attacks. Then, the key exchange protocol is proved with leakage resilient security model to determine whether its security proof can provide the security properties it claimed or to find out any unexamined flaw during protocol building. It is an interesting work to review the meaningful security properties provided by these security models. This work review how a leakage resilient security model for a key exchange protocol has been evolved over years according to the increasing security requirement which covers a different range of attacks. The relationship on how an adversary capability in the leakage resilient security model can be related to real-world attack scenarios is studied. The analysis work for each leakage resilient security model here enables a better knowledge on how an adversary query addresses different leakage attacks setting, thereby understand the motive of design for a cryptographic primitive in the security model.

Hierarchical attribute based encryption with continuous leakage-resilience

Attribute based encryption (ABE) is widely applied in cloud computing settings due to its fine-grained access control. Most ABE schemes do not consider the side channel attacks which may leak the secret information of cryptosystems. Leakage-resilient cryptography aims to model security for various side channel attacks. In this paper, we first give the formal definition and security model of hierarchical attribute based encryption (HABE) with continuous leakage-resilience. Furthermore, we present a ciphertext-policy HABE scheme with continuous leakage-resilience. The proposed scheme is resilient to master key leakage and secret key leakage. We prove the security of our scheme under composite order bilinear group assumptions by using dual system encryption techniques. The performance of leakage-resilience is analyzed theoretically. If the depth of our proposed scheme is 1, the relative leakage ratio is almost up to 1/3. In addition, we give the performance comparison through experiments.

Leakage-Resilient Certificate-Based Signature Resistant to Side-Channel Attacks

Certificate-based cryptography is an attractive public-key setting, and it not only simplifies certificate management in the traditional public-key cryptography but also eliminates the key escrow problem inherent in the identity-based cryptography. Recently, leakage-resilient cryptography resistant to side-channel attacks has received significant attention from cryptographic researchers. By side-channel attacks, adversaries could obtain partial information of secret and private keys involved in cryptographic algorithms by perceiving execution time or energy consumptions of each algorithm invocation. The certificate-based signature (CBS) is a class of important public-key signature. Up to date, there exists no leakage-resilient CBS (LR-CBS) scheme resistant to side-channel attacks. In this paper, the first LR-CBS scheme is proposed and it possesses overall unbounded leakage property, namely, it permits adversaries to continuously obtain partial information of secret or private keys involved in the associated algorithm invocations. The security analysis is given to prove that the proposed LR-CBS scheme is existential unforgeability against adaptive chosen-message attacks for adversaries in the generic bilinear group model.