155 results about "Quantum signature" patented technology

The invention provides a post quantum cryptography and national cryptography algorithm mixed signature and verification method and device, and relates to the technical field of cryptography application. The method comprises the following steps: performing hash operation on a to-be-signed message plaintext by adopting a preset national cryptographic hash algorithm to obtain a first hash value, and performing mixed signature operation on the first hash value and the to-be-signed message plaintext according to a national cryptographic signature private key component and a post-quantum signature private key component in a mixed signature private key of a sender to obtain a second hash value; and obtaining a national secret signature value and a post-quantum signature value, packaging the national secret signature value and the post-quantum signature value to obtain a mixed signature value corresponding to the to-be-signed message plaintext, and sending the mixed signature value and the to-be-signed message plaintext to a receiver. Due to the fact that the mixed signature value sent by the sending party to the receiving party comprises the national secret signature value and the post-quantum signature value, an attacker needs to crack the national secret signature value and the post-quantum signature value at the same time to achieve identity anti-fake attack, and guarantee is provided for integrity and authenticity of message plaintext transmission.

The invention relates to the technical field of platform management, and discloses an internet-based financial service platform management method, which comprises the following steps: collecting static identity data, dynamic behavior data, biological characteristics and environmental parameters of a user in real time, and generating a multi-dimensional dynamic identity chain after desensitization aggregation; the attack-defense engine simulates an attack strategy by using a diffusion model, analyzes causal contradictions and topology aggregation coefficients in combination with a space-time diagram convolutional network, marks high-risk nodes, and iteratively optimizes model parameters everyday through reinforcement learning; quantum signature and alliance chain verification are adopted in the transaction link, transactions involving high-risk countries or transactions with the fund abnormal transfer probability exceeding 95% are automatically frozen, and compliance channels are switched; the biological monitoring module analyzes the physiological signal in real time and triggers a multi-mode challenge; and aggregating edge node data through a federated learning framework, and automatically rolling back to a historical optimal version when the false alarm rate of the updated global risk control model exceeds the standard. According to the invention, efficient management of the financial service platform can be realized.

The invention discloses a quantum cryptography migration resisting method based on an FIDO2 protocol, electronic equipment and a medium. The method comprises the steps that interaction between an authenticator and a client side achieves key packaging based on a Kyber algorithm; a registration stage and an authentication stage are completed through three-party interaction of the authenticator, the client and the server; in the registration stage, an identity certificate is created on a server for the first time by a user, and a public key is generated and stored by using a Diithium anti-quantum signature algorithm or a Falcon anti-quantum signature algorithm; in the authentication stage, a user logs in a server, and identity authentication is carried out by using the credential created in the registration stage. According to the method, the Kyber anti-quantum encryption algorithm and the Dilliimum / Falcon anti-quantum digital signature algorithm are introduced into the FIDO2 protocol, so that the confidentiality and integrity of communication and the reliability of authentication can be guaranteed continuously in a quantum computing environment, and the anti-quantum security of the FIDO2 protocol is improved remarkably.

The invention relates to the technical field of electronic information technology anti-counterfeiting, in particular to an electronic seal verification method based on a quantum true random number and an anti-quantum multi-dimensional dynamic code, which comprises the following steps of: generating a true random number by introducing a quantum random number generator, constructing a quintuple scene space, and fusing multi-dimensional data of a user, a seal, a file, time and a geographic position; an SPHINCS + anti-quantum signature algorithm is adopted, and block chain evidence storage is combined, so that a four-layer nested dynamic security system of quantum random injection, multi-source data binding, anti-quantum encryption and block chain evidence storage is formed. Meanwhile, a whole-process credible system from application, auditing to stamping and verification is constructed, compared with a traditional electronic seal, the quantum-attack-resistant electronic seal has remarkable breakthrough in the aspects of random number safety, quantum attack resistance, data fusion degree, tampering detection sensitivity and the like, and a credible electronic seal solution in the quantum era is provided for finance and other scenes.

The invention discloses a post-quantum signature method, a post-quantum signature verification method and a related device, OTS signature data and LMS signature data can be generated by using a Chinese Hash algorithm, and the LMS signature data can resist quantum attacks and meet Chinese encryption requirements. The post-quantum signature method comprises the following steps: after selecting an OTS private key from an LMS private key, generating OTS signature data according to a Chinese Hash algorithm, the OTS private key, a first reference random number and a message, generating a Hash tree according to the LMS private key, generating an LMS public key according to a root node of the Hash tree, determining a brother node set in the Hash tree according to a target leaf node corresponding to the OTS private key, and sending the brother node set to the OTS private key; generating LMS signature data according to the serial number of the OTS private key, the OTS signature data, the LMS feature value and the brother node set; and sending the LMS public key, the LMS signature data and the message to a signature verification device.

The invention provides a multi-party collaborative anti-quantum signature method and system based on homomorphic Hash, and relates to the technical field of digital signatures, the system comprises a plurality of participants and a verification party, and the method is specifically as follows: each participant generates a pair of public and private keys, the public key is composed of a total public key, a total signature matrix and a total Hash matrix of all participants; based on the public and private keys, all participants perform multi-party signature on the target message through a homomorphic hash function, and a generated signature is composed of total hash challenge values of all participants, total response vectors of all participants and commitment value high-order difference values of all participants; according to the signature, the verifier recovers the high order of the total commitment value, further calculates the total hash challenge value, and verifies the validity of the signature through comparison of the total hash challenge value; according to the method, multi-user collaborative signature is realized, and the security of resisting malicious opponents under a quantum computer is obtained.

The invention discloses an anti-quantum security data transmission authentication method based on an SLH-DSA, and aims to cope with communication security challenges in a quantum computing environment. The device generates a key pair locally in a registration phase and applies for an anti-quantum identity certificate from a certificate agent. In the authentication stage, a user exchanges an identity certificate and a key parameter to realize identity authentication and negotiate a shared key. In the data transmission stage, a session key is dynamically derived by using a shared key, data is encrypted in combination with a national secret symmetric encryption algorithm SM4, and integrity verification is performed by using an anti-quantum signature algorithm. The method gives consideration to security, lightweight and performance, has quantum attack resistance and forward security, is suitable for application scenarios with high data security requirements, such as industrial Internet of Things and smart power grids, and has good application prospects and popularization values.

The invention discloses a remote proving method for a trusted data space, which relates to the technical field of computer and data security, and comprises the following steps: initializing a trusted environment and generating a dynamic identity key for a heterogeneous terminal containing a CPU (Central Processing Unit), a GPU (Graphic Processing Unit) and an FPGA (Field Programmable Gate Array) based on a hardware security module and a physical unclonable function, and generating a challenge value by a verifier by using a quantum random number; the method comprises the following steps of: firstly, transmitting to a proving party through quantum encryption and a neuromorphic photonic network, constructing proving by the proving party by adopting a tensor decomposition zero-knowledge proving protocol, finishing multi-stage verification on a verification party in combination with a space-time cause and effect graph and quantum signature aggregation, and finally realizing dynamic trust evaluation and adaptive strategy adjustment through a quantum Bayesian network, reinforcement learning and biological feedback. According to the method, the remote attestation performance of the trusted data space is improved, dynamic trust evaluation is realized by means of the quantum Bayesian network and biological feedback, attacks are effectively resisted, real-time requirements are met, trust is accurately evaluated, and the development of the trusted data space is promoted.

The invention discloses a digital certificate authentication method and device based on a post-quantum algorithm, equipment and a medium, and relates to the field of information security, and the method comprises the steps: obtaining digital certificate application information of a target entity for auditing; if the digital certificate request passes the verification, analyzing the digital certificate request submitted by the target entity, and obtaining a key encryption key of the digital certificate request by using a post-quantum key encapsulation algorithm after verification; then obtaining user information of the target entity and an active cryptographic algorithm public key value in the digital certificate request, generating an anti-quantum-attack digital certificate according to the user information and a public key, and signing and issuing the anti-quantum-attack digital certificate to the target entity; the digital certificate comprises a post-quantum key packaging public key, a post-quantum signature public key and an active-service public key password public key ciphertext of the entity. According to the application, the public key data of the traditional public key cryptographic algorithm can be protected, the quantum computing attack is resisted, the public key plaintext of the traditional public key cryptographic algorithm can be obtained in the service system, and the influence on the original service using the traditional public key cryptographic algorithm is extremely low.

The invention relates to the technical field of quantum computers, in particular to an anti-quantum signature method and system used between two terminals and electronic equipment. The initial to-be-verified commitment value of each terminal is obtained by homomorphic Hash commitment operation of the signature calculation result of the public key information stored in the terminal private key and encryption based on the commitment key, so that the confidentiality and randomness of the public commitment value are effectively ensured, and only two terminals need to interact and share the initial to-be-verified commitment value in the process. Compared with conventional data encryption, transmission and decryption operations, the data transmission frequency is reduced, and the signature verification effect is improved.

The invention provides an anonymous Internet of Things identity authentication method and device based on an anti-quantum computing password. A key generation device sends a public key to a message sending end and a message receiving end, and sends a private key to an authentication server; the authentication server generates a blind parameter based on the private key, and sends the blind parameter to the message sending end; the message sending end performs blind operation based on the blind parameter, the public key and the to-be-transmitted message to obtain a blind message, and sends the blind message to the authentication server; the authentication server performs post-quantum signature on the blind message based on the private key to obtain a to-be-solved blind signature, and sends the to-be-solved blind signature to the message sending end; the message sending end carries out blind resolution operation on the to-be-resolved blind signature to obtain a target signature; the message sending end sends the to-be-transmitted message and the target signature to a message receiving end; and the message receiving end performs post-quantum verification on the target signature based on the public key, and if verification succeeds, processing is performed based on the to-be-transmitted message. Through the scheme of the invention, the user data security can be improved.

The invention relates to the technical field of data processing, in particular to an artificial intelligence-assisted password security policy dynamic adaptive adjustment method, which comprises the following steps of: acquiring login operation data to generate an encryption feature gradient; the aggregation server generates a dynamically updated federal reference model through a double-channel model updating mechanism, wherein a long-term model updating channel is fused with historical parameters to maintain system stability, and a short-term behavior time window channel dynamically adjusts recent gradient weight based on a time decay function. A graph computation engine generates a spatio-temporal composite risk score. And the risk decision engine generates a third-level response instruction. A risk misjudgment event is verified based on a response result, high-value training samples are screened to drive incremental training, space-time sensitive parameters are finely adjusted by a meta-learning optimizer, and noise is injected into a quantum gradient jammer to defend attacks. Optimized parameters are synchronized to a system module, and a closed-loop adaptive system is formed by combining homomorphic encryption transmission, model digital watermarking, quantum source hardware binding and anti-quantum signature.

The invention discloses a method for applying an anti-quantum certificate to a TLS1.2 handshake process, which realizes the application of an MLDSA anti-quantum signature algorithm and an MLKEM anti-quantum key encapsulation mechanism in the handshake process by expanding a cipher suite and a signature algorithm field of a TLS1.2 protocol. The method specifically comprises the following steps: defining a cipher suite supporting MLKEM and an MLDSA signature algorithm enumeration value; the client declares algorithm support in ClientHello, and the server responds and returns a certificate chain containing the double-antibody quantum certificate; after the client verifies the certificate, the pre-master key is encapsulated by using MLKEM, and the server de-encapsulates the derived session key; and the two parties send Finished messages to each other to complete handshake. According to the scheme, on the premise that a TLS1.2 basic framework is not changed, the communication security is enhanced through double-resistance quantum algorithm integration, international and national cryptographic algorithms are supported, quantum computing attacks can be resisted, meanwhile, adaptation of the international and national cryptographic algorithms is supported, compatibility with an existing system is ensured, the security risk of a traditional cryptographic algorithm in a quantum environment is solved, and the security risk of the traditional cryptographic algorithm in the quantum environment is reduced. And a standardized solution is provided for anti-quantum upgrade of the TLS1.2 protocol.

The invention relates to the technical field of quantum signature, in particular to a two-party collaborative post-quantum decryption method, which comprises the following steps of: generating a collaborative Aigis-sig signature secret key part; a public key and respective private keys are calculated through interaction of the client and the server, and encryption is carried out through the public key to generate a ciphertext; carrying out two-party collaborative decryption calculation; plaintext recovery is completed through cooperation of the client and the server. According to the invention, a two-party cooperative decryption mode is adopted, that is, the server-side cipher machine firstly performs partial decryption, and then the client-side completes the final decryption process. The decryption process is distributed on two different calculation subjects, so that even if an attacker steals data of one of the calculation subjects, the plaintext cannot be directly recovered, and the anti-attack capability is effectively enhanced.

The invention discloses a multi-level security authentication method and system, and relates to the technical field of the Internet of Things, and the method comprises the steps: enabling a new node to verify the anti-quantum signature legality and behavior feature trust score of an identity certificate through a neighbor node according to a dynamic authentication path, and outputting a device layer authentication result; according to the equipment layer authentication result, selecting a verification node in an authentication path to carry out identity recheck and trust score weighted updating on a new node to obtain a neighbor layer authentication result and a whole network trust score; based on the neighbor layer authentication result, the core node verifies the whole network trust score and anti-quantum signature legality of the new node through the dynamic authentication atlas, and performs updating and anomaly detection on the dynamic atlas; through anti-quantum signature verification and behavior feature trust scoring, the ability of new node authentication to resist quantum computing attack and malicious node disguise is enhanced.

The invention discloses a network threat evidence fixation technology for solving source tracing evidence chain deficiency, and belongs to the field of network security. Aiming at the problems of distributed log time sequence chaos, easy tampering of virtual environment metadata, difficult association of cross-platform attack traces and the like existing in the traditional evidence obtaining technology, the invention constructs a technical system of dynamic data capture and space-time fusion modeling, evidence chain reconstruction and distributed verification, anti-quantum storage and security solidification. A self-adaptive sensor network is constructed through an improved BFGS algorithm, a space-time fusion evidence model is provided, and an improved MPT structure and a lattice-based cryptographic algorithm are designed in combination with a time sequence Petr i network and a Byzantine fault-tolerant protocol. Tests show that the attack feature capture rate reaches 91%, the cross-platform attack association accuracy rate reaches 93%, the quantum signature resistance speed is 1200 times per second, and the problem of source tracing evidence chain deficiency is effectively solved.

The invention provides a quantum security signature method which can be applied to the technical field of quantum communication. The quantum security signature method comprises the following steps: in response to a received quantum security signature request for target plaintext data, performing risk assessment on the quantum security signature request according to a preset security policy, and determining a risk level corresponding to the quantum security signature request; determining a target post-quantum signature algorithm and an algorithm parameter set corresponding to the quantum security signature request according to the risk level; according to the algorithm parameter set, performing dynamic function reconstruction on the computing resource for executing the signature operation to obtain the reconstructed computing resource; and performing quantum security signature on the target plaintext data by using the reconstructed computing resources and a target post-quantum signature algorithm to obtain a quantum security signature result. The invention further provides a quantum security signature device and equipment, a storage medium and a program product.

The invention provides a signature method, a verification method and device, electronic equipment, a computer readable storage medium and a computer program product, and the signature method comprises the steps: carrying out the signature of a transaction request based on a mixed signature mode under the condition that a client supports a quantum signature algorithm, the mixed signature mode is a mode of performing signature based on a quantum signature algorithm and then performing signature based on a traditional signature algorithm; under the condition that the client does not support the quantum signature algorithm, the transaction request is signed based on a traditional signature algorithm. Therefore, even if the signature algorithm versions between the client and the transaction verification node are inconsistent, verification can be carried out through a traditional signature algorithm, and the occurrence of transaction verification interruption is reduced.

The invention provides an optimization method, device and equipment for a quantum key distribution handshake protocol, and the method comprises the steps: preparing a standard quantum state and building a spinning peak shifting time sequence, and achieving the time division multiplexing and parallel processing of a plurality of quantum channels; constructing a handshake mode with successful pre-matched codebook storage history, and supporting rapid mode matching; the authentication sub-process and the negotiation sub-process are decoupled through a parallel verification architecture and are executed in parallel on different time phases; the quantum signature token and the key negotiation parameters are fused to generate a handshake state matrix, and global representation of the handshake process is achieved; performing success probability pre-judgment based on quantum entanglement association strength analysis, and triggering a pre-generation mechanism of a candidate key pool; performing quantum state weaving and fusion on the key by using local, non-local and mixed entanglement distribution adjustment to generate an enhanced key; and finally, optimal distribution is completed in combination with a spinning peak shifting time sequence, so that parallelization of authentication and negotiation, intellectualization of resource distribution and real-time state monitoring are realized.

The invention discloses an anti-quantum transaction processing method and device based on a block chain, a medium and equipment. The method comprises the following steps: partitioning transaction information of a target transaction based on an algorithm type of each target signature algorithm configured by a first terminal to obtain a plurality of message blocks; each target signature algorithm at least comprises a traditional signature algorithm and an anti-quantum signature algorithm; based on each target signature algorithm and the target private key of each target signature algorithm, performing signature processing on each message block by adopting a nested signature mode to obtain a signature message block corresponding to each message block; and sending each signature message block to a first server, and broadcasting each signature message block through the first server, so that each second server in the block chain performs signature verification on each signature message block based on a target public key corresponding to the target private key. The problem that an old system node which is not upgraded cannot participate in transaction verification is solved.

Data compression with quantum-resistant intrusion detection, that measures in real-time the probability distribution of an encoded data stream and analyzes entropy characteristics across multiple bit-scale windows to detect both classical and quantum-generated intrusions. The system compares the probability distribution to a reference probability distribution and uses statistical algorithms to determine divergence between distributions while simultaneously analyzing entropy cascade patterns characteristic of quantum computing sources. When divergence exceeds configured thresholds or quantum-generated characteristics are detected, the system generates intrusion alerts identifying the threat type. The system comprises encoding and decoding machines, an intrusion detection engine that performs multi-scale entropy analysis, a codebook training engine that creates quantum-resistant codebooks using entropy-stratified training algorithms, and databases including a quantum signature database storing compression patterns of known quantum algorithms. The codebook training engine adaptively retrains encoding algorithms upon detecting new quantum patterns, maintaining system effectiveness against evolving quantum threats.

The invention belongs to the technical field of information security, and discloses a hybrid collaborative signature method fusing SM2 and a post-quantum cryptography algorithm, and the method comprises the steps that a client generates a first session key based on a first post-quantum signature key pair and a first post-quantum asymmetric encryption and decryption key pair, generating a first public key ciphertext corresponding to the first public key parameter based on the first session key; the server decrypts the first public key ciphertext based on the first session key, and applies for obtaining a digital signature certificate after generating a common public key through an SM2 algorithm; performing collaborative signature according to a second session key generated by a second post-quantum signature key pair and a second post-quantum asymmetric encryption and decryption key pair, calculating by the server to obtain a first signature component intermediate value, and calculating by the client to obtain a second signature component; the client calculates a target signature value according to the second signature component, the first signature component intermediate value ciphertext and the digital signature certificate; through the method, the safety of the hybrid collaborative signature is improved.

The embodiment of the invention relates to a key generation method and device, a computer program product and electronic equipment, and relates to the technical field of network and security, and the method comprises the steps: adding a quantum security key exchange password group in an encrypted communication request sent by a client based on a quantum security transport layer security protocol, algorithm information in the quantum security key exchange password group is configured; adding an anti-quantum signature algorithm in the encrypted communication request, and selecting a key negotiation algorithm in the encrypted communication request to determine a first key exchange field corresponding to the client; selecting a key negotiation algorithm in a response message sent by the server in response to the encrypted communication request to determine a second key exchange field corresponding to the server; and performing key negotiation according to the first key exchange field and the second key exchange field to generate a session key. The security of the protocol can be improved.

The invention relates to the technical field of information security, and discloses an anti-quantum multi-key binding digital certificate implementation mode, which comprises a main body public key information field (SubjectPublic Key Info) used for storing a first anti-quantum algorithm public key, the type of the first anti-quantum algorithm public key is determined through an associated algorithm identifier, and the type of the first anti-quantum algorithm public key is stored in the main body public key information field. Comprising an anti-quantum signature public key or an anti-quantum encryption public key, and further comprises a certificate extension for storing a second anti-quantum algorithm public key. According to the anti-quantum multi-key binding digital certificate implementation mode, an anti-quantum signature public key (SubjectPublic Key Info) and an encryption public key (extension item) are separately stored in a single X.509 certificate, signature and encryption dual-function integration is achieved, the dual-certificate requirement is avoided, the extension item adopts an ASN.1 standard and a critical = FALSE mechanism, traditional RSA / ECC public key mixed storage is compatible, transitional smooth migration is supported, and the anti-quantum multi-key binding digital certificate implementation mode is suitable for application and popularization. Through key isolation, anti-quantum hash / KEM verification and unified management, 50% of certificate operation and maintenance cost is reduced, quantum security protection is enhanced, and the method is suitable for high-sensitivity scenes such as the Internet of Things and 5G.

The invention provides an anti-quantum cryptography application method and device based on an FIDO 2.0 standard and a related product, and relates to the technical field of information security. According to the method, on the basis of an FIDO2.0 standard specification, two groups of keys, namely a traditional key and an anti-quantum key, are generated by using cryptographic algorithm key derivation through the same random number seed; the two groups of keys are tightly coupled to serve as a composite key for FIDO authentication to be used for a subsequent authentication signature process, so that the anti-quantum security is enhanced; when the composite signature is used, the anti-quantum signature covers the traditional password signature, so that a strong dependency chain of two password algorithms is formed, and the security authentication strength is enhanced; when the signature is verified, the composite public key is used for verifying the traditional signature and the anti-quantum signature, authentication can be passed only when the traditional signature and the anti-quantum signature pass, double security protection of the traditional password technology and the anti-quantum password technology is achieved, and the attack risk of the traditional password in quantum computing is effectively resisted.

The invention discloses a quantum signature and identity authentication method based on a quantum public key, relates to the technical field of quantum signature and quantum identity authentication, and solves the problems that existing digital signature and identity authentication are low in security, and eavesdropping and potential quantum computing threats cannot be detected. The method is composed of a registration stage, an initialization stage, a signature stage, an identity verification stage and a signature verification stage, a public key encryption algorithm based on quantum gate transformation is designed in the method and is applied to a quantum public key identity authentication stage, on one hand, encryption processing is conducted on a public key, and on the other hand, the identity authentication process is conducted. Therefore, the security of the signature identity is ensured. Meanwhile, a quantum signature method based on unitary operation is used, so that the security and quantum attack resistance of the signature stage are ensured. Finally, the security analysis shows that the security of the method depends on the undistinguishability of an unknown quantum sequence and a quantum man-in-the-middle attack threat model. According to the method, the security of signature and identity authentication is effectively improved.

The invention relates to the field of information security, and discloses a post-quantum certificate management method, a certificate management platform and equipment. In the invention, the method comprises the following steps: acquiring a digital certificate application file of a client, the digital certificate application file comprising an SPHINCS + public key; according to the SPHINCS + public key, a post-quantum signature certificate is generated; configuring an encryption key pair for the digital certificate application file, and generating a post-quantum encryption certificate according to the encryption key pair; and sending the post quantum signature certificate and the post quantum encryption certificate to the client. According to the method, an anti-quantum dual-certificate system separation system is constructed, keys are respectively generated through multiple key generation algorithms, the problems of key multiplexing and compatibility are solved, signature and encryption key pairs are thoroughly isolated through a dual-certificate architecture, and systematic risks caused by leakage of a single key are avoided.

The present invention relates to the field of information security technology, and in particular to a distributed quantum-resistant digital signature method and system, the method comprising: a mobile terminal and a cloud server jointly negotiate a signature public key, independently generate their own signature private keys, and use their own signature private keys to perform signature calculations, and obtain their own intermediate vectors and send them to a trusted cryptographic device; the trusted cryptographic device aggregates and calculates the received data, and returns the calculation results to the mobile terminal and the cloud server; the mobile terminal and the cloud server use the data returned by the trusted cryptographic device to generate partial signature fragments respectively; the cloud server sends the partial signature fragments generated by itself to the mobile terminal, and the mobile terminal verifies the partial signature fragments sent by the cloud server. After the verification is passed, the partial signature fragments of the two parties are aggregated to generate a complete signature. The present invention can improve the security of quantum-resistant signatures through distributed signatures of the mobile terminal, the cloud server and the trusted cryptographic device.

The present invention relates to a method for securing against fault attacks the execution of the verification of a Crystals-Dilithium post-quantum digital signature σ of a message M comprising a challenge seed c̃, a test vector z and a hint vector of polynomials h generated with a secret key sk = (ρ, K, tr, s1, s2, t0), said digital signature verification taking as inputs, said digital signature σ, said message M and a public key pk = (ρ, t1) and comprising a step of ensuring that a fault attack aiming at verifying one of the conditions P1, P2 and P3 does not lead to accept a forged signature, where : ct1.2d=0, ‖ct1.2d‖∞<βand‖LowBitsq(Az−ct1.2d,2γ2‖∞<γ2−β, ‖ct1.2d‖∞<γ2.