CN112329025A - Power terminal bypass safety analysis method and power terminal bypass safety analysis system - Google Patents

Power terminal bypass safety analysis method and power terminal bypass safety analysis system Download PDF

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CN112329025A
CN112329025A CN202011296994.8A CN202011296994A CN112329025A CN 112329025 A CN112329025 A CN 112329025A CN 202011296994 A CN202011296994 A CN 202011296994A CN 112329025 A CN112329025 A CN 112329025A
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energy consumption
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CN112329025B (en
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赵东艳
王喆
唐晓柯
胡毅
胡晓波
成嵩
李德建
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State Grid Corp of China SGCC
State Grid Information and Telecommunication Co Ltd
Beijing Smartchip Microelectronics Technology Co Ltd
Electric Power Research Institute of State Grid Ningxia Electric Power Co Ltd
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State Grid Information and Telecommunication Co Ltd
Beijing Smartchip Microelectronics Technology Co Ltd
Electric Power Research Institute of State Grid Ningxia Electric Power Co Ltd
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Abstract

The invention provides a method and a system for analyzing the bypass safety of a power terminal, and belongs to the technical field of power terminal safety. The method comprises the following steps: s1) selecting one of the intermediate values in the attack algorithm intermediate value set, and measuring the energy consumption curve of the attack object when decrypting and/or encrypting different data according to the selected intermediate value to obtain the corresponding energy consumption curve matrix when the different data are encrypted and/or decrypted; s2) calculating the assumed intermediate values of different data and different assumed keys to obtain an assumed intermediate value matrix; s3) mapping the matrix of hypothetical intermediate values to a matrix of hypothetical energy consumption values for different hypothetical keys; s4) obtaining an optimal key by comparing the assumed energy consumption value matrix and the energy consumption curve matrix of each assumed key; s5) recovering the optimal key and determining whether the optimal key is the correct key. The method can improve the finding efficiency and the finding accuracy of the leaked key of the power terminal bypass attack method.

Description

Power terminal bypass safety analysis method and power terminal bypass safety analysis system
Technical Field
The invention relates to the technical field of power terminal safety, in particular to a power terminal bypass safety analysis method and a power terminal bypass safety analysis system.
Background
With the construction of smart power grids and ubiquitous power internet of things, power grids develop towards the directions of informatization, networking, intellectualization and the like, and the quantity of various power terminal devices is rapidly increased. The electric power terminal is widely distributed and various, the probability of malicious attack is greatly increased, and aiming at the technical development of the electric power terminal equipment, novel attack technical means are also endless, which poses great threat to the long-term safety and reliability of the electric power terminal. For the smart grid, once the safety and reliability of the terminal equipment cannot be guaranteed, the safe and stable operation of the whole power system can be affected, huge loss can be brought to power enterprises, and serious harm can be brought to the country and the society. Protecting the smart grid security must begin with protecting the power terminal equipment. Currently, a security attack method is commonly used for testing and discovering security holes of the power terminal.
The bypass attack of the power terminal is the most common power terminal security attack method at present, but the discovery efficiency and the discovery accuracy of the leaked key both reach the technical bottleneck, and breakthrough progress cannot be achieved.
Disclosure of Invention
The embodiment of the invention aims to provide a power terminal bypass security analysis method and a power terminal bypass security analysis system, so as to further improve the finding efficiency and the finding accuracy of a leaked key of a power terminal bypass attack method.
In order to achieve the above object, a first aspect of the present invention provides a power terminal bypass safety analysis method, including: s1) selecting one of the intermediate values in the attack algorithm intermediate value set, and measuring the energy consumption curve of the attack object when decrypting and/or encrypting different data according to the selected intermediate value to obtain the corresponding energy consumption curve matrix when the different data are encrypted and/or decrypted; s2) calculating hypothetical intermediate values for different data and different hypothetical keys, obtaining a matrix of hypothetical intermediate values S3) mapping the matrix of hypothetical intermediate values to a matrix of hypothetical energy consumption values for different hypothetical keys; s4) comparing the assumed energy consumption value matrix of each assumed key with the energy consumption curve matrix of the assumed key according to a deep learning algorithm to obtain a comparison matrix, and acquiring the assumed key with the highest matching value in the comparison matrix as an optimal key; s5) determining whether the optimal key is the correct key, and outputting risk information according to the determination result.
Optionally, in step S1), selecting one of the median values of the attack algorithm, where the median value is a function f (d, k); wherein d is known data; k is partial information of the key.
Optionally, in step S1), the measuring, according to the selected intermediate value, energy consumption when the attack object decrypts and/or encrypts different data, and obtaining an energy consumption curve matrix corresponding to the encrypted and/or decrypted different data includes: measuring energy consumption curves of the attack object when different data are decrypted and/or encrypted to obtain a plurality of energy consumption curves corresponding to the decrypted and/or encrypted data; and integrating a plurality of the energy consumption curves into an energy consumption curve matrix when different data are encrypted and/or decrypted.
Optionally, in step S2), the calculating assumed middle values of different data and different assumed keys to obtain an assumed middle value matrix includes: calculating intermediate values of different data to obtain all possible assumed intermediate values; a matrix of hypothesized intermediate values is obtained from the set of all possible hypothesized intermediate values.
Optionally, in step S4), the obtaining a comparison matrix by comparing the assumed energy consumption value matrix of each assumed key with the energy consumption curve matrix of the assumed key includes: converting different positions of a curve in the energy consumption curve matrix of the assumed key into characteristic values according to a preset algorithm; comparing the assumed energy consumption value matrix of the assumed key with the characteristic values of different positions to obtain a plurality of comparison relation values; and sorting the plurality of contrast relation values into the contrast matrix.
Optionally, the preset algorithm is a convolutional neural network algorithm in which a convolutional layer and a pooling layer are added on the basis of a perceptron basic algorithm.
Optionally, the convolutional layer and the pooling layer are a one-dimensional convolution kernel and a one-dimensional pooling layer; wherein the size value and the compensation value of the convolutional layer and the pooling layer are adjusted according to an application environment.
A second aspect of the present invention provides a power terminal bypass security analysis system, including: the acquisition unit is used for selecting one intermediate value in the intermediate value set of the attack algorithm, measuring an energy consumption curve when the attack object carries out decryption and/or encryption of different data according to the selected intermediate value, and obtaining an energy consumption curve matrix corresponding to the different data which are encrypted and/or decrypted; the processing unit is used for calculating the assumed intermediate values of different data and different assumed keys, obtaining an assumed intermediate value matrix, mapping the assumed intermediate value matrix into the assumed energy consumption value matrices of the different assumed keys, and obtaining a comparison matrix by comparing the assumed energy consumption value matrix of each assumed key with the energy consumption curve matrix of the assumed key; and the execution unit is used for acquiring the assumed key with the highest matching degree value in the comparison matrix as an optimal key, judging whether the optimal key is a correct key or not, and outputting risk information according to a judgment result.
In another aspect, the present invention provides a computer-readable storage medium having instructions stored thereon, which when executed on a computer, cause the computer to perform the above-mentioned power terminal bypass security analysis method.
According to the technical scheme, the leakage information in the power terminal equipment is obtained through the bypass attack test, the assumed secret key attack is carried out according to the known information of the leakage information, because different data are processed, different energy losses exist, and whether the assumed secret key is the correct secret key can be judged by comparing the energy consumption value matrix and the energy consumption curve matrix of the assumed secret key. When the assumed energy consumption value matrix is compared with the energy consumption curve matrix, the discriminator is operated through a deep learning algorithm, and the finding efficiency and the finding accuracy of the leaked key are effectively improved.
Additional features and advantages of embodiments of the invention will be set forth in the detailed description which follows.
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The accompanying drawings, which are included to provide a further understanding of the embodiments of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the embodiments of the invention without limiting the embodiments of the invention. In the drawings:
fig. 1 is a flowchart illustrating steps of a method for analyzing bypass safety of a power terminal according to an embodiment of the present invention;
fig. 2 is a system configuration diagram of a power terminal bypass safety analysis system according to an embodiment of the present invention;
fig. 3 is a control flowchart of a power terminal bypass safety analysis method according to an embodiment of the present invention.
Description of the reference numerals
10-an acquisition unit; 20-a processing unit; 30-execution unit.
Detailed Description
The following detailed description of embodiments of the invention refers to the accompanying drawings. It should be understood that the detailed description and specific examples, while indicating the present invention, are given by way of illustration and explanation only, not limitation.
Fig. 2 is a system configuration diagram of a power terminal bypass safety analysis system according to an embodiment of the present invention, and as shown in fig. 2, the power terminal bypass safety analysis system according to the embodiment of the present invention includes: the acquisition unit 10 is used for selecting one of the intermediate values in the attack algorithm intermediate value set and measuring an energy consumption curve when the attack object carries out decryption and/or encryption of different data; the processing unit 20 is configured to calculate assumed intermediate values of different data, obtain an assumed intermediate value matrix, and obtain an assumed energy consumption value matrix according to the assumed intermediate value matrix; is further used for comparing the energy consumption curve matrix and the assumed energy consumption value matrix of each assumed key to obtain a comparison matrix; and the execution unit 30 is configured to obtain the assumed key with the largest matching degree value in the comparison matrix as an optimal key, determine whether the optimal key is a correct key, and output determination result information.
Fig. 1 is a flowchart illustrating steps of a power terminal bypass security analysis method according to an embodiment of the present invention. As shown in fig. 1, an embodiment of the present invention provides a power terminal bypass security analysis method, which is applied to bypass security analysis and detection of a power terminal device, where the power terminal bypass security analysis is to record bypass information when the power terminal device performs security operation on a large amount of different data, utilize information leakage during operation of the power terminal, and extract security information hidden in the leaked information, such as security sensitive information, such as a secret key, an operation intermediate value, and the like, by using a security analysis method of mathematical statistics according to a leakage model in combination with known plaintext/ciphertext information. Namely, leakage information of a power terminal bypass can be cracked by an attack algorithm as known information through a secret key, so that power terminal equipment is attacked, once the safety and reliability of the terminal equipment cannot be guaranteed, the safe and stable operation of the whole power system can be influenced, huge loss can be brought to power enterprises, and serious harm can be brought to the country and the society. Therefore, the existing power terminal equipment needs to perform bypass attack training, so as to find out the possible attacked loophole and perform targeted loophole repairing.
The artificial intelligence is developed rapidly, the deep learning analysis technology is widely applied to the directions of behavior cognition, image recognition, voice recognition, dynamic track recognition and the like as the conventional algorithm technology of the artificial intelligence, and if the deep learning analysis technology method is introduced into the power terminal bypass attack method, the accuracy and efficiency of discovering and hiding the leakage information by the bypass attack method are obviously improved. Therefore, the invention provides a power terminal bypass security analysis method, which introduces a bypass attack method through a block cipher algorithm and comprises the following steps:
step S10: and selecting one of the intermediate values in the intermediate value set of the attack algorithm.
Specifically, the block cipher algorithm divides a digital sequence represented by a plaintext message code into n-length groups, and each group is converted into an output digital sequence with equal length under the control of a secret key. When an attack algorithm is carried out, the whole process of decrypting data by the secret key cannot be directly obtained, one intermediate value in the intermediate value set needs to be obtained firstly, and the whole decryption process is restored through the intermediate value. To ensure that the intermediate value conforms to the currently executed algorithm, the intermediate value v must satisfy the function v ═ f (d, k), where d is known data; k is partial information of the key. Since the bypass attack is a method for rapidly breaking the password by bypassing the tedious analysis of the encryption algorithm and combining the information leaked in the operation realized by the hardware of the password algorithm with the statistical theory, the known data includes the plaintext information and the ciphertext information leaked by the bypass, such as execution time, power consumption and electromagnetic radiation. Leakage information in a bypass of the power terminal equipment is acquired through an acquisition module of the acquisition unit 10, and a middle value of an execution algorithm is acquired through the acquired leakage information and key information.
Step S20: and measuring the energy consumption curve of the attack object during decryption and/or encryption of different data according to the intermediate value to obtain a corresponding energy consumption curve matrix.
Specifically, the attack object has energy consumption in both the decryption and encryption processes of data through the attack algorithm, and the energy consumption value is directly related to the processed data, that is, the energy consumption is different when different data decryption/encryption is performed. Performing encryption/decryption operation on certain data through the intermediate value obtained in step S10), the acquisition unit 10 acquires the energy consumption value of the key point in real time during the operation of the attack object, and obtains the energy consumption trajectory during the decryption/encryption of the data, thereby obtaining the encrypted/decrypted energy consumption curve of the data. Then, the encryption/decryption operation of the other data is performed by the intermediate value obtained in step S10), and the encrypted/decrypted energy consumption curve of the other data is obtained. And analogizing in turn, respectively obtaining energy consumption curves of the D kinds of data which are encrypted/decrypted to obtain D energy consumption curves, and sorting the energy curves into an energy curve matrix which is marked as T.
Step S30: and calculating the assumed intermediate values of different data to obtain an assumed intermediate value matrix.
Specifically, the power terminal bypass attack method is to perform key assumption by using known information and then perform encrypted data cracking by assuming the key, which is a repeated training for multiple timesThe process of training, namely extracting the correct key from a plurality of assumed keys through a plurality of training. Obtaining the assumed value of K of all assumed keys, and recording the assumed value as K, wherein K is (K)1,…,kk). Then, the assumed median v of each key for performing different data decryption/encryption is obtained according to the median function relation in step S10), and the calculation formula is:
v=f(di,ki)
where i is 1, …, D, indicating the serial number of the encrypted/decrypted data; obtaining the assumed intermediate value of all assumed keys for different data encryption/decryption, and recording the assumed intermediate value as a matrix V, V ═ V { (V)i,j=f(di,kj)|i=1,…,D,j=1,…,K}。
Step S40: the matrix of hypothetical intermediate values is mapped to a matrix of hypothetical energy consumption values for different hypothetical keys.
Specifically, the power consumption curve of the correct key in data decryption/encryption is fixed, so that it is determined whether the assumed key is the correct key, which can be obtained by comparing the power consumption curves of the same data in decryption/encryption, and the higher the matching degree is, the more likely the assumed key is the correct key. Therefore, the obtained matrix of assumed intermediate values needs to be converted into a matrix of assumed energy consumption values. The energy consumption value is directly related to the processed data, and the relationship is represented as charging and discharging of a load capacitor at a circuit level; at the register stage, the state of the flip-flop inside the register is inverted; at the instruction level is the hamming distance of the data before and after the instruction is executed. Therefore, the power consumption expression mode is required to be used for converting the assumed intermediate value matrix and the assumed energy consumption value matrix, and the traditional energy consumption model has a Hamming distance power consumption model or a Hamming weight power consumption model. Taking a hamming distance power consumption model as an example, mapping the assumed intermediate value to the assumed energy consumption matrix is realized by the power consumption change caused by the register byte state inversion, and mapping the assumed intermediate value matrix V to the assumed energy consumption value matrix H through the hamming distance power consumption model.
Step S50: and comparing the assumed energy consumption value matrix and the energy consumption curve matrix of each assumed key to obtain a comparison matrix, and acquiring the optimal key with the highest matching value.
Specifically, after obtaining the assumed energy consumption value matrix H of all assumed keys processing different data, comparing the assumed energy consumption value matrix H corresponding to each key assumption with the energy curve matrix T recorded at each position to obtain a matrix R, where the largest value of the R is the matrix with the high matching degree, i.e., the key assumption is determined to be correct. Converting different positions of the curve in the energy consumption curve matrix into characteristic values according to a preset algorithm; comparing the assumed energy consumption value matrix with the characteristic values of different positions to obtain a plurality of comparison relation values; and arranging the plurality of contrast relation values into a contrast matrix. The calculation of the comparison relation value is executed by a discriminator, the traditional discriminator has Pearson correlation and a multivariate Gaussian analysis function, in order to effectively improve the extraction efficiency and accuracy of the optimum key of the discriminator, the original discrimination algorithm is preferably replaced by a convolutional neural network algorithm in a deep learning algorithm, and a convolutional layer and a pooling layer are added on the basis of a multilayer perceptron. Performing feature mapping by using a convolution function conv, wherein the relational expression of the feature mapping is as follows:
Figure BDA0002785700380000081
wherein M is(i)=conv(X-1,Ki) (ii) a X is the feature mapping layer currently in need of computation, i.e. NMA set of feature mappings; m(i)Representing the ith feature map; x-1An input layer representing a previous layer of the network, i.e., a current layer; kiRepresents the ith convolution kernel; conv represents the feature map computed using a given convolution kernel input convolution. Obtaining a feature mapping set according to the relational expression, and obtaining the optimal result through the following relational expression:
Figure BDA0002785700380000082
wherein f is a passing training set, DtrainTrained models, DtestFor the test set, the accuracy is defined as the proportion of samples the model predicts correctly during the test. And obtaining a plurality of comparison relation values R according to the relation to obtain a comparison relation value matrix R, and performing transverse comparison judgment on all R, wherein the greater the R is, the higher the matching degree of the assumed energy consumption value matrix and the energy curve matrix is, and the selected assumed key corresponding to the maximum value of R is the correct key.
Preferably, for bypass security analysis, a one-dimensional convolution kernel and a one-dimensional pooling layer are employed. For a high-noise environment of the power terminal, the size of a convolution kernel is preferably 3, and the compensation is preferably 2; the pooling layer is preferably a maximum pooling layer, the filter size is preferably 2 and the compensation is preferably 2. Through the setting, the high-noise environment of the power terminal can be effectively stripped, the situation that irrelevant information is used as known information to carry out a bypass attack method is avoided, and the influence of the high-noise environment on the bypass attack method is reduced.
Step S60: and recovering the optimal key and judging whether the optimal key is a correct key.
Specifically, the optimal key is obtained by comparing the assumed energy consumption value matrix H corresponding to each key hypothesis with the energy curve matrix T recorded at each position, and although the comparison relationship value of the optimal key is the largest, the degree of matching between the assumed energy consumption value matrix and the energy curve matrix is also the highest, the extracted optimal key may not be the correct key because of the power loss model error and the matching error. In order to verify whether the optimal key is the correct key, preferably, a testing step is added in the method, the obtained optimal key is restored to be the complete key, the same data processing is performed on the complete key and the corresponding correct key, whether the energy consumption tracks are the same when the two are subjected to the same data processing is judged, and if the energy consumption curves are matched to be qualified, the obtained optimal key is the correct key. The current key can be extracted through bypass attack, that is, the current bypass leakage information includes current key information, the power terminal equipment has a key leakage risk, and a risk report is output, so that relevant personnel can repair the bypass information according to the problem, the power terminal is prevented from continuously leaking the current key information, and the security holes of the power terminal are reduced.
In a possible implementation manner, as shown in fig. 3, a certain power device terminal bypass attack test is performed, a certain intermediate value in an attack algorithm is selected, and an energy curve matrix is obtained according to the current intermediate value. The intermediate function has a relation v ═ f (d)i,ki) Where k is key information, k ∈ {1,2, …,255,256 }. And according to the acquired known information leaked from the bypass, sequentially carrying out different data processing on the 256 assumed keys to obtain an energy consumption curve of each assumed key for processing different data, and obtaining an assumed energy consumption value matrix. And calculating comparison relationship values of all the assumed energy consumption value matrixes and the energy consumption curve matrixes by a deep learning method of the discriminator to obtain corresponding matrixes, and selecting the assumed key corresponding to the maximum value of the comparison relationship values as the optimal key. And restoring the optimal key into a complete key, checking whether the optimal key is a correct key, and outputting risk information when the optimal key is judged to be the correct key.
Embodiments of the present invention also provide a machine-readable storage medium, which stores instructions that, when executed on a computer, cause the computer to execute the above-mentioned power terminal bypass security analysis method.
Those skilled in the art will appreciate that all or part of the steps in the method for implementing the above embodiments may be implemented by a program, which is stored in a storage medium and includes several instructions to enable a single chip, a chip, or a processor (processor) to execute all or part of the steps in the method according to the embodiments of the present invention. And the aforementioned storage medium includes: a U-disk, a removable hard disk, a Read-Only Memory (ROM), a Random Access Memory (RAM), a magnetic disk or an optical disk, and other various media capable of storing program codes.
While the embodiments of the present invention have been described in detail with reference to the accompanying drawings, the embodiments of the present invention are not limited to the details of the above embodiments, and various simple modifications can be made to the technical solution of the embodiments of the present invention within the technical idea of the embodiments of the present invention, and the simple modifications are within the scope of the embodiments of the present invention. It should be noted that the various features described in the above embodiments may be combined in any suitable manner without departing from the scope of the invention. In order to avoid unnecessary repetition, the embodiments of the present invention will not be described separately for the various possible combinations.
In addition, any combination of the various embodiments of the present invention is also possible, and the same should be considered as disclosed in the embodiments of the present invention as long as it does not depart from the spirit of the embodiments of the present invention.

Claims (10)

1. A power terminal bypass safety analysis method is characterized by comprising the following steps:
s1) selecting one of the intermediate values in the attack algorithm intermediate value set, and measuring the energy consumption curve of the attack object when decrypting and/or encrypting different data according to the selected intermediate value to obtain the corresponding energy consumption curve matrix when the different data are encrypted and/or decrypted;
s2) calculating the assumed intermediate values of different data and different assumed keys to obtain an assumed intermediate value matrix;
s3) mapping the matrix of assumed intermediate values to a matrix of assumed energy consumption values for different assumed keys;
s4) comparing the assumed energy consumption value matrix of each assumed key with the energy consumption curve matrix of the assumed key according to a deep learning algorithm to obtain a comparison matrix, and acquiring the assumed key with the highest matching value in the comparison matrix as an optimal key;
s5) determining whether the optimal key is the correct key, and outputting risk information according to the determination result.
2. The power terminal bypass security analysis method according to claim 1, wherein step S1) selects one of a set of intermediate values of the attack algorithm, the intermediate value being a function f (d, k); wherein d is known data; k is partial information of the key.
3. The power terminal bypass safety analysis method according to claim 2, wherein the known data comprises: plaintext information and/or ciphertext information.
4. The electric power terminal bypass security analysis method according to claim 1, wherein in step S1), the measuring the energy consumption of the attack object in decrypting and/or encrypting different data according to the selected intermediate value to obtain the corresponding energy consumption curve matrix when the different data is encrypted and/or decrypted includes:
measuring energy consumption curves of the attack object when different data are decrypted and/or encrypted to obtain a plurality of energy consumption curves corresponding to the decrypted and/or encrypted data;
and integrating a plurality of the energy consumption curves into an energy consumption curve matrix when different data are encrypted and/or decrypted.
5. The power terminal bypass security analysis method according to claim 1, wherein in step S2), the calculating assumed middle values of different data and different assumed keys to obtain an assumed middle value matrix includes:
calculating intermediate values of different data to obtain all possible assumed intermediate values;
a matrix of hypothesized intermediate values is obtained from the set of all possible hypothesized intermediate values.
6. The power terminal bypass security analysis method according to claim 1, wherein in step S4), the obtaining a comparison matrix by comparing the matrix of assumed energy consumption values of each assumed key with the matrix of energy consumption curves of the assumed key includes:
converting different positions of a curve in the energy consumption curve matrix of the assumed key into characteristic values according to a preset algorithm;
comparing the assumed energy consumption value matrix of the assumed key with the characteristic values of different positions to obtain a plurality of comparison relation values;
and sorting the plurality of contrast relation values into the contrast matrix.
7. The power terminal bypass safety analysis method according to claim 6, wherein the preset algorithm is a convolutional neural network algorithm with a convolutional layer and a pooling layer added on the basis of a perceptron base algorithm.
8. The power terminal bypass security analysis method of claim 7, wherein the convolutional layer and the pooling layer are a one-dimensional convolutional kernel and a one-dimensional pooling layer; wherein the size value and the compensation value of the convolutional layer and the pooling layer are adjusted according to an application environment.
9. A power terminal bypass security analysis system, the system comprising:
the acquisition unit is used for selecting one intermediate value in the intermediate value set of the attack algorithm, measuring an energy consumption curve when the attack object carries out decryption and/or encryption of different data according to the selected intermediate value, and obtaining an energy consumption curve matrix corresponding to the different data which are encrypted and/or decrypted;
the processing unit is used for calculating the assumed intermediate values of different data and different assumed keys, obtaining an assumed intermediate value matrix, mapping the assumed intermediate value matrix into the assumed energy consumption value matrices of the different assumed keys, and obtaining a comparison matrix by comparing the assumed energy consumption value matrix of each assumed key with the energy consumption curve matrix of the assumed key;
and the execution unit is used for acquiring the assumed key with the highest matching degree value in the comparison matrix as an optimal key, judging whether the optimal key is a correct key or not, and outputting risk information according to a judgment result.
10. A computer-readable storage medium having instructions stored thereon, which when executed on a computer, cause the computer to perform the power terminal bypass security analysis method of any one of claims 1 to 8.
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