SEC: A Practical Secure Erasure Coding Scheme
for Peer-to-Peer Storage System1
Jing Tian, Zhi Yang, Yafei Dai
CNDS Lab, Peking University
{tianjing, yangzhi, dyf}@https://www.doczj.com/doc/7f1922261.html,
Abstract. Though conventional block cipher is widely used in Peer-to-Peer
storage systems, it is insufficient to ensure the confidentiality of long-term ar-
chive data because of the inherent Peer-to-Peer storage vulnerability that data is
stored on other untrusted peers. There are two potential security weak spots: (1)
sensitive data may be exposed from the encrypted data stored on an adversary
peer when the weakness of block cipher is discovered or there is a leakage of
secret key; (2) even when a user realizes the security threats, he/she cannot de-
stroy the encrypted data stored on an adversary peer to minimizing the loss.
This paper proposes a novel secure erasure code (SEC) scheme to solve the
above security problems. The SEC scheme encodes the sensitive data into sev-
eral fragments, and then stores the fragments onto different peers. In theory,
SEC ensures unconditional confidentiality at the fragment level, which means
that an adversary cannot obtain any portion of sensitive data from local stored
encrypted fragment even if he/she has the secret key and infinite computation
power. By leveraging the large scale property of Peer-to-Peer systems, SEC
further makes it infeasible for the adversary to collect enough fragments for de-
crypting. SEC can be used alone or together with block cipher to solve these
potential security problems. The performance results show that the SEC scheme
is practical for the real-world applications.
1 Introduction
In recent years, the Peer-to-Peer technology has been shown to be promising in con-structing large scale persistent storage. Many Peer-to-Peer storage systems are pro-posed, such as [1] [2] [3] [4] [5] [6]. Extensive research efforts have focused on pro-viding stable overlay network and long-term data durability. However, there has been little research reported on the confidentiality of the data.
In Peer-to-Peer storage systems, the user’s data is stored on disks of other un-trusted peers, and any peer can leave the system at any time. Consequently, there is a great challenge for the confidentiality and durability of user’s data. The conventional approach used in several proposed systems [1] [2] [6] is to encrypt the user’s data to 1This work is supported by National Grand Fundamental Research 973 program of China under Grant No.2004CB318204, National Natural Science Foundation of China under Grant No.90412008
ensure its confidentiality, and then store the encrypted data using either a replication or an erasure code [7] strategy. However, we believe that only the block cipher en-cryption is insufficient because of the inherent Peer-to-Peer storage system vulner-ability that data is stored on other untrusted peers. The potential security weak spots are: (1) sensitive data may be exposed from the encrypted data stored on an adversary peer when the weakness of block cipher is discovered or there is a leakage of secret key; (2) even when a user realizes the security threats, the user cannot destroy the encrypted data stored on an adversary peer to minimizing the loss. In other words, once the encrypted data is stored, the owner is put in a passive position in the face of various security threats. These problems set great barrier for the applications of the Peer-to-Peer storage system.
To solve these problems, this paper proposes a novel secure erasure code (SEC) scheme for the Peer-to-Peer storage systems. The SEC scheme is in fact a customized Reed-Solomon erasure code. By customizing, SEC provides a strong strength data encryption as well as the data redundancy at the same time. The SEC first encrypts data into several fragments, and then distributes the fragments out as independent objects to different peers by the underlying Peer-to-Peer storage protocol. The SEC scheme guarantees that less than a threshold number of fragments do not give any information about any portion of the encrypted data theoretically. This property pre-vents adversary from decrypting the sensitive data only from the local stored frag-ment even if the weakness of the cipher is discovered or the adversary obtains the secret key. By the system assumption detailed in section 4, the SEC scheme further makes it infeasible that the adversary can collect enough fragments of one object for decrypting throughout the Peer-to-Peer system. Even though the adversary may col-lect enough fragments, the strong strength encryption of SEC protects the confidenti-ality of data. Thus, SEC provides a three-layer security. The SEC alone can be used as an encryption scheme, while it can also be used with conventional block cipher to solve security problems in Peer-to-Peer storage systems.
The body of this paper is organized as follows. After introducing some preliminary knowledge in section 2, we describe the whole SEC scheme in section 3. In section 4, the system model and a three-layer security analysis framework are presented, and the optimal parameter of the SEC is suggested according to a detailed analysis of the security. The performance results are given in section 5 which proves the practicality of the SEC scheme. In section 6, we show the two application modes of the SEC. After introducing the related works in section 7, we conclude this paper in section 8. 2 Reed-Solomon Codes Preliminaries
As opposed to replication scheme, erasure codes provide a more flexible redundancy scheme to tolerate storage component failures. Erasure codes divide a data object into m fragments, and encode them into n fragments, where all the fragments have same size and n m ≥, so the redundancy rate r is n/m . When decoding, erasure codes only need any t (t m ≥) fragments out of all the n encoded fragments to reconstruct the data. The original data fragments can be viewed as a vector 12(,,...,)m D D D D =, and the
encoding function Enc maps the original data fragments onto the encoded fragments ()Enc D E =, 12(,,...,)n E E E E =. The decoding function Dec maps a subset of E back to the original data, 12(,,...,)t i i i Dec E E E D =.
Reed-Solomon codes are an important subclass of erasure codes. Reed-Solomon codes are linear codes , and that means the encoding function is linear. Consequently, the encoding function of a Reed-Solomon code can be described by a vector-matrix multiplication ()Enc D DG = of a generator matrix G and the original data fragments vector D , where G is an m n × matrix.
Reed-Solomon codes are also maximal distance separable (MDS) codes, where MDS means that any m encoded data fragments are sufficient to recover the data. It is easy to see that the MDS property requires that any m m × square submatrix of G is non-singular. Consequently, the decoding function can also be described by vector-matrix multiplication 1(')''Dec E E G ?=, where E’ is an m elements sub vector of E and G’ is the submatrix with m corresponding rows of G .
To achieve a low computation cost, designers usually employ the systematic Reed-Solomon codes. A code is called systematic if and only if its generator matrix has the form (I m |R ), where I m is an m m × identity matrix. Hence, the m original data frag-ments also appear in the encoded fragments.
3 The SEC Scheme
The SEC scheme is composed of two parts, the encryption part and the fragment naming part. The encryption part is in fact a customized version of Reed-Solomon codes, which encodes the original data into several encrypted fragments. Then the fragment naming part assigns the encrypted fragments with some meaningless names and distributes them out. Recall that the Reed-Solomon code need the G’-1 to recon-struct the original data, then if we keep the generator matrix G as the secret key, the adversary cannot get the original data even when he/she has collected enough en-coded fragments. Based on this fact, SEC encrypts the data by creating a secret gen-erator matrix G SEC from the user specified key. The fragment naming part further makes it impractical for the adversary to collect enough fragments. In this section, we first show how to construct the required G SEC , and then describe the fragment naming algorithm.
3.1 The Secure Generator Matrix
The Reed-Solomon codes do work in an infinite Z field. However, the real domain and range of the computation in our computer are binary words of a fixed length L . Hence, we must perform the Reed-Solomon codes over a Galois Field with 2L ele-ments, denoted by GF (2L ) [8]. The elements of GF (2L ) are the integers from zero to 2L -1, and each element is represented by a L bits word. To ensure the correctness of our computation, the GF (2L ) should at least contain more than n+m elements[9], namely 2L >m+n . In the following discussion, all the arithmetic is over the GF (2L ).
However, the detailed computation over Galois Field is beyond the scope of this pa-per. Over the GF (2L ), we define our G SEC over GF (2L ) as follows
Definition 1 An m n × matrix G over GF (2L ) is called a secure generator matrix of the SEC scheme if and only if G is non-singular and it does not contain any (1)(1)m m ?×? singular submatrix .
One important subclass of non-singular matrices over GF (2L ) is Cauchy matrices, and we construct our G SEC by customizing the Cauchy matrix. Here, we give the defi-nition of Cauchy matrix.
Definition 2 Let 1{,,}m x x … and 1{,,}n y y … be two sets of elements in GF(2L ). If the two sets satisfy the following constraint
(1) ,{1,,}, i j x x i j m i j ≠?∈≠…
(2) ,{1,,}, i j y y i j n i j ≠?∈≠…
(3) {1,,}, {1,,}i j
x y i m j n ≠?∈?∈…… then, the matrix
11
12121
222121111111
11n n m m m n x y x y x y x y x y x y x y x y x y ????+++??????+++????????+++??
is called a Cauchy matrix over GF (2L ).
A Cauchy matrix has the following property, and the proof of it can be found in
[10].
Theorem 1 Every square submatrix of a Cauchy matrix is non-singular .
3.2 The Construction of G SEC
G SEC requires any (1)(1)m m ?×? square submatrix to be non-singular, and a Cauchy matrix satisfies this condition. However, it is impractical to keep such a big matrix G SEC as secret key. Contrarily, we generate the G SEC from a user specified secret key. In this subsection, we introduce how to construct the G SEC by customizing a Cauchy matrix according to a user specified l k bit secret key k G .
Since the G SEC is an m n × matrix over GF (2L ), we also need m and n to be algo-rithm parameters besides k G . Here, we require l k to be a multiple of m+n , and L=l k /(m+n). Subsequently, the k G can be segmented into a vector consisting of m+n elements, where each element is a bit string of size L . By using the pseudo random function, we get a new vector of m+n distinct elements from the prior vector. Finally, these m+n elements can be used to generate the G SEC . The pseudocode is in Fig1. In the key generation procedure, there are two constraints of the parameters as fol-lows,
22log ()log ((1))L m n m r >+=×+ (1)
()(1)k l L m n L m r =×+=××+ (2)
To encode the data, we should segment the original data into vectors, each of which has m bit strings of size L. Hence, for each vector, the vector-matrix multiplica-tion yields an encoded vector of size n. Thus, all the ith elements in all encoded vec-
Fig1: The pseudocode to construct the G SEC
tors form the ith encoded fragment, and then we get n encoded fragments.
3.3 Fragments Naming and Placement
To keep the whole data secure, we store each encoded fragment as a new object indi-vidually in the Peer-to-Peer storage system to make it impractical for the adversary to collect enough fragments. Here, we employ a strong one-way hash function Hash, e.g. MD5 or SHA1, to hide the relationship between any two fragments of the same origi-nal data. We use SHA1 in this paper. Besides, we also need a bit vector V for each
original data and a separate key k N . The filename of the ith fragment f i is generated as follows for i=1, 2, …,n (|| is the bit concatenation):
10(||||)""
i i N f Hash f k V f ?=??=? (3) The V can be a concatenation of some properties of the original data, e.g. the path-name and the last modified time of a file. Though the k N can be the same with the key k G for G SEC , it is not suggested to do that.
After assigned meaningless names, all the fragments should be distributed out ran-domly or by their names, and be stored on different peers throughout the whole Peer-to-Peer system using the underlying protocol.
4 The Security of SEC
Unlike the block cipher, the SEC scheme provides data confidentiality at three layers. In this section, after giving the system model we present a three-layer security analy-sis framework for the SEC scheme, and then give a detailed analysis of the confiden-tiality based on the analysis framework, and finally we suggest the optimal parameter selection according to the analysis.
4.1 System Model
In the following analysis, we distinguish the two security levels, the information-theoretic security and computational security. The information-theoretic security, also called “unconditional security”, means that the ciphertext gives no information about the original data even if the adversary has infinite computational power. Compared with information-theoretic security, the computational security is a little weaker, and it means that no polynomial-time algorithm can reconstruct the original data from the ciphertext.
By classifying the adversary’s intent, there are two adversary models for Peer-to-Peer storage systems, the passive adversary and the active adversary. A passive adver-sary only tries to peek at the data stored on the disk of local computer, and the adver-sary does not make an effort to compromise more computers for more information. In a Peer-to-Peer storage system, every user can be a passive adversary since the data are all stored on peers. Therefore, the passive adversaries are very dangerous though they seem somewhat na?ve. It is clear that the conventional block cipher scheme is only computational secure in the face of passive adversary. An active adversary has more powerful ability. Besides accessing the local stored data, the active adversary may request other files in the system, or compromise other peers to learn the desired files. In particularly, an active adversary tries to collect all the fragments for one data object in a system using the SEC scheme. SEC provides very strong computational security in the face of the active adversary.
In this paper, we make the following system assumptions of the underlying Peer-to-Peer system: (1) it provides anonymous routing [12] [13]; (2) it is Byzantine-fault-
tolerant [14] [15]; (3) it is Sybil attack resistant [16]. These system assumptions are hot issues of Peer-to-Peer security, which have been extensively studied.
4.2 Three-layer Security Framework
The SEC scheme provides the following three-layer securities,
Fragment Security: The fragment security is the base layer security, which pro-vides an information-theoretic secure at fragment level. This ensures that less than m fragments reveal nothing about any fragment of the original data theoretically. It has been proved in [17] that less than m shares definitely give some information about the secret when the size of the share is smaller than the secret. However, less than m fragments can give no information about any single fragment of original data because the encoded fragments and the original fragments are in the same size.
Distribution Security: The SEC scheme assigns the fragments with different meaningless names, which makes it impractical for the adversary to find out all the fragments of one data object throughout the large scale Peer-to-Peer system. Encryption Security: Though an adversary may collect enough m fragments, he/she has to find out the secret matrix G SEC to decrypt encoded fragments.
The fragment security is important since a single fragment is theoretically mean-ingless for the adversary. Therefore, only with the local stored fragment the adversary cannot get any portion of the sensitive data even if the weakness of the cipher is dis-covered or the adversary gets the secret key. Thus, the passive adversary is no longer a threat for the sensitive data in SEC scheme, while the data stored on passive adver-sary is the core reason of the potential security problems in conventional scheme. Subsequently, fragment security enforces active adversary to collect enough frag-ments, which results in distribution security. Based on the system assumptions, an adversary can only try all the candidate combinations of fragments in the whole sys-tem for decrypting, which is shown to be impractical in next subsection. Because the adversary cannot get all the fragments of the sensitive data easily, there is a chance for the user to remove other fragments when there is security threat. Eventually the strong encryption security of SEC protects the sensitive data even if the adversary collects enough fragments.
Without anonymous routing, the owner information of the fragments will be learned by adversaries, which will significantly weaken the distribution security. If there is Byzantine fault or Sybil attack, by attacking the routing tables or using multi-ple identifiers the adversary can lead more upload traffic onto themselves, and then the adversary can reduce the number of trials of candidate fragment combinations by the timing information of the concurrent uploads. This also weakens the distribution security.
Though we can store the encrypted fragments of conventional block cipher by the simple segmenting, this simple scheme does not provide fragment security and the distribution security. This is because the adversary can obtain the portion of the sensi-tive data by decrypting a single fragment, and it is not necessary for the adversary to collect all the fragments. Therefore, the fragment security and the distribution security are particular properties of the SEC scheme.
4.3 Analysis of Three-layer Security
With respect to the three-layer security framework, we prove the information-theoretic property of fragment security layer, and give a quantitative analysis on strength of the distribution security layer and the encryption security layer. Fragment Security
To prove the information-theoretic property of fragment security, we first give the following lemma.
Lemma 2 Let G be an m m × matrix. There is no zero element in the inverse ma-trix G -1 if G does not contain any (1)(1)m m ?×? singular submatrix .
Proof: According to the computation of inverse matrix,
*1||
G G G ?= if there is no (1)(1)m m ?×? singular submatrix in G , there is no zero element in G *, so there is no zero element in G -1. Q.E.D of Lemma 2.
Theorem 3 Let the m n × matrix G be the secure generator matrix of SEC . The SEC scheme provides information-theoretic security at fragment level i.e. the adver-sary cannot obtain any information about any fragment of the original data from less than m fragments even when the adversary has got the matrix G .
Proof: Let 12(,,...,)m D D D be the original data vector, and 12(,,...,)m i i i E E E be any
encoded fragments vector with m element. Let G’ be the m m ×submatrix of G SEC with the columns i 1, i 2,…i m . Therefore, we have 12'112(,,...,)(,,...,)m
m i i i D D D E E E G ?=. Ac-cording to lemma 2, the G’-1 does not have any zero element. As a result, any frag-ment of the original data D j must be computed from enough m encoded fragments, and less than m encoded fragments give no information about any fragment of the original data in information theoretic sense. Q.E.D of Theorem 3. Distribution Security
Since the SEC provides information-theoretic security at the fragment level, the active adversary will try to collect enough fragments throughout the whole system. However, meaningless fragment naming and random placement make it infeasible to collect sufficient m fragments. For a (m, n ) SEC scheme, if there are N possible frag-ments in the system, there are C (N,m ) possible combinations of m fragments while only C (n,m ) are valid ones. Thus, we get the number of trials for the adversary as following,
(,)(,)(,)(,)
C N m C N m C n m C m r m =× (4) If the adversary knows the V in (3), then the number of trails is as following, (,)min{,2}(,)
k N l C N m C n m (5) where N k l is the length of k N .
The SEC’s distribution security leverages the huge possible candidate fragments in Peer-to-Peer system to keep the high security of the data object. From (4), we find that the distribution security will strengthen dramatically along with the number of candidates N and that the larger redundancy rate will result in worse distribution secu-rity. In a real Peer-to-Peer storage system, we can improve the distribution security by increasing m while N is a relative stable property of the system.
Encryption Security
If the worst case happens, that the adversary collects enough m fragments, the SEC would falls back on its third line of defense, the encryption security. Here, we give the analysis on the number of trials for the brute force attack.
Lemma 4 Let 1212{,,...,,,,...,}m n x x x y y y , and 1212{',',...,',',',...,'}m n x x x y y y be two sequences over GF (2L ), where m>1 and n>1. If
='' {1,,}, {1,,}i j i j x y x y i m j n ++?∈?∈……
then =',{1,,}i i x x i m ?∈… and ='
, {1,,}j j y y j n ?∈… Proof: Because of that m>1 and n>1, the following linear equation system always exists (a, b, c and d are constants)
111112122121
2222''''''''x y a x y x y b x y x y c x y x y d x y +==+??+==+??+==+??+==+?
(6) It is clear that there is only one unique solution for both four variables linear equa-tion systems in (6). Hence, we get 11'x x = and 11'y y =. Consequently, we get =', {1,,}j j y y j n ?∈… since 11=''j j x y x y ++ and 11'x x =. For the same reason, we have 'i i x x = {1,,}i m ?∈…. Q.E.D of Lemma 4.
From Lemma 4, we yield the following theorem,
Theorem 5 A Cauchy matrix is determined by one and only one unique sequence 11{,...,,,...,}m n x x y y
Suppose that the adversary has collected the m encoded fragments 12(,,...,)m
i i i E E E , then we have 1
1111111(,...,)(,...,)1
1m m m i i m i i m i m i x y x y D D E E x y x y ????++????=??????++??
(7) where 12(,,...,)m D D D is the original data vector. Theorem 5 implies that there is
only one sequence 11{,...,,,...,}m
m i i x x y y which can generate the m m × matrix used in (7). As a result, the adversary can only guess that very sequence over GF (2L ) for the decryption. Since there are 2m × different elements in the sequence, the ad-versary must try P(2L , 2m) times in the worst case. From (2) we yield,
(1)(2, 2)(2,2)k l L r m
P m P m +×= (8)
From (8), it is obvious that for the given l k and r , the strength is determined by m ,
5 Performance Evaluation
Although the analysis has shown that the SEC scheme can provide strong strength confidentiality for Peer-to-Peer storage systems, it is important for a real applied scheme to consume less of the end user’s CPU cycles in a Peer-to-Peer environment. In this section, we report some performance results of our implementation in Java language. The experiment results are collected on a desktop computer, with a Cel-eron(R) CPU running at 2.4GHz, and 512MB of main memory, which are the typical settings of the desktop computer.
Fig 2: Optimal m for different key length l k and
number of candidate fragments N with r=1
where L is 8 for all key lengths.
Computation Overheads
There are some computation overheads before the main vector-matrix multiplica-tion in the encoding and decoding phases of the SEC scheme. First of all, the SEC generates the log and inverse log tables for every elements in GF(2L ). Then the SEC generates the encoding matrix from the specified key for the encoding, or the inverse matrix for the decoding. We present the performance for a 512-bit key in table 1. It is clear that the computation overheads are negligible.
Table 1. The computation overheads, l k =512, r=1, m=16, L=16 log/inverse log tables in GF(2L ) encoding matrix from key inverse matrix for decoding
31ms 0.157ms 0.047ms
Encoding and Decoding Throughput
Now we focus on the main computation part of the SEC, the vector-matrix multi-plication over GF (2L ). We collect the throughput information for encoding a 100M data object using different l k and different r , and plot them in Fig 3. From the experi-ment results, we get
1 k r encoding throughput l +∝ (10)
which can be also deduced from [11]. The throughput for the decoding procedure is the same as the encoding procedure since the vector-matrix computations are the same.
From (10) we get the following implications. First, the encoding and decoding throughputs are both independent with the parameter m when the key length is fixed. Therefore, we can make use of the optimal parameter m discussed in the prior section without influencing the throughput. Secondly, the larger r will result in better throughputs though it decreases the security strength. Thirdly, we can use a smaller key length to achieve better performance, when the SEC is used together with a block cipher and the block cipher provides strong enough encryption security.
Fig 3 shows that the performance of the SEC is good enough. When using a 128-bit key with r=1, the SEC achieves a throughput of 20MB/s . Alternatively speaking, the SEC scheme only costs 7% of CPU cycles for a typical wide area peer with a 10Mb/s bandwidth. If short key length and large redundancy rate are used, the SEC scheme can even achieve a throughput over 200MB/s .
6 Application of SEC Scheme
Stand-alone Encryption
The SEC scheme provides information-theoretic security against the passive ad-versary and computational security against the active adversary. Therefore, the SEC alone can be used as an encryption scheme while providing the data redundancy as well. The encryption security of the SEC scheme can be very strong with r=1, e.g. the
number of trials for a 128-bit key can be 2127.99. Though the encryption security drops
when increasing r, the distribution security of SEC still keeps the overall security very strong. For instance, if the designer uses a SEC scheme with r=2 and l k≤128 in a system with N=104, the upper bound of the encryption security is only 288. However,
when we use a 120-bit key with the parameter m=8, the overall number of trials for
the active adversary can reach 2150.8.
The parameters of the SEC scheme can be adaptively configured to satisfy the
given requirements of the performance, the security and the redundancy. First, ac-
cording to (10) SEC determines the upper bound of l k for the given r and the per-
formance criteria. Second, according to the estimated parameter N of the system, SEC
looks for the optimal m and l k that satisfy both (1) and (2), where the overall security
of (9) meets the designer’s requirement. If the optimal m is found, the performance,
the redundancy and the security are all satisfied by the selected parameters.
When used in stand-alone mode, the vector-matrix multiplication may be attacked
by the known-plaintext. To solve this problem, we can simply use the hash value of
the secret key and the object identifier as the encryption key for every data object.
Working with Block Cipher
The SEC scheme can also be used together with the conventional block cipher, since the block cipher is known to have a strong encryption security and high per-formance. For example, the AES with 128-bit key has a throughput of 61MB/s [25]. When using together, the block cipher first encrypts the data object, and then SEC encrypts the encrypted data and distributes the fragments out. As the SEC is mainly responsible for the fragment security and the distribution security, it is not necessary to use a long encryption key length l k, which will result in a better throughput of the SEC. As a result, the overall performance is still good enough.
7 Related Works
The confidentiality is the essential issue for every Peer-to-Peer storage system. Most of the proposed systems, such as [1] [2] [3] [4] [5] and [6], simply employ the stan-dard block cipher algorithm.
Shamir develops a secret sharing algorithm in [19], where less than a threshold fragments give no information of the data. Unfortunately, each share of the secret is the same size as the secret, so it is not space efficient for the storage system. The IDA [20] algorithm is also a secret sharing algorithm, where each share is smaller than the secret. The IDA can be applied in routing for parallel computers and reliable storage.
[21] proposes a combined scheme of XOR secret sharing and replication to eliminate the key management problem in the collaborative environments. Pasis [18] uses the threshold scheme for both confidentiality and durability, where the threshold scheme includes the secret sharing scheme by definition. Unlike the threshold scheme, SEC can work alone as an encryption function because it encrypts data using the user specified secret key. Moreover, with the fragment naming sub-algorithm SEC is de-
signed particularly for the Peer-to-Peer systems because it leverages the large scale distribution feature of Peer-to-Peer system to protect sensitive data.
Rao proposes the idea of joint encryption and error correction schemes in [22] [23]. The proposed scheme keeps the whole generator matrix as the secret key, which makes it infeasible to be employed in a storage system. Xu develops an encryption scheme based on MDS code in [24]. This scheme keeps the indices of the encoded fragments as the secret key, and returns the secret key to user, which results in a large number of keys. In contrast, our SEC generates the generator matrix from a user specified secret key, which eliminates the key management problem.
8 Conclusions
Confidentiality and durability are the essential issues in Peer-to-Peer storage systems. Though the conventional block cipher encryption is widely used to keep the confiden-tiality, we emphasized that the inherent vulnerability of the Peer-to-Peer storage sys-tems makes it dangerous to only use block cipher. The proposed SEC scheme en-crypts data into fragments, and then distributes them out onto different peers. Besides data redundancy, SEC scheme provides a three-layer security, including fragment security, distribution security and encryption security. By a detailed security analysis, we have shown how SEC eliminates the potential security problems of Peer-to-Peer storage systems. Experiments have shown that SEC has a practical throughput. Fi-nally, the SEC can work either in a stand-alone mode or together with conventional block cipher to keep the confidentiality of data.
9Acknowledgement
The authors wish to thank Yong Liu for the helpful discussions.
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特殊角三角函数值表： 函数名 在平面直角坐标系xOy中，从点O引出一条射线OP，设旋转角为θ，设OP=r，P点的坐标为（x，y）有 正弦函数sinθ=y/r余弦函数cosθ=x/r正切函数tanθ=y/x余切函数cotθ=x/y 正弦（sin）:角α的对边比斜边余弦（cos）:角α的邻边比斜边 正切（tan）:角α的对边比邻边余切（cot）:角α的邻边比对边 特殊函数人倒数关系: tanα ?cotα=1sinα ?cscα=1cosα ?secα=1特殊函数人商数关系:tanα=sinα/cosαcotα=cosα/sinα 特殊函数人平方关系：sinα2+cosα2=11+tanα2=secα21+cotα=cscα2 以下关系，函数名不变，符号看象限 sin（π+α）=-sinα cos（π+α）=-cosα tan（π+α）=tanα cot（π+α）=cotα sin（π－α）=sinα cos（π－α）=-cosα tan（π－α）=-tanα cot（π－α）=-cotα sin（2π－α）=-sinα cos（2π－α）=cosα tan（2π－α）=-tanα cot（2π－α）=-cotα 以下关系，奇变偶不变，符号看象限 sin（90°-α）=cosα cos（90°-α）=sinα tan（90°-α）=cotα cot（90°-α）=tanα sin（90°+α）=cosα cos（90°+α）=sinα tan（90°+α）=-cotαcot（90°+α）=-tanα 特殊三角函数人积化和差的关系： sinα ?cosβ=（1/2）*[sin（α+β）+sin（α－β）] cosα ?sinβ=（1/2）*[sin（α+β）－sin（α－β）] cosα ?cosβ=（1/2）*[cos（α+β）+cos（α－β）] sinα ?sinβ=（1/2）*[cos（α+β）－cos（α－β）] 特殊三角函数 - 和差化积公式 sinα+sinβ=2*[sin(α+β)/2]*[cos(α-β)/2] sinα-sinβ=2*[cos(α+β)/2]*[sin(α-β)/2]
角度 函数 0 30 45 60 90 120 135 150 180 270 360 角a 的弧度 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π 3π/2 2π sin 0 1/2 √2/2 √3/2 1 √3/2 √2/2 1/2 0 -1 0 cos 1 √3/2 √2/2 1/2 0 -1/2 -√2/2 -√3/2 -1 0 1 tan √3/3 1 √3 -√3 -1 -√3/3 1、图示法：借助于下面三个图形来记忆，即使有所遗忘也可根据图形重新推出： sin30°=cos60°=2 1 ，sin45°=cos45°=22， tan30°=cot60°=33， tan 45°=cot45°=1 正弦函数 sinθ=y/r 余弦函数 cosθ=x/r 正切函数 tanθ=y/x 余切函数 cotθ=x/y 正割函数 secθ=r/x 余割函数 cscθ=r/y 2、列表法： 说明：正弦值随角度变化，即0? 30? 45? 60? 90?变化；值从0 2 1 22 23 1变化，其余类似记忆． 3、规律记忆法：观察表中的数值特征，可总结为下列记忆规律： ① 有界性：（锐角三角函数值都是正值）即当0°＜α＜90°时， 则0＜sin α＜1； 0＜cos α＜1 ； tan α＞0 ； cot α＞0。 ②增减性：（锐角的正弦、正切值随角度的增大而增大；余弦、余切值随角度的增大而减小），即当0＜A ＜B ＜90°时，则sin A ＜sin B ；tan A ＜tan B ； cos A ＞cos B ；cot A ＞cot B ；特别地：若0°＜α＜45°，则sin A ＜cos A ；tan A ＜cot A 若45°＜A ＜90°，则sin A ＞cos A ；tan A ＞cot A ． 4、口决记忆法：观察表中的数值特征 正弦、余弦值可表示为 2m 形式，正切、余切值可表示为3 m 形式，有关m 的值可归纳成顺口溜：一、二、三；三、二、一；三九二十七． 30? 1 2 3 1 45? 1 2 1 2 60? 3
三角函数特殊值 1、图示法：借助于下面三个图形来记忆，即使有所遗忘也可根据图形重新推出： sin30°=cos60°= 21 sin45°=cos45°=2 2 tan30°=cot60°=3 3 tan 45°=cot45°=1 2 30? 1 2 3 1 45? 1 2 1 2 60? 3
说明：正弦值随角度变化，即0? 30? 45? 60? 90?变化；值从0 2 3 1变化，其余类似记忆． 3、规律记忆法：观察表中的数值特征，可总结为下列记忆规律： ① 有界性：（锐角三角函数值都是正值）即当0°＜α＜90°时， 则0＜sin α＜1； 0＜cos α＜1 ； tan α＞0 ； cot α＞0。 ②增减性：（锐角的正弦、正切值随角度的增大而增大；余弦、余切值随角度的增大而减小），即当0＜A ＜B ＜90°时，则sin A ＜sin B ；tan A ＜tan B ； cos A ＞cos B ；cot A ＞cot B ；特别地：若0°＜α＜45°，则sin A ＜cos A ；tan A ＜cot A 若45°＜A ＜90°，则sin A ＞cos A ；tan A ＞cot A ． 4、口决记忆法：观察表中的数值特征 正弦、余弦值可表示为 2m 形式，正切、余切值可表示为3 m 形式，有关m 的值可归纳成顺口溜：一、二、三；三、二、一；三九二十七． 巧记特殊角的三角函数值 初学三角函数，记忆特殊角三角函数值易错易混。若在理解掌握的基础上，经过变形，使其呈现某种规律，再配以歌诀，则可浅显易记，触目成诵。 仔细观察表1，你会发现重要的规律。
特殊角三角函数值表 函数名 在平面直角坐标系xOy中，从点0引出一条射线0P，设旋转角为0,设OP=r , P点的坐标为（x, y ）有 正弦函数sin 0 =y/r 余弦函数cos 0 =x/r 正切函数tan 0 =y/x 余切函数cot 0 =x/y 正弦（Sin ）：角a的对边比斜边余弦（COS ）:角a的邻边比斜边 正切（tan ）:角a的对边比邻边余切（cot ）:角a的邻边比对边 特殊函数人倒数关系：tan a ?COt a =1 sin a ?CSC a =1 COS a ?SeC a =1 特殊函数人商数关系：tan a =Sin a /COS a COt a =COS a /Sin a 特殊函数人平方关系：sin a 2+COS a 2=1 1+tan a 2=sec a2 1+cot a =CSC a2 以下关系，函数名不变，符号: 看象限 sin (n + a) =-Sin a COS (n + a) =. -COS a tan (n + a) =tan a cot (n + a) =cot a sin (n — a) =sin a cos (n —-a) =-COS a tan (n — a) =-tan a cot (n —-a) =-COt a sin (2 n — a) =-Sin a COS (2 n —a )=COS a tan (2 n — a) =-tan a COt (2 n —a )=-cot 以下关系，奇变偶不变，付号看象限 sin (90° - a) =COS a COS (90°-a ) =sin a tan (90° - a) =COt a cot (90°-a ) =ta n a sin (90° + a) =COS a cos (90°+ a )=sin a tan (90° + a) =- COt a(90°+ a ) =-ta n a 特殊三角函数人积化和差的关系: sin a ?cos 3 = ( 1/2 ) *[si n (a + 3) +sin (a — 3) ] COS a ?si n 3 = ( 1/2 ) *[si n (a + 3) —sin (a — 3) ] COS a ?cos 3 = ( 1/2 ) *[cos (a + 3) +COS (a — 3) ] sin a ?si n 3 = ( 1/2 ) *[cos (a + 3) —COS (a — 3) ] 特殊三角函数-和差化积公式 sin a +sin 3 =2*[sin( a + 3 )/2]*[cos( a - 3 )/2] sin a -sin 3 =2*[cos( a + 3 )/2]*[sin( a - 3 )/2]
1、图示法：借助于下面三个图形来记忆，即使有所遗忘也可根据图形重新推出： sin30°=cos60°=2 1 ，sin45°=cos45°=22，tan30°=cot60°=33，tan45°=cot45°=1 正弦函数sinθ=y/r 余弦函数cosθ=x/r 正切函数tanθ=y/x 余切函数cotθ=x/y 正割函数secθ=r/x 余割函数cscθ=r/y 2、列表法： 说明：正弦值随角度变化，即0?30?45?60?90?变化；值从0 21222 3 1变化，其余类似记忆． 3、规律记忆法：观察表中的数值特征，可总结为下列记忆规律： ① 有界性：（锐角三角函数值都是正值）即当0°＜α＜90°时， 则0＜sin α＜1；0＜cos α＜1；tan α＞0；cot α＞0。 ②增减性：（锐角的正弦、正切值随角度的增大而增大；余弦、余切值随角度的增大而减小），即当0＜A ＜B ＜90°时，则sin A ＜sin B ；tan A ＜tan B ；cos A ＞cos B ；cot A ＞cot B ；特别地：若0°＜ α＜45°，则sin A ＜cos A ；tan A ＜cot A 若45°＜A ＜90°，则sin A ＞cos A ；tan A ＞cot A ． 4、口决记忆法：观察表中的数值特征 正弦、余弦值可表示为 2m 形式，正切、余切值可表示为3 m 形式，有关m 的值可归纳成顺口溜：一、二、三；三、二、一；三九二十七． 函数名正弦余弦正切余切正割余割 符号sincostancotseccsc 正弦函数sin （A ）=a/c 余弦函数cos （A ）=b/c 正切函数tan （A ）=a/b 余切函数cot （A ）=b/a 其中a 为对边，b 为邻边，c 为斜边 三角函数对照表 30? 1 2 1 45? 1 1 2 60?
特殊三角函数数值表集团文件发布号：（9816-UATWW-MWUB-WUNN-INNUL-DQQTY-
两角和公式 sin(A+B) = sinAcosB+cosAsinB sin(A-B) = sinAcosB-cosAsinB cos(A+B) = cosAcosB-sinAsinB cos(A-B) = cosAcosB+sinAsinB tan(A+B) = (tanA+tanB)/(1-tanAtanB) tan(A-B) = (tanA-tanB)/(1+tanAtanB) cot(A+B) = (cotAcotB-1)/(cotB+cotA) cot(A-B) = (cotAcotB+1)/(cotB-cotA) sin3A = 3sinA-4(sinA)^3; cos3A = 4(cosA)^3 -3cosA tan3a = tan a · tan(π/3+a)· tan(π/3-a) 公式 sin(A/2) = √{(1--cosA)/2} cos(A/2) = √{(1+cosA)/2} tan(A/2) = √{(1--cosA)/(1+cosA)} cot(A/2) = √{(1+cosA)/(1-cosA)} tan(A/2) = (1--cosA)/sinA=sinA/(1+cosA) sin(a)+sin(b) = 2sin[(a+b)/2]cos[(a-b)/2]
sin(a)-sin(b) = 2cos[(a+b)/2]sin[(a-b)/2] cos(a)+cos(b) = 2cos[(a+b)/2]cos[(a-b)/2] cos(a)-cos(b) = -2sin[(a+b)/2]sin[(a-b)/2] tanA+tanB=sin(A+B)/cosAcosB sin(a)sin(b) = -1/2*[cos(a+b)-cos(a-b)] cos(a)cos(b) = 1/2*[cos(a+b)+cos(a-b)] sin(a)cos(b) = 1/2*[sin(a+b)+sin(a-b)] cos(a)sin(b) = 1/2*[sin(a+b)-sin(a-b)] 诱导公式 sin(-a) = -sin(a) cos(-a) = cos(a) sin(π/2-a) = cos(a) cos(π/2-a) = sin(a) sin(π/2+a) = cos(a) cos(π/2+a) = -sin(a) sin(π-a) = sin(a) cos(π-a) = -cos(a) sin(π+a) = -sin(a) cos(π+a) = -cos(a) tgA=tanA = sinA/cosA 万能公式
特殊三角函数数值表 Company number：【WTUT-WT88Y-W8BBGB-BWYTT-19998】
两角和公式 sin(A+B) = sinAcosB+cosAsinB sin(A-B) = sinAcosB-cosAsinB cos(A+B) = cosAcosB-sinAsinB cos(A-B) = cosAcosB+sinAsinB tan(A+B) = (tanA+tanB)/(1-tanAtanB) tan(A-B) = (tanA-tanB)/(1+tanAtanB) cot(A+B) = (cotAcotB-1)/(cotB+cotA) cot(A-B) = (cotAcotB+1)/(cotB-cotA) sin3A = 3sinA-4(sinA)^3; cos3A = 4(cosA)^3 -3cosA tan3a = tan a · tan(π/3+a)· tan(π/3-a) 公式 sin(A/2) = √{(1--cosA)/2} cos(A/2) = √{(1+cosA)/2} tan(A/2) = √{(1--cosA)/(1+cosA)} cot(A/2) = √{(1+cosA)/(1-cosA)} tan(A/2) = (1--cosA)/sinA=sinA/(1+cosA) sin(a)+sin(b) = 2sin[(a+b)/2]cos[(a-b)/2] sin(a)-sin(b) = 2cos[(a+b)/2]sin[(a-b)/2] cos(a)+cos(b) = 2cos[(a+b)/2]cos[(a-b)/2] cos(a)-cos(b) = -2sin[(a+b)/2]sin[(a-b)/2] tanA+tanB=sin(A+B)/cosAcosB sin(a)sin(b) = -1/2*[cos(a+b)-cos(a-b)] cos(a)cos(b) = 1/2*[cos(a+b)+cos(a-b)] sin(a)cos(b) = 1/2*[sin(a+b)+sin(a-b)] cos(a)sin(b) = 1/2*[sin(a+b)-sin(a-b)] 诱导公式 sin(-a) = -sin(a) cos(-a) = cos(a) sin(π/2-a) = cos(a) cos(π/2-a) = sin(a) sin(π/2+a) = cos(a) cos(π/2+a) = -sin(a) sin(π-a) = sin(a)
特殊角三角函数值表： ? 函数名 在平面直角坐标系ｘＯy中，从点O引出一条射线OＰ，设旋转角为θ，设OＰ=r,Ｐ点的坐标为(x,ｙ)有?正弦函数sinθ=y/r余弦函数ｃoｓθ=x／r 正切函数tanθ＝ y/x 余切函数cｏｔθ=ｘ／ｙ?正弦（sin):角α的对边比斜边余弦（cos)：角α的邻边比斜边 正切(ｔan）:角α的对边比邻边余切（cot）:角α的邻边比对边 特殊函数人倒数关系: tａnα ?coｔα=１sinα ?csｃα＝1 ｃosα ?secα=1 特殊函数人商数关系:tａｎα=sinα／cｏsαｃotα＝cosα/sinα?特殊函数人平方关系:sinα2+cｏｓα2=1 1+tanα2=sｅｃα21+cotα=cscα2?以下关系,函数名不变，符号看象限 siｎ（π+α)＝－siｎαcos（π+α)=-cｏsα ｔan（π+α）＝ｔａnα coｔ(π＋α)=ｃotα sin（π—α)=ｓinα cos（π—α)=—cosα tａn(π—α)=－taｎαcｏt(π—α）=—ｃotα ｓin(2π－α）=-sinα cos（2π－α）=ｃosα tan(2π－α）=-tａnα cｏt(２π—α）＝－cotα?以下关系,奇变偶不变,符号看象限 siｎ(90°-α)=cosα cｏs(90°-α)=ｓinα tan（90°－α）=cｏｔα coｔ（９０°—α)=tａｎα?ｓin（９0°+α)=cosαｃｏ s(90°+α）=sinα tan（90°+α）=-ｃotαcot（90°＋α）＝—tanα 特殊三角函数人积化和差的关系：?siｎα ?ｃoｓβ=（1/2）*[ｓiｎ（α+β）+sin （α-β）］ cｏsα ?ｓinβ=(1/2)＊[sｉn（α+β)—sin(α－β）］ ｃoｓα ?cｏｓβ=（1/2）*［cｏs（α+β）＋cos（α—β)] sｉnα ?ｓｉnβ=（1/２）＊[ｃoｓ（α+β）-cｏs（α—β）]?特殊三角函数 - 和差化积公式?siｎα＋siｎβ=2*［ｓｉｎ(α＋β）/2］*[coｓ(α－β)/2] sinα-sinβ=2*［ｃos(α+β)／２]＊[ｓin（α—β）/2] cosα＋ｃｏｓβ=2*[ｃos(α＋β）/２］*［ｃｏｓ(α－β）/2］cosα—ｃosβ＝-22*［ｓin（α+β）／2］＊［sin(α－β)/2］
（1）特殊角三角函数值 sin0=0 sin30=0.5 sin45=0.7071 二分之根号2 sin60=0.8660 二分之根号3 sin90=1 cos0=1 cos30=0.866025404 二分之根号3 cos45=0.707106781 二分之根号2 cos60=0.5 cos90=0 tan0=0 tan30=0.577350269 三分之根号3 tan45=1 tan60=1.732050808 根号3 tan90=无 cot0=无 cot30=1.732050808 根号3 cot45=1 cot60=0.577350269 三分之根号3 cot90=0 （2）0°～90°的任意角的三角函数值，查三角函数表。（见下）（3）锐角三角函数值的变化情况 （i）锐角三角函数值都是正值
（ii）当角度在0°～90°间变化时， 正弦值随着角度的增大（或减小）而增大（或减小） 余弦值随着角度的增大（或减小）而减小（或增大） 正切值随着角度的增大（或减小）而增大（或减小） 余切值随着角度的增大（或减小）而减小（或增大） （iii）当角度在0°≤α≤90°间变化时， 0≤sinα≤1, 1≥cosα≥0, 当角度在0°<α<90°间变化时， tanα>0, cotα>0. “锐角三角函数”属于三角学，是《数学课程标准》中“空间与图形”领域的重要内容。从《数学课程标准》看，中学数学把三角学内容分成两个部分，第一部分放在义务教育第三学段，第二部分放在高中阶段。在义务教育第三学段，主要研究锐角三角函数和解直角三角形的内容，本套教科书安排了一章的内容，就是本章“锐角三角函数”。在高中阶段的三角内容是三角学的主体部分，包括解斜三角形、三角函数、反三角函数和简单的三角方程。无论是从内容上看，还是从思考问题的方法上看，前一部分都是后一部分的重要基础，掌握锐角三角函数的概念和解直角三角形的方法，是学习三角函数和解斜三角形的重要准备。 附：三角函数值表 sin0=0, sin15=（√6-√2)/4 , sin30=1/2, sin45=√2/2, sin60=√3/2, sin75=(√6+√2)/2 , sin90=1, sin105=√2/2*(√3/2+1/2) sin120=√3/2 sin135=√2/2 sin150=1/2 sin165=（√6-√2)/4 sin180=0 sin270=-1 sin360=0
创作编号：BG7531400019813488897SX 创作者： 别如克* 只想上传这一个表 下面的都是无用的话 不用看了。 1、图示法：借助于下面三个图形来记忆，即使有所遗忘也可根据图形重新推出： sin30°=cos60°= 2 1 sin45°=cos45°=22 tan30°=cot60°=3 3 tan 45°=cot45°=1 230? 1 2 3 1 45? 1 2 1 2 60? 3
说明：正弦值随角度变化，即0? 30? 45? 60? 90?变化；值从0 23 1变化，其余类似记忆． 3、规律记忆法：观察表中的数值特征，可总结为下列记忆规律： ① 有界性：（锐角三角函数值都是正值）即当0°＜α＜90°时， 则0＜sin α＜1； 0＜cos α＜1 ； tan α＞0 ； cot α＞0。 ②增减性：（锐角的正弦、正切值随角度的增大而增大；余弦、余切值随角度的增大而减小），即当0＜A ＜B ＜90°时，则sin A ＜sin B ；tan A ＜tan B ； cos A ＞cos B ；cot A ＞cot B ；特别地：若0°＜α＜45°，则sin A ＜cos A ；tan A ＜cot A 若45°＜A ＜90°，则sin A ＞cos A ；tan A ＞cot A ． 4、口决记忆法：观察表中的数值特征 正弦、余弦值可表示为 2m 形式，正切、余切值可表示为3 m 形式，有关m 的值可归纳成顺口溜：一、二、三；三、二、一；三九二十七． 创作编号：BG7531400019813488897SX 创作者： 别如克*
文案大全 初中三角函数值表 特殊角三角函数值 sin0=0 sin30=0.5 sin45=0.7071=22 sin60=23=0.866 sin90=1 cos0=1 cos30=23 =0.866 cos45=22=0.70 cos60=0.5 cos90=0 tan0=0 tan30=33 =0.577 tan45=1 tan60=3=1.732 tan90=无 cot0=无cot30=3=1.732 cot45=1 cot60=33 =0.577 cot90=0 （2）0°～90°的任意角的三角函数值，查三角函数表。（见下） （3）锐角三角函数值的变化情况 （i ）锐角三角函数值都是正值 （ii ）当角度在0°～90°间变化时， 正弦值随着角度的增大（或减小）而增大（或减小） 余弦值随着角度的增大（或减小）而减小（或增大）
正切值随着角度的增大（或减小）而增大（或减小） 余切值随着角度的增大（或减小）而减小（或增大）（iii）当角度在0°≤α≤90°间变化时， 0≤sinα≤1, 1≥cosα≥0, 当角度在0°<α<90°间变化时， tanα>0, cotα>0. 附：三角函数值表 sin0=0, sin15=（√6-√2)/4 , sin30=1/2, sin45=√2/2, sin60=√3/2, sin75=(√6+√2)/2 , sin90=1, sin105=√2/2*(√3/2+1/2) 文案大全
sin120=√3/2 sin135=√2/2 sin150=1/2 sin165=（√6-√2)/4 sin180=0 sin270=-1 sin360=0 文案大全
1、图示法：借助于下面三个图形来记忆，即使有所遗忘也可根据图形重新推出： sin30°=cos60°=2 1 ，sin45°=cos45°=22， tan30°=cot60°=33， tan 45°=cot45°=1 正弦函数 sinθ=y/r 余弦函数 cosθ=x/r 正切函数 tanθ=y/x 余切函数 cotθ=x/y 正割函数 secθ=r/x 余割函数 cscθ=r/y 2、列表法： 说明：正弦值随角度变化，即0? 30? 45? 60? 90?变化；值从0 2 1 22 23 1变化，其余类似记忆． 3、规律记忆法：观察表中的数值特征，可总结为下列记忆规律： ① 有界性：（锐角三角函数值都是正值）即当0°＜α＜90°时， 则0＜sin α＜1； 0＜cos α＜1 ； tan α＞0 ； cot α＞0。 ②增减性：（锐角的正弦、正切值随角度的增大而增大；余弦、余切值随角度的增大而减小），即当0＜A ＜B ＜90°时，则sin A ＜sin B ；tan A ＜tan B ； cos A ＞cos B ；cot A ＞cot B ；特别地：若0°＜α＜45°，则sin A ＜cos A ；tan A ＜cot A 若45°＜A ＜90°，则sin A ＞cos A ；tan A ＞cot A ． 4、口决记忆法：观察表中的数值特征 正弦、余弦值可表示为 2m 形式，正切、余切值可表示为3 m 形式，有关m 的值可归纳成顺口溜：一、二、三；三、二、一；三九二十七． 30? 1 2 3 1 45? 1 2 1 2 60? 3
函数名正弦余弦正切余切正割余割符号sin cos tan cot sec csc 正弦函数sin（A）=a/c 余弦函数cos（A）=b/c 正切函数tan（A）=a/b 余切函数cot（A）=b/a 其中a为对边，b为邻边，c为斜边 三角函数对照表