An AreaEfficient Architecture of ReedSolomon Codec for Advanced RAID Systems MinAn Song, IFeng Lan, and SyYen Kuo


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1 An AreaEfficient Architecture of Reedolomon Codec for Advanced RAID ystems MinAn ong, IFeng Lan, and yyen Kuo Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan Abstract In this paper, a simple codec algorithm based on Reed olomon (R) is proposed for erasure correcting in RAID level 6 systems. Unlike conventional R, here this scheme with a mathematical reduction method, called Reduced taticchecksum Table Approach, could improve coding performance. We used Reed olomon which are designed according to characteristics of advanced RAID systems to handle two disk failures in RAID system. Also, this scheme performs all computations with only simple exclusiveor (XOR) operators just the same as EvenOdd. This new XORbased R could adapt to implementation in terms of improving reliability and flexibility.. Introduction In storage systems, especially for large disk arrays, reliability is getting critical while storage systems scale up. In [], it has been demonstrated that disk failures would be a daily event in petabytescale file systems. o, how to improve the capability of detecting or even correcting failures has been a significant issue for large storage systems. RAID systems which could be classified from level to 6 are commonly used to achieve this issue. Unlike other levels, RAID level 6, or socalled RAID 6, not only provide correcting capability, but also could recover at least two disk failures simultaneously. Usually each specific algorithm in RAID 6 performs a particular parities distribution []. Over the last two decades, lots of erasurecorrecting algorithms in RAID 6 have been proposed such as EvenOdd [3], Reedolomon (R) [], and X [5]. R code, a very popular error control code, has been studied in various applications, especially in communication systems [6]. Also, researchers have suggested some R based solutions to avoid hazards happening in RAIDlike systems [7], but those schemes might not be suitable to meet the desire of recovering system as quick as EvenOdd. Therefore, in this paper, we present an XORbased R codec scheme, which uses a reduced staticchecksum table approach, to manipulate the erasuresonly hazard. Basically, in this new scheme, it is similar to the conventional R codec algorithm [] that involves pipeline procedure, which consists of yndrome Calculation (C), Key Equation olver (KE), Chien earch (C), and Forney Algorithm (FA), but without involving the portions of either KE or C. Both KE and C are used to locate errors and need extra cost of finite field operators. For the erasureonly RAID system, system controller is not necessary to locate such errors since the individual disk devices have their own errorcontrol coding mechanisms to recover from errors []. Moreover, usually in large disk arrays, failures of a single storage device could be detected by the storage system controllers and then could be marked as well []. ince device failures can be marked as erasures, erasurecorrecting are usually employed to achieve the information recovery, the failed in disks can be recovered and system still can work as usual without broken. Compared with the traditional R codec scheme, a simpler scheme is proposed in this paper in terms of less cost, improving flexibility and reliability. The rest of this paper is organized as follows. ection describes general ideas in our encoding algorithm for the erasureonly RRAID system and the main feature of our scheme, called reduced staticchecksum table approach, is suggested as well. Our decoding approach will be shown in ection 3. ection gives results of performance analysis and also a comparison in the number of XOR operations with the EvenOdd, the traditional RRAID structure, and the XORbased R code as well. The Hardware implementation of the proposed R decoderencoder is described in ection 5. Finally, ection 6 gives the conclusions.. RRAID Encoder The encoding procedure in our scheme not only follows the rules of a mapping with systematic, but also builds a lookup table with an aspect of the constant multiplier. A mapping, in most R encoders, and the encoder usually generates systematic, namely, message bits of a symbol could be presented explicitly in its corresponding codeword. Equation cw( x) b( x) n k m ( x ) x shows a result after applying the systematic coding method, where cw(x), b(x), and m(x) are codeword, checksum and message respectively. W is the codeword length in the R code [8]. If W=, there could be checksum drives and drives in this system, i.e. b(x)={c, C, C3, C} and m(x)={d, D, D3,., D, D}. This research was supported by the Ministry of Economics, Taiwan under contract number 93EC7A86 Proceedings of the 5 th International Conference on Parallel and Distributed ystems (ICPAD'5) $. 5 IEEE
2 . Basic cheme in R Encoding The R code is a class of linear block [], so its computation must satisfy a linear property, that is to say, we can treat each symbol (drive) independently. In other words, any change in each drive would affect checksum symbols (drives) independently. Here, we deduce the linear property of our RRAID model using constant multipliers as follows: b(x) x m(x) mod g(x) k  x (m m x m x m k x ) mod g(x) (x m mod g(x)) (x m mod g(x)) n  (x m k  mod g(x)) m (x mod g(x)) m (x mod g(x)) n  m k  (x mod g(x)) () For that reason, the effect of each symbol (drive) could be computed separately to see how it works to checksum symbols first. Then the complete checksum symbols must be computed by accumulating the effects of all independent drives. Algorithm for Building Checksum ymbols (Encoding Procedure): tep. Premultiply (or shift) the message polynomial m(x) by x nk. tep. Construction of a staticchecksum table: Computing the item: [m i x nk mod g(x)], where each m i equals to multiplicative identity: in GF(x), would know what the effect is in each location (drive). tep 3. By using the table we built in tep, checksum symbols b(x) would be obtained by multiplying all values of staticchecksum table by the practical value of m(x). It nk can be presented as m ( x mod g( x)) m ( x n k mod g( x)).we are able to easily apply the constant multiplier to operate all computations after constructing the staticchecksum table, because all values of the constant table from tep are fixed.. Reduced taticchecksum Table Approach The encoding process is still very crucial due to operating too many XOR gates, even after constructing the previous lookup table. Therefore, a further work to reduce the number of required XOR gates during the encoding process is proposed in this paper. In case of GF( ), for example, applying the aspects of constant multipliers with only a variable could build a table, called constantmultiplier coefficient table ( abbreviated as CMC table), 3 as Table, where A a a a 3 a is the variable with coefficients a ~ a, and a ' ~ ' a are coefficients of A which is generated after being z multiplied by, where z = : Table. The constant multiplier coefficient table Now, if we take a generator polynomial: g ( x ) ( x )( x ), x ( ) x x x with a capability to tolerate up to two erasures, the checksums b(x)=c x+c could be shown as Table.In order to obtain the sixth column, which indicates as the number of XOR operations after the reduction, in Table, our approach consists of following steps: tep. For each location of the staticchecksum table, first, two values of checksums, C and C, are marked. And then in the CMC table, i.e. Table, each marked value could be represented as parts of a single row. tep. Comparing each part of the two rows, there might be some common terms in both rows, which we marked in tep. If so, we could reduce half of these common terms until there is no more common term between both rows. tep 3. Finally, the value of the sixth column in Table can be accumulated by the rest of XOR operations in each part of the two marked rows in Table. Proceedings of the 5 th International Conference on Parallel and Distributed ystems (ICPAD'5) $. 5 IEEE
3 Table. The reduced staticchecksum table with m(x) = = 9 = 7 C= ( * ) 7 ( * ) C= ( * ) 9 ( * ) Figure illustrates placement in our RRAID system, where C and C are checksum drives, DD3 are drives, and for each column, values of the second row are corresponding symbols to their binary values. For instance, to reduce location D in the tatic Checksum Table tep. C = 5 and C =, therefore, we marked the rows A* 5 and A*. tep. Through comparing the following two marked rows, as we can see, a3 a, a 3 a a a, and a a 3 are all the common terms between the two rows. Hence, after applying this approach, the total XOP operations could be reduced by 5 XOR operations. a, tep 3. The number of required XOR operations after processing step is 5 = 9. Besides, this scheme applies the shortened code method as well to achieve a better performance on coding process []. With this method, active drives are placed on some exact locations first. This disk location arrangement is based on which disk costs fewer XORgates after our reducing approach. That is to say, in the case of Table, to reach higher performance of computations, the locations must be arranged with the order, D8, D3, D, D9, etc. Let s assume that a message polynomial, m(x)= x, has to be stored into an empty RRAID in GF( ). And all in checksum drives could be computed as follows: B, from the location D, x of the Table, we could put * and * into two checksum drives separately. imilarly, by, from D5, x 6 in Table, the stored of the two checksum drives are * 7 and * 9. Therefore, values stored in the two checksumdrives after the above processes are: Figure. Data allocation in RRAID ystem with hortened Code method 3. RRAID Decoder In this section, two cases of decoding algorithm are discussed over GF( ), and they are carried out by a solving equations method, called crammer rule, directly. 3. ingle Failed Disk We take to be one of the roots with consecutive powers in our generator polynomial, i.e., g(x)=(x )(x ). Therefore, in the case of single failed disk condition, the decoding would be performed as easily as the parity scheme of the RAID level 5. From the equation: Failed Drive= =(All Normal Drives), recovering the failed disk needs only to do XOR operations in the rest of active disks together. Assuming that only the drive D has been erased as Figure. Figure. A RAID with only a failed disk The original information of D could be recovered as: D(,)=C(,)+C(,)+D(5,)=++= D(,3)=C(,3)+C(,3)+D(5,3)=++= D(,)=C(,)+C(,)+D(5,)=++= D(,)=C(,)+C(,)+D(5,)=++=. 3. Two Failed Disks at the ame Time In this case, in order to recover two disks which simultaneously fail, the decoding procedure in our scheme could be treated as solving a simultaneous linear equation Proceedings of the 5 th International Conference on Parallel and Distributed ystems (ICPAD'5) $. 5 IEEE
4 with two unknown variables. Here the matrix form of this equation is as follows. i j A B, where i and j are both the very positions of the two failed disks in this condition, and then syndrome: n kz cw is computed from all normal drives. k z Z By the crammer rule, the two variables, A and B, could be represented as follows respectively: A j, i j B. () i i j Furthermore, applying the same idea of the CMC table to build a table fulfilled with inverseelements of i j ( ) in advance would be more efficient. This table can avoid the extra cost of implementation on designing an ALU. In the circuit implementation: could be computed through XOR all the normal drives. For the syndrome, if the implementation of s hardware must be an VLI chip, it could share the same hardware with the encoder designed, both of them could share the same circuit of the multiplier.. Results and Comparisons In order to demonstrate how the encoding performance of our XORbased R algorithm is, we implement both CMC table and reduced staticchecksum table in GF( 8 ) to count the total number of XOR operators. Besides, a disk drive set {7,, 3, 7, 3, 9, 3,, 3} is our experimental example. Here, Figure 3 shows corresponding curves to Table 3. Table 3. # of XOR gates while encoding with the XORbased R, the conventional R and the EvenOdd # of Disk Drives Even Odd XOR based Reed olomon Conventional Reedolomon than what the XORbased R code does. However our approach indeed needs less XOR operators than the conventional R did in [3]. Figure 3. Curves plotted by the # of XOR gates while encoding with the XORbased R, the conventional R and the EvenOdd code. Moreover, here Figure shows that traditionally Even Odd need to be implemented by coding through a 3 dimension structure while our algorithm can be easily implemented through a dimension array structure. For Figure, if there is a byte changed, we need to deal with eight codewords from Page to Page 7. Therefore, the update might be an overhead to EvenOdd, but it does not happen in our scheme because we chose 8 bits to be the length of a codeword in the D array structure. Figure. The 3dimension structure of EvenOdd implementation ( n 8 ) In order to compare R with EvenOdd, we use EvenOdd proposed in [3] directly, which en (m) bytesdisk and m drives, and the estimated number of this EvenOdd code is 8 (m m ). That is there are m*(m) bytes will be encoded. In order to process the same amount of, we multiply the above by (m) directly, and the amount of is also equal to m*(m) bytes. The comparisons are listed as follows. From both Table 3 and Figure 3, we can see, the Even Odd perform a more efficient encoding capability Proceedings of the 5 th International Conference on Parallel and Distributed ystems (ICPAD'5) $. 5 IEEE
5 Figure 5. The number of XOR gates to encode m*(m) bytes Apparently, the calculating speed of R code is slower than EvenOdd. If there are 5 to 53 hard disks, the calculation amount is from. to.75. However, the main point mentioned here is that from the coding framework, Reed olomon proposed in the paper can process in parallel. That is because the encoding process of the proposed Reed olomon Code can calculate the effects of each drive to checksum drives respectively. Finally, we add the effects of each drive to checksum drives. Hence, the framework is suitable for parallel processing. Therefore, calculating process can be speeded up and time can be saved Odd Reed olomon MD code Yes Yes Calculating Medium Easy Complexity Encoding Mapping is easy and intuitive EvenOdd Mapping is done in tree dimension, hard to do addressing. Decoding Processes are simple large amount of buffers (memory) Flexibility Yes Yes Frameworks Parallel process Multiarray parallel process Table. A comparison sheet between R code and Even Update complexity Faulttolerant capability # of checksum > drives Design free Only 5. Hardware Implementation of This R RAID Codec In this section, we use Altera tratix FPGA Device (EPF8C5) to implement R Codec, Figure 6. is the Functional Diagram. fail fail_no Codec Encoder C C Decoder _A _A C _B C fail fail_no Figure 6. GF( ) Codec functional diagram 5. The Encoder Block In the encoder block, we create a module (a multiplication table ) that will help to generate the checksum ( C, C) as soon as there is any written to Hard Drive. D D D D3 C C_d_new C_d3_new C C_d_new C_d3_new 5. The decoder block Xor Xor C_d_new C_d_new C_d3_new C_d3_new Figure 7. Encoder block diagram The decoder block includes two sub modules: (a) FM_decoder: It is a tate Machine to control the path for even one or two Hard Drive errors. (b) path: The path is the function(pgz C C C C C C Proceedings of the 5 th International Conference on Parallel and Distributed ystems (ICPAD'5) $. 5 IEEE
6 algorithmic) to calculate the correct with C,C and other correct Hard Drive Data when there is any Hard Drive Data failed. _A : The correct of the failed hard drive; _A, _AB: The two correct of the failed hard drives. fail C C fail_no FM_decoder en_i en_a en_i en_j en_ en_ en_a en_b path _A _A _B _A _A _B Figure 8. Decoder block diagram During the FPGA implementation, we will use the EDA Tools in Table 5. Table 5. EDA Tools Tool Name Text Editor Modelim Quartus II RTL Coding Function Description Function and Timing imulatiom Altera FPGA Compiler for ynthesis, P&R,Timing Analyzing and Power Estimation wr en no[] no[] no[] no[3] [] [] [] [3] fail[] fail_no[] fail no[] fail[] fail_no[] fail no[3] C[] C[] C[] C[] C[] C[3] C[] C[3] Fig 9. FPGA floorplan Table 6. Pin name description Pin name IO Description I ystem Clock I Reset ignal I Write Enable ignal A[] A[] A[] A[3] A[] A[] A[] A[3] B[] B[] B[] B[3] During the FPGA Implementation, we will use the EDA Tools in Table5. The detail is described as follows: (a) RTL Coding: We use Verilog HDL to create all the design files. (b) Function imulation: Use Modelim to verify the Codec design function. (c) ynthesis and P&R: Use QuartusII to map the Verilog HDL format to Altera Atmos format netlist, and perform the Timing Analyzing. (d) Timing imulation: Use Modelim to verify the Codec design Timing. (e) Power Estimation: Use QuartusII to estimate the internal and IO power. I Write Done ignal I Hard Drive No for Write Data I Data for Write to Hard Drive fail I Hard Drive Fail ignal fail_no I Failed Hard Drive No C O Encoder Checksum Data C O Encoder Checksum Data _A O Decoder Recovery Data _A O Decoder Recovery Data _B O Decoder Recovery Data Proceedings of the 5 th International Conference on Parallel and Distributed ystems (ICPAD'5) $. 5 IEEE
7 Table 7 ummarizes chip characteristics and clarifies whether the structure owns the feature of power efficiency. Device : Table 7. Chip characteristics Total logic elements,5 Actual Time imulation End Time imulation Netlist ize EPF8C MHz ( period = 9.3 ns ) 9. us Total Number of Transitions Total Power 5.3 imulation Waveform 576 nodes.8 mw (A) Encoder If m(x) = Write D: ; D5:. Then results are C: ; C:, as shown in Figure Figure. Encoder simulation waveform (B) Decoder If 5th Hard Drive (Drive_NO:) fail, then the encoder will calculate the correct (DATA_A:) with all other Hard Drive and C, C. as shown in Figure. Figure. Decoder simulation waveform 6. Conclusion In this paper, we proposed an XORonly RRAID algorithm with two auxiliary tables, the CMC table and the reduced staticchecksum table, which not only constructing the XORbased R algorithm, but also speeding our scheme up. The above features also make those advanced RAID systems with our scheme be carried out by merely using regular industrial RAID level 5 controllers, which are capable of performing the XOR calculations very well. Therefore a lower cost controller could be applied in our RAID 6 algorithm in stead of a specific designed controller, which usually cost a lot, needed in other RAID 6 algorithms. We proposed an XORonly RRAID algorithm to optimize the coding circuits is suitable for RAID ystems applications where the accuracy, power, speed, and area issues are crucial. References [] Qin Xin, E.L. Miller, T. chwarz, D.D.E. Long,.A. Brandt, W. Litwin, Reliability mechanisms for very large storage systems, Mass torage ystems and Technologies, 3. (MT 3). Proceedings th IEEEth NAA Goddard Conference, 7 April 3, pp [] P.M. Chen, E.K. Lee, G.A. Gibson, R.H. Katz, and D.A. Patterson, RAID: HighPerformance, Reliability econdary torage, ACM Computing urveys, June 99, pp [3] M. Blaum, J. Brady, J. Bruck, and J. Menon, EVENODD: An efficient scheme for tolerating double disk failures in RAID architectures, IEEE Transactions on Comput., Feb. 995, pp. 9. [] Irving. Reed, Xuemin Chen, ErrorControl Coding For Data Networks, Kluwer Academic Publishers, 999. [5] L. Xu and J. Bruck, Xcode: MD array with optimal encoding, IEEE Transactions on Information Theory,, Jan. 999, pp [6] Telemetry Channel Coding, Recommendation for pace Data ystems tandards, CCD.B3, Blue Book, Issue 3, May 99. [7] J.. Plank, Correction to the 997 Tutorial on Reed olomon Coding, Technical Report UTC35, University of Tennessee, April, 3. [8] J.. Plank, A tutorial on Reedolomon coding for faulttolerance in RAIDlike systems. oftware Practice& Experience, eptember 997, 7(9):995. [9] T.K. Truong, J.H. Jeng, T.C. Cheng, A New Decoding Algorithm for Correcting Both Erasures and Errors of Reedolomon Codes, IEEE Transactions on Communications, March 3, pp [] D.V. arwate, N. R. hanbhag, Highpeed Architectures for Reedolomon Decoders, IEEE Transactions on VLI,, pp [] Lihao Xu, Highly Available Distributed torage ystems, Ph.D. Dissertation, 999. Proceedings of the 5 th International Conference on Parallel and Distributed ystems (ICPAD'5) $. 5 IEEE
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