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Network Working Group                                          D. McGrew
Internet-Draft                                       Cisco Systems, Inc.
Intended status: Informational                                P. Patnala
Expires: September 4, 2010                                    Consultant
                                                               A. Hoenes
                                                                  TR-Sys
                                                           March 3, 2010


                        Threshold Secret Sharing
                        draft-mcgrew-tss-03.txt

Abstract

   Threshold Secret Sharing (TSS) provides a way to generate N shares
   from a value, so that any M of those shares can be used to
   reconstruct the original value, but any M-1 shares provide no
   information about that value.  This method can provide shared access
   control on key material and other secrets that must be strongly
   protected.

   This note defines a threshold secret sharing method based on
   polynomial interpolation in GF(256) and a format for the storage and
   transmission of shares.  It also provides usage guidance, describes
   how to test an implementation, and supplies test cases.

Status of this Memo

   This Internet-Draft is submitted to IETF in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
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   This Internet-Draft will expire on September 4, 2010.



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Copyright Notice

   Copyright (c) 2010 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (http://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.  Code Components extracted from this document must
   include Simplified BSD License text as described in Section 4.e of
   the Trust Legal Provisions and are provided without warranty as
   described in the BSD License.


Table of Contents

   1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  3
     1.1.  Conventions Used In This Document  . . . . . . . . . . . .  3
   2.  Operations . . . . . . . . . . . . . . . . . . . . . . . . . .  4
     2.1.  Create Shares  . . . . . . . . . . . . . . . . . . . . . .  4
     2.2.  Reconstruct Secret . . . . . . . . . . . . . . . . . . . .  4
   3.  Polynomial Interpolation over GF(256)  . . . . . . . . . . . .  5
     3.1.  Field Representation . . . . . . . . . . . . . . . . . . .  5
     3.2.  Share Generation . . . . . . . . . . . . . . . . . . . . .  7
     3.3.  Secret Reconstruction  . . . . . . . . . . . . . . . . . .  8
   4.  Robust Threshold Secret Sharing  . . . . . . . . . . . . . . . 10
     4.1.  RTSS Data Format . . . . . . . . . . . . . . . . . . . . . 10
   5.  Error Correction and Data Recovery . . . . . . . . . . . . . . 13
     5.1.  Data Recovery  . . . . . . . . . . . . . . . . . . . . . . 13
     5.2.  Error Correction . . . . . . . . . . . . . . . . . . . . . 13
     5.3.  A Repetition Code  . . . . . . . . . . . . . . . . . . . . 15
   6.  Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
   7.  Design and Rationale . . . . . . . . . . . . . . . . . . . . . 18
   8.  Testing  . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
   9.  Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 20
   10. Security Considerations  . . . . . . . . . . . . . . . . . . . 21
   11. IANA Considerations  . . . . . . . . . . . . . . . . . . . . . 22
   12. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 23
   13. References . . . . . . . . . . . . . . . . . . . . . . . . . . 24
     13.1. Normative References . . . . . . . . . . . . . . . . . . . 24
     13.2. Informative References . . . . . . . . . . . . . . . . . . 24
   Appendix A.  Mathematical Background . . . . . . . . . . . . . . . 25
   Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 26






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1.  Introduction

   Threshold secret sharing (TSS) provides a way to generate N shares
   from a value, so that any M of those shares can be used to
   reconstruct the original value, but any M-1 shares provide no
   information about that value.  This method does not rely on any
   assumptions about the complexity of solving a particular
   computational problem (such as factoring); it is information-
   theoretically secure.  Each share is slightly longer than the
   original secret.

   In the context of secret sharing, the word "share" means a part of
   something, and "sharing" means the act of breaking up into parts.
   Readers may be confused if they think of "sharing" as meaning "giving
   to or possessing with others".

   TSS is especially useful whenever there is a need to ensure the
   availability of a secret, yet there is a simultaneous need to reduce
   the risk of compromise of the secret.  By dividing the secret into
   multiple shares, and distributing each share to a different trusted
   entity, TSS reduces that risk while providing for the availability of
   the secret.  At the time that the secret is divided into shares, the
   threshold defining a number of shares that are needed to reconstruct
   the secret is set.

   TSS can be applied to any secret key, such as one used to encrypt
   data at rest, or to any private key, such as the signing key used by
   a certificate authority.  It can be used to create a "backup" copy of
   a key, to protect against the loss or corruption of an "active" copy
   of the key.  Alternatively, TSS can be applied to a key, and then the
   original key can be deleted, as a means of enforcing shared access
   control on that key.

1.1.  Conventions Used In This Document

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
   document are to be interpreted as described in [RFC2119].













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2.  Operations

   A threshold secret sharing system provides two operations: one that
   creates a set of shares given a secret, and one that reconstructs the
   secret, given a set of shares.  This section defines the inputs and
   outputs of these operations.  The following sections describe the
   details of TSS based on a polynomial interpolation in GF(256).

2.1.  Create Shares

   This operation takes an octet string S, whose length is L octets, and
   a threshold parameter M, and generates a set of N shares, any M of
   which can be used to reconstruct the secret.

   The secret S is treated as an unstructured sequence of octets.  It is
   not expected to be null-terminated.  The number of octets in the
   secret may be anywhere from zero up to 65,534 (that is, two less than
   2^16).

   The threshold parameter M is the number of shares that will be needed
   to reconstruct the secret.  This value may be any number between one
   and 255, inclusive.

   The number of shares N that will be generated MUST be between the
   threshold value M and 255, inclusive.  The upper limit is particular
   to the TSS algorithm specified in this document.

   If the operation can not be completed successfully, then an error
   code should be returned.

2.2.  Reconstruct Secret

   The reconstruct operation reconstructs the secret from a set of
   shares.

   The number of shares N must be provided as a parameter.

   The only other parameter is the list of shares themselves.  The
   shares should be treated as unstructured octet strings.

   If the operation could be completed successfully, then the secret
   value will be returned.

   If the operation can not be completed successfully, then an error
   code should be returned.






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3.  Polynomial Interpolation over GF(256)

   A finite field is a set of elements with associated addition,
   multiplication, subtraction, and division operations.  Each of those
   operations acts on elements in the field, and returns an element in
   the field.  This specification uses the field GF(256), and each
   element is represented as a single octet.  There are many possible
   ways to represent a finite field; below we define the field
   arithmetic operations as having inputs and outputs that are octets.
   This fixes a particular representation, without explicitly defining
   it, and it avoids the issue of the bit-representation of octets.  In
   this representation, the zero field element is the zero octet, and
   the unity field element is 0x01 (hexadecimal).

3.1.  Field Representation

   Each element of the field GF(256) is represented as an octet.  In the
   following, each octet is represented as a hexadecimal number with a
   leading "0x", as in ANSI/ISO C. The representation of the finite
   field that we use is defined in terms of the addition, subtraction,
   multiplication, and division operations.  We define these operations
   as taking two octets as input and returning a single octet as output.
   In order to distinguish GF(256) arithmetic from integer arithmetic,
   we denote addition and multiplication in GF(256) as (+) and (*),
   respectively.  We also refer to the summation and product operations
   in GF(256) as GF_SUM and GF_PRODUCT, respectively.

   The multiplication in GF(256) and its inverse operation (division)
   are defined in terms of two tables, the EXP table (Figure 1) and the
   LOG table (Figure 2), which define the exponential function and the
   logarithmic function, respectively.  The ith elements of these tables
   are denoted as EXP[i] and LOG[i].  LOG takes a non-zero field element
   as input, and returns an integer, and EXP takes an integer and
   returns a field element.

   The addition operation returns the bitwise exclusive-or of its
   operands.  The subtraction operation is identical, because the field
   has characteristic two.

   The multiplication operation takes two elements X and Y as input and
   proceeds as follows.  If either X or Y is equal to 0x00, then the
   operation returns 0x00.  Otherwise, the value EXP[ (LOG[X] + LOG[Y])
   modulo 255] is returned.

   The division operation takes a dividend X and a divisor Y as input
   and computes X divided by Y as follows.  If X is equal to 0x00, then
   the operation returns 0x00.  If Y is equal to 0x00, then the input is
   invalid, and an error condition occurs.  Otherwise, the value EXP[



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   (LOG[X] - LOG[Y]) modulo 255] is returned.

   The operation of raising a field element X to a power i, where i is a
   positive integer, is denoted as X^i, and it consists of multiplying X
   by itself i times.

         0x01, 0x03, 0x05, 0x0f, 0x11, 0x33, 0x55, 0xff,
         0x1a, 0x2e, 0x72, 0x96, 0xa1, 0xf8, 0x13, 0x35,
         0x5f, 0xe1, 0x38, 0x48, 0xd8, 0x73, 0x95, 0xa4,
         0xf7, 0x02, 0x06, 0x0a, 0x1e, 0x22, 0x66, 0xaa,
         0xe5, 0x34, 0x5c, 0xe4, 0x37, 0x59, 0xeb, 0x26,
         0x6a, 0xbe, 0xd9, 0x70, 0x90, 0xab, 0xe6, 0x31,
         0x53, 0xf5, 0x04, 0x0c, 0x14, 0x3c, 0x44, 0xcc,
         0x4f, 0xd1, 0x68, 0xb8, 0xd3, 0x6e, 0xb2, 0xcd,
         0x4c, 0xd4, 0x67, 0xa9, 0xe0, 0x3b, 0x4d, 0xd7,
         0x62, 0xa6, 0xf1, 0x08, 0x18, 0x28, 0x78, 0x88,
         0x83, 0x9e, 0xb9, 0xd0, 0x6b, 0xbd, 0xdc, 0x7f,
         0x81, 0x98, 0xb3, 0xce, 0x49, 0xdb, 0x76, 0x9a,
         0xb5, 0xc4, 0x57, 0xf9, 0x10, 0x30, 0x50, 0xf0,
         0x0b, 0x1d, 0x27, 0x69, 0xbb, 0xd6, 0x61, 0xa3,
         0xfe, 0x19, 0x2b, 0x7d, 0x87, 0x92, 0xad, 0xec,
         0x2f, 0x71, 0x93, 0xae, 0xe9, 0x20, 0x60, 0xa0,
         0xfb, 0x16, 0x3a, 0x4e, 0xd2, 0x6d, 0xb7, 0xc2,
         0x5d, 0xe7, 0x32, 0x56, 0xfa, 0x15, 0x3f, 0x41,
         0xc3, 0x5e, 0xe2, 0x3d, 0x47, 0xc9, 0x40, 0xc0,
         0x5b, 0xed, 0x2c, 0x74, 0x9c, 0xbf, 0xda, 0x75,
         0x9f, 0xba, 0xd5, 0x64, 0xac, 0xef, 0x2a, 0x7e,
         0x82, 0x9d, 0xbc, 0xdf, 0x7a, 0x8e, 0x89, 0x80,
         0x9b, 0xb6, 0xc1, 0x58, 0xe8, 0x23, 0x65, 0xaf,
         0xea, 0x25, 0x6f, 0xb1, 0xc8, 0x43, 0xc5, 0x54,
         0xfc, 0x1f, 0x21, 0x63, 0xa5, 0xf4, 0x07, 0x09,
         0x1b, 0x2d, 0x77, 0x99, 0xb0, 0xcb, 0x46, 0xca,
         0x45, 0xcf, 0x4a, 0xde, 0x79, 0x8b, 0x86, 0x91,
         0xa8, 0xe3, 0x3e, 0x42, 0xc6, 0x51, 0xf3, 0x0e,
         0x12, 0x36, 0x5a, 0xee, 0x29, 0x7b, 0x8d, 0x8c,
         0x8f, 0x8a, 0x85, 0x94, 0xa7, 0xf2, 0x0d, 0x17,
         0x39, 0x4b, 0xdd, 0x7c, 0x84, 0x97, 0xa2, 0xfd,
         0x1c, 0x24, 0x6c, 0xb4, 0xc7, 0x52, 0xf6, 0x00

     Figure 1: The EXP table.  The elements are to be read from top to
     bottom and left to right.  For example, EXP[0] is 0x01, EXP[8] is
    0x1a, and so on. Note that the EXP[255] entry is present only as a
         placeholder, and is not actually used in any computation.








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            0,    0,   25,    1,   50,    2,   26,  198,
           75,  199,   27,  104,   51,  238,  223,    3,
          100,    4,  224,   14,   52,  141,  129,  239,
           76,  113,    8,  200,  248,  105,   28,  193,
          125,  194,   29,  181,  249,  185,   39,  106,
           77,  228,  166,  114,  154,  201,    9,  120,
          101,   47,  138,    5,   33,   15,  225,   36,
           18,  240,  130,   69,   53,  147,  218,  142,
          150,  143,  219,  189,   54,  208,  206,  148,
           19,   92,  210,  241,   64,   70,  131,   56,
          102,  221,  253,   48,  191,    6,  139,   98,
          179,   37,  226,  152,   34,  136,  145,   16,
          126,  110,   72,  195,  163,  182,   30,   66,
           58,  107,   40,   84,  250,  133,   61,  186,
           43,  121,   10,   21,  155,  159,   94,  202,
           78,  212,  172,  229,  243,  115,  167,   87,
          175,   88,  168,   80,  244,  234,  214,  116,
           79,  174,  233,  213,  231,  230,  173,  232,
           44,  215,  117,  122,  235,   22,   11,  245,
           89,  203,   95,  176,  156,  169,   81,  160,
          127,   12,  246,  111,   23,  196,   73,  236,
          216,   67,   31,   45,  164,  118,  123,  183,
          204,  187,   62,   90,  251,   96,  177,  134,
           59,   82,  161,  108,  170,   85,   41,  157,
          151,  178,  135,  144,   97,  190,  220,  252,
          188,  149,  207,  205,   55,   63,   91,  209,
           83,   57,  132,   60,   65,  162,  109,   71,
           20,   42,  158,   93,   86,  242,  211,  171,
           68,   17,  146,  217,   35,   32,   46,  137,
          180,  124,  184,   38,  119,  153,  227,  165,
          103,   74,  237,  222,  197,   49,  254,   24,
           13,   99,  140,  128,  192,  247,  112,    7

    Figure 2: The LOG table.   The elements are to be read from top to
    bottom and left to right.  For example, LOG[1] is 0, LOG[8] is 75,
        and so on.  Note that the LOG[0] entry is present only as a
         placeholder, and is not actually used in any computation.

3.2.  Share Generation

   We first define how to share a single octet.

   The function f takes as input a single octet X that is not equal to
   0x00, and an array A of M octets, and returns a single octet.  It is
   defined as

      f(X, A) =  GF_SUM A[i] (*) X^i
                 i=0,M-1



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   Because the GF_SUM summation takes place over GF(256), each addition
   uses the exclusive-or operation, and not integer addition.  Note that
   the successive values of X^i used in the computation of the function
   f can be computed by multiplying a value by X once for each term in
   the summation.

   To create N shares from a secret, with a threshold of M, the
   following procedure, or any equivalent method, is used:

      For each share, a distinct Share Index is generated.  Each Share
      Index is an octet other than the all-zero octet.  All of the Share
      Indexes used during a share generation process MUST be distinct.

      Each share is initialized to the Share Index associated with that
      share.

      For each octet of the secret, the following steps are performed.
      An array A of M octets is created, in which the array element A[0]
      contains the octet of the secret, and the array elements A[1],
      ..., A[M-1] contain octets that are selected independently and
      uniformly at random.  For each share, the value of f(X,A) is
      computed, where X is the Share Index of the share, and the
      resulting octet is appended to the share.

   After the procedure is done, each share contains one more octet than
   does the secret.  The share format can be illustrated as

        +---------+---------+---------+---------+---------+
        |    X    | f(X,A)  | f(X,B)  | f(X,C)  |   ...   |
        +---------+---------+---------+---------+---------+

   where X is the Share Index of the share, and A, B, and C are arrays
   of M octets; A[0] is equal to the first octet of the secret, B[0] is
   equal to the second octet of the secret, and so on.

3.3.  Secret Reconstruction

   We define the function L_i (for i from 0 to M-1, inclusive) that
   takes as input an array U of M pairwise distinct octets, and is
   defined as

                                U[j]
      L_i(U) = GF_PRODUCT   -------------
               j=0,M-1, j!=i  U[j] (+) U[i]

   Here the product runs over all of the values of j from 0 to M-1,
   excluding the value i.  (This function is equal to ith Lagrange
   function, evaluated at zero.)  Note that the denominator in the above



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   expression is never equal to zero because U[i] is not equal to U[j]
   whenever i is not equal to j.

   We denote the interpolation function as I. This function takes as
   input two arrays U and V, each consisting of M octets, and returns a
   single octet; it is defined as

      I(U, V) =  GF_SUM  L_i(U) (*) V[i].
                 i=0,M-1

   To reconstruct a secret from a set of shares, the following
   procedure, or any equivalent method, is used:

      If the number of shares provided as input to the secret
      reconstruction operation is greater than the threshold M, then M
      of those shares are selected for use in the operation.  The method
      used to select the shares can be arbitrary.

      If the shares are not equal length, then the input is
      inconsistent.  An error should be reported, and processing must
      halt.

      The output string is initialized to the empty (zero-length) octet
      string.

      The octet array U is formed by setting U[i] equal to the first
      octet of the ith share.  (Note that the ordering of the shares is
      arbitrary, but must be consistent throughout this algorithm.)

      The initial octet is stripped from each share.

      If any two elements of the array U have the same value, then an
      error condition has occurred; this fact should be reported, then
      the procedure must halt.

      For each octet of the shares, the following steps are performed.
      An array V of M octets is created, in which the array element V[i]
      contains the octet from the ith share.  The value of I(U, V) is
      computed, then appended to the output string.

      The output string is returned.

   After the procedure is done, the string that is returned contains one
   fewer octet than do the shares.







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4.  Robust Threshold Secret Sharing

   A robust TSS system, or RTSS, is one that provides security even when
   one or more of the shares that are provided to the reconstruction
   algorithm may be crafted by a malicious adversary.  In addition, an
   RTSS system will detect unintentional corruption of the shares.

   We provide robustness by adding a pre-processing step to the TSS
   share generation step, and a post-processing step to the TSS secret
   reconstruction step.  The pre-processing consists of taking the
   secret S, then appending a hash H(S) to it.  The post-processing step
   consists of verifying that the reconstructed secret has the form S ||
   H(S), where the symbol || denotes the concatenation operation.  The
   hash function must be collision-resistant; all RTSS implementations
   MUST support the SHA-256 hash algorithm [SHS].

   If the robust reconstruction operation fails, and the number of
   shares that are available is greater than the threshold, then the
   operation MAY be tried on a different set of shares.

   An RTSS system can perform an additional operation that verifies the
   validity of a set of shares.  This operation has the same inputs as
   the Reconstruct operation.  Its output consists of an indication
   whether or not the secret could be reconstructed, but the secret
   itself is not returned.  This operation may be useful in a situation
   in where the availability of a secret must be verified, for example,
   as part of an audit.

4.1.  RTSS Data Format

   We use a data format with the following fields, in order:

   Identifier.  This field contains 16 octets.  It identifies the secret
      with which a share is associated.  All of the shares associated
      with a particular secret MUST use the same value Identifier.  When
      a secret is reconstructed, the Identifier fields of each of the
      shares used as input MUST have the same value.  The value of the
      Identifier should be chosen so that it is unique, but the details
      on how it is chosen are out of scope of this document.

   Hash Algorithm Identifier.  This field contains a single octet that
      indicates the hash function used in the RTSS processing, if any.
      A value of zero indicates that no hash algorithm was used, no hash
      was appended to the secret, and no RTSS check should be performed
      after the reconstruction of the secret.  Other values are defined
      in the table below.





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   Threshold.  This field contains a single octet that indicates the
      number of shares required to reconstruct the secret.  This field
      MUST be checked during the reconstruction process, and that
      process MUST halt and return an error if the number of shares
      available is fewer than the value indicated in this field.

   Share Length.  This field is two octets long.  It contains the number
      of octets in the Share Data field, represented as an unsigned
      integer in network byte order.

   Share Data.  This field has a length that is a variable number of
      octets.  It contains the actual share data.

   This format is illustrated in Figure 3.

        0                   1                   2                   3
        0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
       +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
       |                                                               |
       |                          Identifier                           |
       |                                                               |
       |                                                               |
       +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
       | Hash Alg. Id. |   Threshold   |         Share Length         |
       +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
       :                                                               :
       :                          Share Data                           :
       :                                                               :
       +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

                          Figure 3: Share Format.

   The correspondence between the Hash Algorithm Identifier field and
   the hash algorithm used in RTSS is defined by the table below.  Each
   hash function outputs a fixed number of octets; the length of the
   output of each hash is indicated in the table.















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     +-----------------+---------------------------+-----------------+
     | Hash Algorithm  | Hash Algorithm Identifier | Length (octets) |
     +-----------------+---------------------------+-----------------+
     | NULL_HASH       |                         0 |               0 |
     |                 |                           |                 |
     | SHA-1 [SHS]     |                         1 |              20 |
     |                 |                           |                 |
     | SHA-256 [SHS]   |                         2 |              32 |
     |                 |                           |                 |
     | RESERVED        |                     3-127 |  not applicable |
     |                 |                           |                 |
     | Vendor specific |                   128-255 |  not applicable |
     +-----------------+---------------------------+-----------------+






































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5.  Error Correction and Data Recovery

   TSS and RTSS are suitable for the protection of long-term key
   material.  In such applications, it is highly desirable to provide
   protection against the accidental corruption of the shares.  This
   section defines data formats that can be used to protect shares.
   These formats are optional extensions to the basic TSS and RTSS
   systems.

5.1.  Data Recovery

   To protect against the corruption of the filesystem that is holding
   the shares, a "magic number" can be used as the initial part of the
   share data format [FILESIG].  A magic number is a constant data
   string that is chosen arbitrarily, but which is unlikely to appear in
   other contexts, and thus can be used to recognize a data format when
   it appears in an arbitrary data stream.  The use of a magic number in
   the data format for a share greatly simplifies the task of finding a
   share after a filesystem has been corrupted.

   The 8-octet magic number f628f91b52023d11 (hexadecimal) SHOULD be
   used.  The number was selected randomly from a uniform distribution.

5.2.  Error Correction

   To protect against data corruption in the underlying media, an error-
   correcting code (ECC) can be used.  An ECC system consists of an
   encoding function, which maps the data to a codeword, and a decoding
   function, which maps a (possibly corrupted) codeword to the data.
   The simplest such code is a repetition code, in which multiple copies
   of the data are stored.  In this specification, all ECCs must be
   systematic, that is, the data must appear as the initial bytes of the
   codeword.  This property allows an implementation of the ECC to avoid
   the implementation of the full decoding algorithm.

   We use a data format that incorporates the following fields, in
   order:

   Encoding Type.  This field is four octets long.  It contains an
      unsigned integer in network byte order that denotes the type of
      the encoding, i.e. the algorithm that was used during the encoding
      process.

   Data Length.  This field is four octets long.  It contains an
      unsigned integer in network byte order that denotes the number of
      octets in the Data field.





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   Redundancy Length.  This field is four octets long.  It contains an
      unsigned integer in network byte order that denotes the number of
      octets in the Redundancy field.

   Data.  This field has a length that is a variable number of octets,
      which is indicated by the Data Length field.  It contains the data
      that is intended to be conveyed by the code.  If no data
      corruption has occurred, then this field will contain the data
      that was originally encoded.

   Redundancy.  This field has a length that is a variable number of
      octets, which is indicated by the Redundancy Length field.  It
      contains information that can be used to check whether or not
      there are any errors in the Data field, and to correct some errors
      that may have occurred.

   This format is illustrated in Figure 4.

                    +--------------------------------+
                    |         Encoding Type          |
                    |           (4 octets)           |
                    +--------------------------------+
                    |          Data Length           |
                    |           (4 octets)           |
                    +--------------------------------+
                    |       Redundancy Length        |
                    |           (4 octets)           |
                    +--------------------------------+
                    |                                |
                    ~             Data               ~
                    |   (variable number of octets)  |
                    |                                |
                    +--------------------------------+
                    |                                |
                    ~          Redundancy            ~
                    |   (variable number of octets)  |
                    |                                |
                    +--------------------------------+

                    Figure 4: Error Correction Format.

   If a code has a free parameter, the value of that parameter MUST be
   inferable from the values of the Data Length and Redundancy Length
   fields.







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5.3.  A Repetition Code

   This section defines a format for a repetition code, which is a
   particular error correcting code that is conceptually simple and easy
   to implement.

   The value of the Encoding Type field is equal to 0000001
   (hexadecimal).

   The Redundancy field contains R copies of the Data field, where R is
   an even number.  The Redundancy Length is equal to the Data Length
   times R. The value of R MAY be equal to zero, in which case no error
   detection or correction is possible (but implementation is simple).
   The value of R SHOULD be at least two.

   For example, if the data that is encoded is equal to 68656c6c6f
   (hexadecimal), then the ECF data with R=2 would be

      <- ET -><- DL -><- RL -><- Data -><--- Redundancy --->
      00000001000000050000000a68656c6c6f68656c6c6f68656c6c6f

   To check the Data field for errors, that field should be compared
   with each of its copies in the redundancy field.

   The Repetition Code can be decoded by using majority-logic decoding.
   Considering both the Data and Redundancy fields, there are R+1
   (possibly corrupted) copies of the original data, where R+1 is an odd
   number.  The decoding process independently considers each octet of
   the Data field, and the corresponding octets of the copies that
   appear in the Redundancy field.  That is, the ith octet of the Data,
   plus octets i, L+i, 2L+i, ... , RL+i, are analyzed independent from
   all other octets, where L is the value of the Data Length field.  The
   following algorithm is applied to these octets.  The binary
   representation of each octet is considered.  For each bit in that
   representation, if more of the copies have a "1" in that position
   than have a "0" in that position, then that position is decoded to
   the value "1"; otherwise, it is decoded to "0".  This process is
   repeated for all of the bit position.  After all of the bits in the
   octet have been decoded, the value of the ith octet in the output of
   the decoding algorithm is computed, using the same binary
   representation as before.

   For example, if the data that was encoded in the previous example was
   corrupted to the value

      <- ET -><- DL -><- RL -><- Data -><--- Redundancy --->
      00000001000000050000000a68656c6c2f68656c6cef68656c6c6f
                                      **        **        **



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   then decoding would proceed as follows.  The fifth octet of the Data
   field is equal to 2f, while the fifth and tenth octets of the
   Redundancy field are equal to ef and 6f, respectively.  Using a bit
   representation with the most significant bit on the left, the octets
   and the "majority" octet are as follows:

                                 hex   binary
       octet from Data           2f    00101111
       octet from first copy     ef    11101111
       octet from second copy    6f    01101111
       ----------------------------------------
       majority                  6f    01101111

   Thus the fifth octet in the output of the decoding algorithm will be
   6f.




































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6.  Format

   This section summarizes the order of processing for when secret
   sharing is performed using the facilities for robustness (RTSS),
   error correction (ECC), and data recovery (Magic Number), and
   clarifies the relationships between data formats.  This processing
   can be viewed as a layered model, as illustrated in Figure 5.  (Note
   that we have not adhered to a strictly layered model, for the sake of
   simplicity, since the format defined by RTSS is used after the shares
   are generated.)

   When RTSS is used, it is applied to the secret before the sharing
   operation (and is removed from the secret after the reconstruction
   operation).  The RTSS data format MUST be used.

   When ECC is used, it is applied to the RTSS data after the sharing
   operation, so that the ECC Data field contains the entire RTSS Data
   Format.

   When a Magic Number is used, it is added after the ECC formatting is
   done, and it is prepended to the Error Correction Format.

                   Secret                       Secret
                      |                            ^
                      v                            |
             +------------------+         +------------------+
             |   Append Hash    |         |   Verify Hash    |
             +------------------+         +------------------+
                      |                            |
             +------------------+         +------------------+
             | Generate Shares  |         |Reconstruct Secret|
             +------------------+         +------------------+
                      |                            |
             +------------------+         +------------------+
             |   ECC Encoding   |         |   ECC Decoding   |
             +------------------+         +------------------+
                      |                            |
             +------------------+         +------------------+
             | Add Magic Number |         |Strip Magic Number|
             +------------------+         +------------------+
                      |                            ^
                      v                            |
                    Shares ----------------> Shares

                 Figure 5: The combined processing model.






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7.  Design and Rationale

   In this implementation, the secret and the shares are octet strings.
   Each octet is treated as an element of the finite field GF(256).  The
   share-generation algorithm is applied to each octet of the secret
   independently.  Similarly, the octets are treated independently
   during the reconstruction of the secrets from the shares.

   Shamir's original description treats the secret as a large integer
   modulo a large prime number [shamir].  The advantages of using a
   vector over GF(256) are that the computations are more efficient and
   the encoding is simpler.  Multiplication and inversion over GF(256)
   can be done with two table lookups and two exors, using two fixed
   tables of 256 bytes each.  One limitation of the GF(256) approach is
   that the number of shares that can be generated cannot be greater
   than 255; this limitation is unlikely to be important in practice,
   since fewer than ten shares are typically used.

   The reconstruction of the secret is done using Lagrange interpolation
   polynomials.  This method is simple and easily tested.  For large
   thresholds, this method is less efficient than an optimal method
   would be.  However, performance is still good, and it is expected
   that the reconstruction of the secret will not be a performance-
   critical operation.



























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8.  Testing

   As with every crypto algorithm, it is essential to test an
   implementation of TSS or RTSS for correctness.  This section provides
   guidance for such testing.

   The Secret Reconstruction algorithm can be tested using Known Answer
   Tests (KATs).  Test cases are provided in Section 9.

   The Share Generation algorithm cannot be directly tested using a KAT.
   It can be indirectly tested by generating secret values uniformly at
   random, then applying the Share Generation process to them to
   generate a set of shares, then applying the Share Reconstruction
   algorithm to the shares, then finally comparing the reconstructed
   secret to the original secret.  Implementations SHOULD perform this
   test, using a variety of thresholds and secret lengths.

   The Share Index (the initial octet of each share) can never be equal
   to zero.  This property SHOULD be tested.

   The random source must be tested to ensure that it has high min-
   entropy.





























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9.  Test Cases

   This section provides test cases that can be used to validate an
   implementation of the Secret Reconstruction algorithm.  All values
   are in hexadecimal.

   algorithm -  The algorithm used in the test case.

   secret -  The secret value to be split into shares.

   threshold -  The number of shares required to reconstruct a secret;
      above, this value is associated with the variable M.

   num. shares -  The number of shares included in the example; above,
      this value is associated with the variable N.

   share index -  A share index.  Each test case has multiple distinct
      share values, and each share is associated with a distinct share
      index.

   share -  A share value, which corresponds to the share index value
      immediately above it.


         algorithm = TSS
            secret = 7465737400
     threshold (M) = 2
   num. shares (N) = 2
       share index = 1
             share = B9FA07E185
       share index = 2
             share = F5409B4511



















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10.  Security Considerations

   It is crucial for security that the source of randomness used in the
   share generation process by cryptographically strong; it MUST be
   suitable for generating cryptographic keys.  [RFC4086] provides
   guidance on the selection and implementation of random sources.

   A TSS implementation SHOULD be tested as described in Section 8.

   The confidentiality of the shares generated by TSS should be
   protected, since the exposure of too many shares will undermine the
   security of the system.  Note that, in this regard, share values are
   more comparable to secret keys than to ciphertext.






































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11.  IANA Considerations

   This document has no actions for IANA.
















































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12.  Acknowledgements

   Thanks to Brian Weis and Jack Lloyd for constructive feedback.
















































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13.  References

13.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119, March 1997.

   [RFC4086]  Eastlake, D., Schiller, J., and S. Crocker, "Randomness
              Requirements for Security", BCP 106, RFC 4086, June 2005.

   [SHS]      "FIPS 180-3: Secure Hash Standard,", Federal Information
              Processing Standard (FIPS) http://csrc.nist.gov/
              publications/fips/fips180-2/fips180-3.pdf, 2008.

13.2.  Informative References

   [FILESIG]  Kessler, G., "File Signatures Table", Web
              page http://www.garykessler.net/library/file_sigs.html,
              2007.

   [POLY]     Seroussi, G., "Table of Low-Weight Binary Irreducible
              Polynomials", Hewlett-Packard Computer Systems Laboratory
              Technical Report HPL-98-135, 1998.

   [shamir]   Shamir, A., "How to share a secret", Communications of the
              ACM (22): 612-613, 1979.

























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Appendix A.  Mathematical Background

   In abstract algebra, a finite field is an algebraic structure for
   which the operations of addition, subtraction, multiplication and
   division are defined and satisfy certain axioms.

   The field GF(256) has exactly 256 elements in it.  There is only one
   field with that number of elements, but there are many different ways
   in which the elements of the field can be represented.  This document
   uses a polynomial representation in which the field polynomial is the
   unique irreducible polynomial with minimum weight of degree 8 over
   GF(2) [POLY], hence it is the 'canonical' choice for a polynomial
   base representation of GF(256).  This field representation is also
   used by the Advanced Encryption Standard (AES).





































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Authors' Addresses

   David A. McGrew
   Cisco Systems, Inc.
   510 McCarthy Blvd.
   Milpitas, CA  95035
   US

   Email: mcgrew@cisco.com
   URI:   http://www.mindspring.com/~dmcgrew/dam.htm


   Praveen Patnala
   Consultant

   Email: praveenpatnala@yahoo.com


   Alfred Hoenes
   TR-Sys
   Gerlinger Str. 12
   Ditzingen  D-71254
   Germany

   Email: ah@TR-Sys.de


























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