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
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Copyright Notice
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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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