Threshold Secret Sharing

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 posessing 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.

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 .

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 ouputs of these operations. The following sections describe the details of TSS based on a polynomial interpolation in GF(256).

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 TSS_MAX_SECRET_LENGTH. 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. 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 could not be completed successfully, then an error code should be returned.

The reconstruct operation reconstructs the secret from a set 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, then the secret value will be returned. If the operation could not be completed successfully, then an error code should be returned.

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.

The elements of the field GF(256) are represented as octets. 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. These operations are defined in terms of two tables, the EXP table () and the LOG table (), which define the exponential function and the logarithmic function, respectively. The ith element of these tables are denoted as EXP[i] and LOG[i]. The addition operation returns the bitwise exclusive-or of its operands. The subtraction operation is identical, because the field has character 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[ LOG[X] - LOG[Y] modulo 255] is returned.

We first define how to share a single octet. The function f takes as input a single octet X and an array A of D+1 octets, and returns a single octet. It is defined as

To create N shares from a secret, with a threshold of M, the following procedure, or any equivalent method, is used: Each share is initialized to the empty (zero-length) octet string. For each share, a distinct x-value is generated. Each x-value is an octet other than the all-zero octet. All of the x-values used during a share generation process MUST be distinct. For each share, the x-value associated with that share is prepended to the share. For each octet of the secret, the following steps are performed. An array A of M+1 octets is created, in which the array element A[0] contains the octet of the secret, and the array elements A[1], ..., A[M] 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 x-value 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

where X is the x-value of the share, and A, B, and C are arrays of M+1 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.

We denote the ith Lagrange function as L_i. This function takes as input a single octet X, and an array U of M octets, and is defined as

Here the product runs over all of the values of j from 0 to M, excluding the value i. We denote the interpolation function as I. This function takes as input a single octet Z, two arrays U and V, each consisting of M octets, and returns a single octet; it is defined as

To reconstruct a secret from M shares, the following procedure, or any equivalent method, is used: 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+1 octets is created, in which the array element V[i] contains the octet from the ith share. The value of I(0x00, 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 does the secret.

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. 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.

We use a data format with the following fields, in order: This field contains a variable number of octets. It identifies the secret with which a share is associated. When a secret is reconstructed, each of the shares used as input should have the same value identifier. This field contains a single octet. This field contains a single octet. This field contains a single octet. This field contains a single octet. This field is four octets long. This field is a single octet in length. It contains the finite field element that is used as the "X" coordinate for the share. This field has a length that is a variable number of octets. It contains the actual share data. This format is illustrated in .

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.

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. 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.

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: 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. 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. 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. 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. This field has a length that is a variable number of octets, which is indicated by the Reduncancy 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 .

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.

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

<- 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

<- DL -><- RL -><- Data -><--- Redundancy ---> 00000001000000050000000a68656c6c2f68656c6cef68656c6c6f ** ** ** ]]> 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:

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

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 . (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 operaation). 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.

Shares ]]>

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. The advantages of using a vector over GF(2^8) are that the computations are more efficient and the encoding is simpler. Multiplication and inversion over GF(2^8) can be done with two table lookups and two exors, using two fixed tables of 256 bytes each. One limitation of the GF(2^8) 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.

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. Test cases are provided for known-answer tests. The Share Index field can never be equal to zero. This property SHOULD be tested. There is a simple consistency test that can be run on an implementation that uses the Lagrange form of the interpolation polynomial. Each function L_i(X,U) as defined above has the property that

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

This section provides test cases that can be used to validate an implementation. All values are in hexadecimal. The algorithm used in the test case. The secret value to be split into shares. The number of shares required to reconstruct a secret. A share index. Each test case has multiple share values, and each share is associated with a share index. A share value.

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. provides guidance on the selection and implementation of random sources. A TSS implementation SHOULD be tested as described in . 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.

This document has no actions for IANA.

Thanks to Brian Weis for constructive feedback.