Contents

• XOR Seed Splitting: A Theoretical Introduction
  • Deterministic Shares
  • Named Splits
  • Master Shares
• Seed Splitting with MMGen Wallet

XOR Seed Splitting: A Theoretical Introduction

The bitwise exclusive-or operation (usually denoted as XOR, or “”) has interesting properties that make it very useful in cryptography.

Suppose we have two bytes, a and b:

To XOR the two bytes, we compare each of their bits. If the bit is the same for both a and b, the corresponding bit of the result is 0. If it differs, the result is 1:

Thus XOR can be thought of logically as “one or the other but not both”, or arithmetically as addition modulo 2 without carry, since 1 plus 1 equals 0 in base-2 arithmetic.

As is clear from our above example, switching the order of a and b has no effect on the result. So XOR, like addition, is commutative:

And like addition, grouping has no effect on the result. XOR is associative:

But unlike addition, XOR has an extra property: invertibility. The result can be switched with any of the operands, making XOR sort of like addition and subtraction rolled into one. Thus, if

then

and so forth.

This last property makes XOR very handy for encryption and decryption. Given a plaintext P and a random value r with the same bit length as P, we encrypt P by XOR’ing it with r to obtain the ciphertext C:

To decrypt, we just XOR the ciphertext with r to recover the plaintext:

The randomness of the ciphertext is guaranteed to be no less than that of the random value. Thus if r is perfectly random, then C is perfectly undecipherable, given no knowledge of r. This is the principle underlying the one-time pads used by spies and diplomats before the computer age, as well as modern stream ciphers.

To demonstrate how this can be used for seed splitting, all we do is change the names of the variables:

Here seed is analogous to the plaintext P, share₁ is a random value with the same bit length as seed, and share₂ is the resulting ciphertext. Just as the C reveals nothing about P in the previous example, share₂ reveals nothing about seed without knowledge of share₁. To recover the seed, we just XOR the two shares. Since XOR is commutative, the order in which we combine them isn’t important:

Thanks to XOR’s associativity, splits of any arbitrary length n can be created by using n-1 random shares, with the n-th share being the result of the chained XOR operations:

Perform an n-way split:

Join shares 1 through n to recover the seed:

Knowledge of any combination of n-1 shares reveals nothing about the seed.

Deterministic Shares

So we’ve seen that the mathematics behind XOR seed splitting is basically trivial. In practice, though, there are several issues that need to be resolved. For example, how do we obtain the random values for the shares? The easy answer is to just use the random number generator provided by our operating system. Generating the values deterministically is a better solution, however, providing us with two key advantages: 1) we avoid reliance on such factors as the quality of our OS RNG, the underlying hardware, and the entropy pool; and 2) we gain reproducibility—the ability to generate identical shares repeatedly, and as a consequence, to generate shares independently of each other.

This latter feature is especially useful. Suppose I want to do a 2-way split of my seed for backup purposes, giving one share to my friend Bob and storing the other share at some location accessible to me but unknown to Bob. With deterministic shares, I can generate Bob’s share now, giving it to Bob, and my own share later, once I’ve determined a good location for its safekeeping. And if either of us loses our share, it can just be regenerated.

OK, so now that we’re sold on the idea of deterministic shares, how do we go about creating them? A naive approach would be to just generate a secure cryptographic hash of our seed using the SHA256 algorithm and use that directly as the first share:

This would work fine for a 2-way split: assuming SHA256 is secure and our seed is strongly random (if either assumption is false, we’re in big trouble anyway), then share₁ is strongly random too and reveals nothing about our seed. And being derived from the seed alone, it’s regenerable on demand by the seed’s owner.

For n-way splits where n is greater than 2, however, we run into a problem when attempting to generate the additional random values. We might be tempted to just keep hashing:

But you may have already spotted the mistake here: the owner of the first share can generate all the successive shares up to n-1. Without the final n’th share he can’t recover the seed, but the whole benefit of having the additional shares has been nullified.

Important disclaimer: there are other reasons, beyond the scope of this discussion, why using a bare hash of the seed as our random number source might not be a good idea. Bear in mind that this is a simplified theoretical introduction, and the examples presented herein are not suitable for implementation in real production code.

The above example illustrates what happens when we violate the golden rule of the wallet developer: never derive a secret from another secret that someone besides the wallet’s owner could potentially gain access to. This goes for the private keys of the addresses in a wallet, which could be compromised in a security breach. And it certainly goes for seed shares, which are intended for distribution to others from the outset.

The solution to this problem is to derive the shares directly from the seed, but with an added identifier that’s unique to each share. The HMAC message-digest algorithm is ideally suited for this:

Using these unique pseudorandom values, we can now split and rejoin our seed in the manner described at the end of the previous section.

Named Splits

Now, we’d like to use seed splitting as part of our backup strategy, entrusting shares of our seed with various people we know. Multiple 2-way splits seems like the best approach—if one of our trustees loses their share, moves to another city, or gets run over by a bus, then we have the others left as a fallback.

However, we have no mechanism as yet for generating multiple splits: using the deterministic method outlined above, we can only generate the same set of pseudorandom shares over and over again. The obvious solution here is to give each split a name: for our split with Bob, we’ll add “bob” to the share identifier, for our split with Alice, we’ll add “alice”, and so forth. Using this approach, we can create an arbitrary number of uniquely named splits.

Create a 2-way split with Bob:

Create a 2-way split with Alice:

In addition, we should handle the case of multiple splits with different length but the same name. To exclude the reuse of shares in this case, we’ll add an additional identifier field specifying the total number of shares in the split.

Create a 3-way split “friends” with Bob and Alice:

Create a 4-way split “friends” with Bob, Alice and Carol:

Thus we’ve ensured the uniqueness of all shares across all possible splits.

Master Shares

As the number of splits we create grows, the question of how to store our shares becomes especially problematic. Each new split creates another new share that must be securely stored somewhere. What we need is some mechanism to generate our share of each split from a single master share. This master share would need to be generated and stored just once, saving us a great deal of trouble.

Having multiple master shares could be useful too. For example, if we ever wanted to revoke our splits, all we’d have to do is destroy our copy of the current master share, ensuring that none of the splits made with its participation could be joined. To create new splits, we’d use the next master share.

This is all easy to implement using the tools we’re already familiar with from the preceding sections. Using HMAC-SHA256, we’ll generate a range of indexed master shares from our seed as follows:

Using master share #1 as our share, our 4-way split “friends” with Bob, Alice and Carol now looks like this:

And rejoining the seed looks like this:

The rejoining process now involves more than a simple XOR’ing of shares: The name of the split must be input to the join function, so it should something that’s easy to memorize. If you really don’t trust yourself to remember the word “friends”, you could write it down somewhere.

Also note that an additional field, master<n>, has been appended to the share identifiers. This is to ensure that the shares of each master share split are unique, and differ from their non-master-share counterparts.

Seed Splitting with MMGen Wallet

MMGen Wallet implements the seed splitting and joining functionality described above via the commands mmgen-seedsplit and mmgen-seedjoin. Usage examples can be found on the mmgen-seedsplit help screen.

Shares can be made from and exported to all supported MMGen Wallet formats. This means you can split a BIP39 seed phrase, for example, and output the share back to BIP39 in one easy command:

# Create share 1 of a 2-way split of the provided BIP39 seed phrase:
$ mmgen-seedsplit -o bip39 sample.bip39 1:2

Each share of a split has a unique share ID. The share IDs are displayed by mmgen-seedsplit so that the user may record them for later reference. They may also be viewed with the mmgen-tool list_shares command:

# List the share IDs of a 2-way named split 'alice' of your default wallet:
$ mmgen-tool list_shares 2 id_str=alice

Seed: 71CA5049 (256 bits)
Split Type: 2-of-2 (XOR)
ID String: alice

Shares
------
1: D0BBD210
2: 25F0BD65

# List the share IDs of a 3-way default split of provided BIP39 seed phrase:
$ mmgen-tool list_shares 3 wallet=sample.bip39

Seed: 03BAE887 (128 bits)
Split Type: 3-of-3 (XOR)
ID String: default

Shares
------
1: 83B9AF74
2: 109485F4
3: 424522DC

Share IDs are handy for checking the correctness of shares when rejoining a split. Let’s say you’ve decided to rejoin your 2-way split with Alice, whose share you exported to BIP39 format. This can be done by contacting Alice by phone, for example, and having her read the mnemonic phrase to you. If the ID of Alice’s share as displayed by mmgen-seedjoin matches the value you recorded when making the split, then you know Alice has given you the correct phrase.

Note: when recovering shares over an insecure channel like the telephone, it’s advisable to destroy all copies of your share once you’ve rejoined the seed to safeguard against a possible eavesdropping attacker.

Ordinary named splits can easily be rejoined even without the MMGen software. First, each share must be converted to hexadecimal data. If your shares are in BIP39 format, for example, there are command-line tools available to do this. Then a single line of Python code is all that’s required to finish the job:

$ python3
>>> seed_hex = hex(int(share1_hex,16) ^ int(share2_hex,16)) # rejoin a 2-way split

(Note that the XOR operator in Python is ^.)

Unfortunately, rejoining master-share splits is considerably harder to do at the Python command prompt. This is because converting the master share into the temporary share used to make the split involves an additional step, as you’ll recall from the above discussion. In addition, this step is implemented by MMGen Wallet somewhat differently than as described above. For advanced users, an example will be provided in a future version of this document.