A demo of the methods involved in this article is here.

And so there has been a leak of information at the White House. Donald Trump calls in his Cyber Security leads, and tells them, “I know one of you leaked the information, but I can’t tell which one”. How can Donald tell that one of his leads has leaked the information, but not know which one? Well, this can be achieved with a ring signature, and which provides anonymity, unforgivably and collusion resistance.

A ring signature is a digital signature that is created by a member of a group which each have their own keys. It is then not possible to determine the person in the group who has created the signature. The method was initially created by Ron Rivest, Adi Shamir, and Yael Tauman in 2001, and in their paper they proposed the White house leak dilemma.

Creating the ring

In a ring signature we define a group of entities who each have their own public/private key pairs of (P1, S1), (P2, S2), …, (Pn, Sn). If we want an entity i to sign a message (message), they use their own secret key (si), but the public keys of the others in the group (m,si,P1…Pn). It should then be possible to check the validity of the group by knowing the public key of the group, but not possible to determine a valid signature if there is no knowledge of the private keys within the group.

So let’s say that Trent, Bob, Eve and Alice are in a group, and they each have their own public and secret keys. Bob now wants to sign a message from the group. He initially generates a random value v, and then generates random values (xi) for each of the other participants, but takes his own secret key (si) and uses it to determine a different secret key, and which is the reverse of the encryption function.

He now takes the message and takes a hash of it, and thus creates a key (k). This key will be used with symmetric encryption to encrypt each of the elements of the ring (Ek), and then each element of the ring uses an EX-OR function from the previous element (Figure 1).

Each of the random values for the other participants are then encrypted with the public key of the given participant. Bob then computes the value of ys in order to create the ring (the result of the ring must equal v). He will then inverse this value to produce the equivalent private key (xs). Bob now releases the overall signature, and the random x values, along with the computed secret key. To check the signature, the receive just computes the ring and checks that the result matches the sent signature.

The basic method are:

1. Generate encryption with k=Hash(message).

2. Generate a random value (u).

3. Encrypt u to give v=Ek(u).

4. For each person (apart from the sender):

4.1 Calculate e=si^{Pi} (mod Ni) and where si is the random number generated for the secret key of the ith party, and Pi is the public key of the party.

4.2 Calculate v=v⊕e

5. For the signed party (z), calculate sz=(v⊕u)^d (mod Nz) and where d is the secret key of the signing party.

We will end up with the signature (v=Ek(u)), and which completes the ring.

I have created a demonstration of the original method here, and here is an outline presentation of the method [slides]:

A demo of the methods involved in this article is here:

The basic method involves creating Bob creating fake private keys for the other people in the ring:

The verification of the ring is then:

Rings Signatures in Monero

The major problem with the Bitcoin network, is that the amount of a transaction and the sender and receive of the funds are not private, and someone who knows someones address can trace their transactions. This is the case because the blockchain needs to check that the sender has enough funds to pay the recipient.Thus many cryptocurrencies are looking for ways of anonymising the transaction. Ethereum, for example, uses zk-Snarks to hide identities.

One method of preserving identity was proposed by Rivest et al and uses RSA encryption. Unfortunately, it is not efficient for modern systems, thus Greg Maxwell’s defined an elliptic curve method as a new way of creating the ring signature: the Borromean ring signature [paper].

The cryptocurrency Monero then adopted the method for anonymising transactions, but have since migrated to a new method: Multi-layered Linkable Spontaneous Anonymous Group signature. This method hides the transaction amount and the identity of the payer and recipient [paper]. It is now known as RingCT (Ring Confidential Transactions), and was rolled-out in January 2017 and mandatory for all transactions from September 2017.

Conclusions

The major problem with the Bitcoin network is that the amount of a transaction and the sender and receiver of the funds are not private, and someone who knows someone’s address can trace their transactions. This is the case because the blockchain needs to check that the sender has enough funds to pay the recipient. Thus many cryptocurrencies are looking for ways of anonymising the transaction.