Preserving Privacy: Borromean Ring Signing

I am a great fan of the work of Greg Maxwell, so here’s one of his classics [here]:

A Borromean ring, in maths, is defined as three closed curves that cannot be separated from each other but can be unknotted when one of the rings is cut or removed.

Ring signatures

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 that 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.

Borromean ring signing

With the Borromean Ring signer method, we can create a number of rings, and then sign a message with one of the private keys on each of the rings. In this case, we will create one ring with three keys and another ring with three keys, and we will then sign with one key from each ring. This should be verified. We will then replace one of the private keys so that there will be no public key on the rings, and which should negatively verify. The code used is [here]:

import sys
from ecpy.curves import Curve
from ecpy.keys import ECPrivateKey
import borromean
import binascii
import secrets
m1 = 'Hello'
if (len(sys.argv)>1):
m1=str(sys.argv[1])
print ("Message: ",m1)
m1=m1.encode()
cv = Curve.get_curve('secp256k1')seckey0 = ECPrivateKey(secrets.randbits(32*8), cv)
seckey1 = ECPrivateKey(secrets.randbits(32*8), cv)
seckey2 = ECPrivateKey(secrets.randbits(32*8), cv)
seckey3 = ECPrivateKey(secrets.randbits(32*8), cv)
seckey4 = ECPrivateKey(secrets.randbits(32*8), cv)
seckey5 = ECPrivateKey(secrets.randbits(32*8), cv)
pubkey0 = seckey0.get_public_key()
pubkey1 = seckey1.get_public_key()
pubkey2 = seckey2.get_public_key()
pubkey3 = seckey3.get_public_key()
pubkey4 = seckey4.get_public_key()
pubkey5 = seckey5.get_public_key()
print(f"Key ring 1: secret keys: {seckey0.d},{seckey1.d},{seckey2.d}")
print(f"Key ring 2: secret keys: {seckey3.d},{seckey4.d},{seckey5.d}")
borromean = borromean.Borromean()pubring1 = [pubkey0, pubkey1, pubkey2]
pubring2 = [pubkey3, pubkey4, pubkey5]
secring1 = [seckey0, seckey1, seckey2]
secring2 = [seckey3, seckey4, seckey5]
s1=1
s2=1
pubset = (pubring1 , pubring2)
secset = [secring1[s1] , secring2[s2]]
secidx = [s1,s2]
sigma = borromean.sign(m1, pubset, secset, secidx )
print ("\ne0=",binascii.hexlify(sigma[0]))
print ("Signature (s):")
for s in sigma[1]:
print (binascii.hexlify(s))
rtn=borromean.verify( m1, sigma, pubset)print ("\nChecking with valid key (Key 2 in Ring 1, Key 2 in Ring 2: ",rtn)
seckey1 = ECPrivateKey(secrets.randbits(32*8), cv)
pubring1 = [pubkey0, pubkey1, pubkey2]
pubring2 = [pubkey3, pubkey4, pubkey5]
secring1 = [seckey0, seckey1, seckey2]
secring2 = [seckey3, seckey4, seckey5]
s1=1 # seckey1 wrong for pubkey1
s2=1 # seckey3 okay for pubkey3
pubset = (pubring1 , pubring2)
secset = [secring1[s1] , secring2[s2]]
secidx = [s1,s2]
sigma = borromean.sign(m1, pubset, secset, secidx )
rtn=borromean.verify(m1, sigma, pubset)
print ("Checking with non-valid key (Key 2 in Ring 1) and valid Key 2 in Ring 2: ",rtn)

A sample run [here]:

Message:  Hello
Key ring 1: secret keys: 29875636843252127507955771140493761058745571354380727422781883818399443737006,97218495476827164225347692618071712330690381390998678394112169358916634197973,115676391249222958503558958505064309953121480304377021022672178145871691693856
Key ring 2: secret keys: 23755527786469626298119838665469887802365884775146742464420183004417306811346,37301517688390233590722677956221422785988140595590490774518256329076480299102,97160227373003091865407598081510266465466948946312479036443319324502948855317
e0= b'd0551fa12507e4338e9fe1460dcae7ee09e769857c615b08dee8fc3b7eba11cd'
Signature (s):
b'53a5f4657992ff875fc5a9bc23f682fed5e9f8d8b5150882ef8a0b884959e564'
b'2df3bde7e1e142491d06e71988fa56fceded189b3620be13f9fdbf18f1eaf42e'
b'b5cb16690e7d096a3e774319796d369a5a01f0a69dfbb93b55d38a3729620522'
b'5c135236bbba5ef39aad2d7db19535df384eac5c038ae310140abfeb1386e889'
b'b86d0352a7c4b6ab044522625ec0914862f69b0ea5370eebf7715ba10fa61267'
b'30937a6fc60b0cc2eb558dfc52fbf890bd1fd1a4c18030108f2af5e4b9834f89'
Checking with valid key (Key 2 in Ring 1, Key 2 in Ring 2: True
Checking with non-valid key (Key 2 in Ring 1) and valid Key 2 in Ring 2: False
Verified!

The code is here:

https://replit.com/@billbuchanan/ringsig01#main.py

Ring signatures have been used in a range of cryptocurrency application, and are used to anonymise the sender and the recipient of a transaction. This overcomes the problem of Bitcoin, which only does pseudo-anonymity. Here are more details:

Professor of Cryptography. Serial innovator. Believer in fairness, justice & freedom. EU Citizen. Auld Reekie native. Old World Breaker. New World Creator.

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