ring signature

{{Missing information|Definition|the definition. This section currently describes some useful properties of a ring signature but not the mathematical definition|date=April 2022}}

Suppose that a set of entities each have public/private key pairs, (P₁, S₁), (P₂, S₂), ..., (P_n, S_n). Party i can compute a ring signature σ on a message m, on input (m, S_i, P₁, ..., P_n). Anyone can check the validity of a ring signature given σ, m, and the public keys involved, P₁, ..., P_n. If a ring signature is properly computed, it should pass the check. On the other hand, it should be hard for anyone to create a valid ring signature on any message for any set without knowing any of the private keys for that set.{{cite journal| last=Debnath| first=Ashmita| author2=Singaravelu, Pradheepkumar |author3=Verma, Shekhar |title=Efficient spatial privacy preserving scheme for sensor network|journal=Central European Journal of Engineering|date=19 December 2012| volume=3|issue=1|pages=1–10|doi=10.2478/s13531-012-0048-7|s2cid=137248994|doi-access=free}}

File:Ring-signature.svg

In the original paper, Rivest, Shamir, and Tauman described ring signatures as a way to leak a secret. For instance, a ring signature could be used to provide an anonymous signature from "a high-ranking White House official", without revealing which official signed the message. Ring signatures are right for this application because the anonymity of a ring signature cannot be revoked, and because the group for a ring signature can be improvised.

Another application, also described in the original paper, is for deniable signatures. Here the sender and the recipient of a message form a group for the ring signature, then the signature is valid to the recipient, but anyone else will be unsure whether the recipient or the sender was the actual signer. Thus, such a signature is convincing, but cannot be transferred beyond its intended recipient.

There were various works, introducing new features and based on different assumptions:

;Threshold ring signatures:{{cite book|last1=E. Bresson|last2=J. Stern|last3=M. Szyd lo|title=Advances in Cryptology — CRYPTO 2002 |chapter=Threshold Ring Signatures and Applications to Ad-hoc Groups |series=Lecture Notes in Computer Science|date=2002|volume=2442|pages=465–480| doi=10.1007/3-540-45708-9_30| isbn=978-3-540-44050-5| chapter-url=https://www.di.ens.fr/~bresson/papers/BreSteSzy02.pdf|doi-access=free}} Unlike standard "t-out-of-n" threshold signature, where t of n users should collaborate to sign a message, this variant of a ring signature requires t users to cooperate in the ring signing protocol. Namely, t parties S₁, ..., S_t ∈ {P₁, ..., P_n} can compute a (t, n)-ring signature, σ, on a message, m, on input (m, S₁, ..., S_t, P₁, ..., P_n).

;Linkable ring signatures:{{cite book|last1=Liu|first1=Joseph K.|last2=Wong|first2=Duncan S.|title=Computational Science and Its Applications – ICCSA 2005 |chapter=Linkable Ring Signatures: Security Models and New Schemes | journal=ICCSA| date=2005 |volume=2| pages=614–623|doi=10.1007/11424826_65|series=Lecture Notes in Computer Science|isbn=978-3-540-25861-2}} The property of linkability allows one to determine whether any two signatures have been produced by the same member (under the same private key). The identity of the signer is nevertheless preserved. One of the possible applications can be an offline e-cash system.

;Traceable ring signature:{{cite journal| last1=Fujisaki|first1=Eiichiro| last2=Suzuki| first2=Koutarou|title=Traceable Ring Signature|journal=Public Key Cryptography| date=2007| pages=181–200}} In addition to the previous scheme the public key of the signer is revealed (if they issue more than one signatures under the same private key). An e-voting system can be implemented using this protocol.

Most of the proposed algorithms have asymptotic output size $O(n)$; i.e., the size of the resulting signature increases linearly with the size of input (number of public keys). That means that such schemes are impracticable for real use cases with sufficiently large $n$ (for example, an e-voting with millions of participants). But for some application with relatively small median input size such estimate may be acceptable. CryptoNote implements $O(n)$ ring signature scheme by Fujisaki and Suzuki in p2p payments to achieve sender's untraceability.

More efficient algorithms have appeared recently. There are schemes with the sublinear size of the signature,{{cite journal|last1=Fujisaki|first1=Eiichiro|title=Sub-linear size traceable ring signatures without random oracles|journal= IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences|date=2011|volume=95|issue=1|pages=393–415| doi=10.1587/transfun.E95.A.151|bibcode=2012IEITF..95..151F}} as well as with constant size.{{cite book|last1=Au|first1=Man Ho|last2=Liu|first2=Joseph K. |last3=Susilo| first3=Willy| last4=Yuen|first4=Tsz Hon|title=Progress in Cryptology - INDOCRYPT 2006 |chapter=Constant-Size ID-Based Linkable and Revocable-iff-Linked Ring Signature |series=Lecture Notes in Computer Science | date=2006|volume=4329|pages=364–378| doi=10.1007/11941378_26|isbn=978-3-540-49767-7| url=https://ro.uow.edu.au/cgi/viewcontent.cgi?article=10250&context=infopapers}}

The original paper describes an RSA based ring signature scheme, as well as one based on Rabin signatures. They define a keyed "combining function" $C\_\{k,v\}(y\_1,y\_2,\backslash dots,y\_n)$ which takes a key $k$, an initialization value $v$, and a list of arbitrary values $y\_1,\; \backslash dots\; y\_n$. $y\_i$ is defined as $g\_i(x\_i)$, where $g\_i$ is a trap-door function (i.e. an RSA public key in the case of RSA based ring signatures).

The function $C\_\{k,v\}(y\_1,y\_2,\backslash dots,y\_n)$ is called the ring equation, and is defined below. The equation is based on a symmetric encryption function $E\_k$:

: $C\_\{k,v\}(y\_1,y\_2,\backslash dots,y\_n)\; =\; E\_k(y\_n\; \backslash oplus\; E\_k(y\_\{n-1\}\; \backslash oplus\; E\_k(\backslash dots\; \backslash oplus\; E\_k(y\_1\; \backslash oplus\; v)\; \backslash dots)))$

It outputs a single value $z$ which is forced to be equal to $v$. The equation $v\; =\; C\_\{k,v\}(y\_1,y\_2,\backslash dots,y\_n)$

can be solved as long as at least one $y\_i$, and by extension $x\_i$, can be freely chosen. Under the assumptions of RSA, this implies knowledge of at least one of the inverses of the trap door functions $g^\{-1\}\_i$ (i.e. a private key), since $g^\{-1\}\_i(y\_i)\; =\; x\_i$.

Generating a ring signature involves six steps. The plaintext is signified by $m$, the ring's public keys by $P\_1,P\_2,\backslash dots,P\_n$.

1. Calculate the key $k=\backslash mathcal\{H\}(m)$, using a cryptographic hash function. This step assumes a random oracle for $\backslash mathcal\{H\}$, since $k$ will be used as key for $E\_k$.
2. Pick a random glue value $v$.
3. Pick random $x\_i$ for all ring members but yourself ($x\_s$ will be calculated using the signer's private key), and calculate corresponding $y\_i=g\_i(x\_i)$.
4. Solve the ring equation for $y\_s$
5. Calculate $x\_s$ using the signer's private key: $x\_s=g\_s^\{-1\}(y\_s)$
6. The ring signature now is the $(2n+1)$-tuple $(P\_1,P\_2,\backslash dots,P\_n;v;x\_1,x\_2,\backslash dots,x\_n)$

Signature verification involves three steps.

1. Apply the public key trap door on all $x\_i$: $y\_i=g\_i(x\_i)$.
2. Calculate the symmetric key $k=\backslash mathcal\{H\}(m)$.
3. Verify that the ring equation holds $C\_\{k,v\}(y\_1,y\_2,\backslash dots,y\_n)=v$.

Here is a Python implementation of the original paper using RSA. Requires 3rd-party module PyCryptodome.

import os

import hashlib

import random

import Crypto.PublicKey.RSA

import functools

class Ring:

"""RSA implementation."""

def __init__(self, k, L: int = 1024) -> None:

self.k = k

self.l = L

self.n = len(k)

self.q = 1 << (L - 1)

def sign_message(self, m: str, z: int):

self._permut(m)

s = [None] * self.n

u = random.randint(0, self.q)

c = v = self._E(u)

first_range = list(range(z + 1, self.n))

second_range = list(range(z))

whole_range = first_range + second_range

for i in whole_range:

s[i] = random.randint(0, self.q)

e = self._g(s[i], self.k[i].e, self.k[i].n)

v = self._E(v ^ e)

if (i + 1) % self.n == 0:

c = v

s[z] = self._g(v ^ u, self.k[z].d, self.k[z].n)

return [c] + s

def verify_message(self, m: str, X) -> bool:

self._permut(m)

def _f(i):

return self._g(X[i + 1], self.k[i].e, self.k[i].n)

y = map(_f, range(len(X) - 1))

y = list(y)

def _g(x, i):

return self._E(x ^ y[i])

r = functools.reduce(_g, range(self.n), X[0])

return r == X[0]

def _permut(self, m):

msg = m.encode("utf-8")

self.p = int(hashlib.sha1(msg).hexdigest(), 16)

def _E(self, x):

msg = f"{x}{self.p}".encode("utf-8")

return int(hashlib.sha1(msg).hexdigest(), 16)

def _g(self, x, e, n):

q, r = divmod(x, n)

if ((q + 1) * n) <= ((1 << self.l) - 1):

result = q * n + pow(r, e, n)

else:

result = x

return result

To sign and verify 2 messages in a ring of 4 users:

size = 4

msg1, msg2 = "hello", "world!"

def _rn(_):

return Crypto.PublicKey.RSA.generate(1024, os.urandom)

key = map(_rn, range(size))

key = list(key)

r = Ring(key)

for i in range(size):

signature_1 = r.sign_message(msg1, i)

signature_2 = r.sign_message(msg2, i)

assert r.verify_message(msg1, signature_1) and r.verify_message(msg2, signature_2) and not r.verify_message(msg1, signature_2)

Monero{{cite web |title=Evaluation of Bulletproof Implementation |url=https://ostif.org/wp-content/uploads/2018/10/OSTIF-QuarksLab-Monero-Bulletproofs-Final2.pdf |publisher=Quarkslab |ref=18-06-439-REP}} and several other cryptocurrencies use this technology.{{cn|date=February 2024}}

• Witness-indistinguishable proofs

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Category:Public-key cryptography

Category:Digital signature schemes