1. Introduction
In this part, we introduce Monero’s original signature scheme as described in van Saberhagen’s seminal Cryptonote paper [2]. The scheme is an adaptation of the Traceable Ring Signature introduced by Fujisaki and Suzuki [1]. The most recent version of Monero implements a different signature known as RingCT. It modifies the original scheme to accomodate confidential transactions. We will discuss it in detail in parts 7, 8 and 9.
Security analysis of ring schemes consisted primarily in proving a) correctness, b) resilience against EFACM attacks in the RO model (unforgeability), and c) anonymity (i.e., signer ambiguity according to e.g., definition # 1 or # 2 as previously described in part 3). However, none of these security metrics tells if 2 signatures were generated by the same user or not. Doing so does not necessarily break the anonymity of the signer, but rather establishes a relationship between pairs of signatures. Identifying whether 2 signatures are linked or not is essential when dealing with electronic cash for example. In this case, the network must not tolerate the double spending of the same unit of electronic currency on 2 different transactions. In an electronic cash setting, the message typically consists of an unspent transaction output (also known as UTXO) and the objective is to make sure that the owner of a UTXO does not sign it twice (i.e., double spend it). Whenever this happens, the incident must be flagged and proper measures taken.
Monero in particular, and cryptocurrencies in general are prone to the double spending problem. This motivates the need to have an additional security requirement to tell if 2 signatures were issued by the same user. This must be done without releasing the identity of the user. We refer to the new requirement as linkability. It can commonly be achieved by adding to the ring signature a new signer-specific component known as a tag or a key-image.
Formally, we define a linkable ring signature scheme as a set of 4 algorithms:
- The signer’s key generation algorithm
(as described in part 1)
- The ring signing algorithm
(as described in part 1).
- The ring verification algorithm
(as described in part 1)
- The ring linkability algorithm
. Its input consists of a set of tags (key-images) and a given signature
. It checks if
‘s tag is included in the tag set. If so, it outputs Linked. Otherwise, it outputs Independent and adds the new tag to the set.
2. Cryptonote’s original linkable ring signature
Cryptonote is an application layer protocol that supports a number of cryptocurrencies. The first implementation of the Cryptonote protocol dates back to July 2012 and consisted of a cryptocurrency known as Bytecoin (different than Bitcoin). In April 2014, Monero was launched as a fork of Bytecoin.
The Cryptonote scheme [2] relies on a large finite group generated by a particular elliptic curve whose equation is given by:
For a concise introduction to elliptic curve cryptography, one could consult this post. The above equation is a polynomial over where
is a very large prime and
is a pre-defined element of
. To simplify the notation, we refer to the group generated by this elliptic curve as
. We note the following:
- Elements of
are pairs
that satisfy the above equation.
- Elliptic curve groups in general and
in particular have a well defined addition operation that we denote by
.
contains a special element
(not necessarily unique) that we refer to as the base point. The base point has order
, where
is a very large prime. That means that adding
to itself
times yields the identity element
of
. In other terms,
. We simply write
(the notation
serves as a reminder that this is scalar multiplication associated with
).
- We let
denote the group generated by
under the
operation of
. We also let
- Solving the Discrete Logarithm (DL) problem on
(and more generally on
) is thought to be intractable.
Cryptonote’s signature uses 2 distinct hash functions and
(modeled as 2 ROs). With a slight divergence from [2], we first introduce a hash function
before we define
. The reason will become clearer in section 4 when we build the signing simulator to prove Cryptonote scheme’s resilience against EFACM.
takes an element
and outputs a tuple
. Here
is a random element chosen according to a uniform distribution over
. We then let
. So
, takes an element
and returns an element
where
is randomly chosen in
.
Note that [2] defines as a map from
to
. Here we restricted the domain and the range to
instead. This is because as we will see shortly,
is applied to public keys. Public keys are elements of
that are scalar multiples of the base point
. Moreover, the scalar is never equal to order(
)
(we impose this constraint when we introduce the key generation algorithm
next). We are then justified in restricting the domain to
. The range is arbitrarily defined to be
, which is permissible since it preserves the injective nature of the map.
The scheme is defined by a set of 4 algorithms:
- The key generation algorithm
. On input
(
is the security parameter that by design we require to satisfy
, it produces a pair
of matching secret and public keys.
is randomly chosen in
, and
is calculated as
. (Note that
and
are both elements of
while
is an element of
.
In addition to the
key pair,
computes
.
is known as the key image (or tag). It is signer-specific since it depends only on the signer’s private and public keys. It allows the ring linkability algorithm
to test for independence between different signatures.
is modeled as a PPT Turing machine.
- The ring signing algorithm
. Suppose a user
decides to sign a message
on behalf of the ring of users
.
has a key pair given by
and a key-image (or tag) given by
.
does the following:
, choose random
,
. assign
and
.
- Choose random
. Assign
and
.
- Assign
.
, assign
and
.
- Set
and
Here
denotes regular scalar multiplication in modulo
arithmetic.
outputs a signature
.
is a PPT algorithm.
-
The ring verification algorithm
. Given a ring signature
, a message
, and the set
of public keys of the ring members
- (Verification equations
to
):
computes
-
- (Verification equation
):
checks whether
If equality holds, the signature is valid and
outputs True. Else, it outputs False.
-
- (Verification equations
- The ring linkability algorithm
.It takes a
-verified valid signature
. It checks if the key-image
was used in the past by comparing it to previous key-images stored in a set
. If a match is found, then with overwhelming probability the 2 signatures were produced by the same key pair (as will be justified in the exculpability section), and
outputs Linked. Otherwise, its key-image is added to
and
outputs Independent.
3. Security analysis – Correctness
Let be a
-generated signature. We compute:
-
-
, if
(since
dictates that
and
).
, if
(since
dictates that
). And since
, the resulting quantity becomes
.
We can then easily see that
.
-
-
-
, if
(since
dictates that
and
).
-
, if
(since
dictates that
). And since
(by
‘s construction), the resulting quantity becomes
.
We can then easily see that
. Subsequently, we get
(by construction of
).
Hence
-generated signatures are valid.
-
4. Security analysis – Unforgeability vis-a-vis EFACM
For unforgeability proofs, we follow the 5-step approach outlined earlier in part 1 (Recall that for ring signatures, we prove resilience against EFACM with respect to a fixed ring attack as described in part 3).
Step 1: To prove that this scheme is secure against EFACM in the RO model, we proceed by contradiction and assume that there exists a PPT adversary such that:
for non-negligible in
.
Step 2: Next, we build a simulator such that it:
- Does not have access to the private key of any signer.
- Has the same range as the original signing algorithm
(i.e., they output signatures taken from the same pool of potential signatures over all possible choices of RO functions and random tapes
and
).
- Has indistinguishable probability distribution from that of
over this range.
The reason we introduced as opposed to introducing only
is that the simulator makes use of the random element
in order to set
to the desired value. In other words, the simulator needs to have access to the random element
that is used in the calculation of
in order to ensure that
equates to
.
By construction, the output of will satisfy the verification equation. Moreover, it does its own random assignments to what otherwise would be calls to RO
(i.e.,
bypasses RO
). Next, note the following:
does not use any private key.
and
both have a range
s.t.
where
and
have the same probability distribution over
. Indeed,
, we have:
- For
The first factor is the probability of choosing the exact
value in the set
that is equal to
. The second factor is the probability of choosing the exact
values given by the
‘s and
‘s
.
- For
:
Note that the range of
is equal to
by construction of
. And so the first factor is the probability of choosing the exact
value in the set
that is equal to
. The second factor is the probability of choosing the exact
values given by the
‘s and
‘s
.
- For
With adequately built, we conclude that (refer to section 6 of part 1 for a justification):
, for
non-negligible in
.
Step 3: We now show that the probability of faulty collisions is negligible (refer to section 6 of part 1 for an overview). The 2 tyes of collisions are:
: A tuple
that
encounters — recall that
makes its own random assignment to
and bypasses RO
— also appears in the list of queries that
sends to RO
. A conflict in the 2 values will happen with overwhelming probability and the execution will halt.
: A tuple
that
encounters — recall that
makes its own random assignment to
— is the same as another tuple
that
encountered earlier — here too,
would have made its random assignment to
. Since the tuples are identical (i.e.,
), the assignments must match (i.e.,
. However, the likelihood that the 2 are equal is negligible. Hence they will be different with overwhelming probability and the execution will halt.
The aforementioned collisions must be avoided. In order to do so, we first calculate the probability of their occurence. We assume that during an EFACM attack, can make a maximum of
queries to RO
, a maximum of
queries to RO
, and a maximum of
queries to
.
,
, and
are all assumed to be polynomial in the security parameter
, since the adversary is modeled as a PPT Turing machine.
.
Recalling that and
are polynomial in
, we conclude that
is negligible in
Next, we compute
(since by design)
Recalling that is polynomial in
, we conclude that
is negligible in
Putting it altogether, we find that the below quantity is negligible in :
This allows us to conclude that the below quantity is non-negligible in (refer to section 6 of part 1 for a justification):
Step 4: In this step, our objective is to show that if is a successful tuple that generated a first EFACM forgery, then the following quantity is non-negligible in
:
Here is an appropriate index that we will define in the proof. To further simplify the notation, we let
and
for all
. (
and
denote respectively the
query to
and to
).
Let’s take a closer look at
Any successful forgery must pass the verification equation given by:
where and
Notice that this equation takes the ‘s as argument before verifying the equality. So we distinguish between 3 scenarios (without loss of generality, we assume that all
-queries sent to RO
are distinct from each-other. Similarly, all
-queries sent to RO
are distinct from each-other. This is because we can assume that
keeps a local copy of previous query results and avoid redundant calls):
- Scenario 1:
was successful in its forgery, and
- No collisions occured, and
- It never queried RO
on input
.
- Scenario 2:
was successful in its forgery, and
- No collisions occured, and
- It queried RO
on input
during execution, and
such that it queried RO
on input
after it had queried RO
on input
.
- Scenario 3:
was successful in its forgery, and
- No collisions occured, and
- It queried RO
on input
during execution, and
, it queried RO
on input
before it queried RO
on input
.
The probability of scenario 1 is upper-bounded by the probability that picks its
‘s for
such that
matches the value of
. Here,
is the value that RO
returns to
(the verification algorithm) when verifying the validity of the forged signature. And since
can be any value in the range of
(which was defined to be
) we get:
, which is negligible in
.
In scenario 2, let be an index such that
queried RO
on input
after it had queried RO
on input
. Note that during the verification process,
will calculate
and hence will make a call to
on input
(remember that
is derived from
). The probability that the resulting
matches the
argument previously fed to
is upper-bounded by
(since the range of
). Moreover,
can be any index in
. We get:
, which is negligible in
So we assume that a successful forgery will likely be of the Scenario 3 type.
, which is non-negligible in
Note that can send queries to RO
and RO
in any order it chooses to. This gives 2 different ways of referencing the index of a particular query sent to RO
. One way is to count the index as it appeared in the sequence of cumulative queries sent to both
and
. In this case, indices take on values in
. The other way, is to do the counting with respect to
queries only causing indices to take on values in
. If
is the index counted in the cumulative numbering system (i.e., the former system), we let
be the equivalent index in the latter system. Clearly,
We define to be the index of the query
sent by
to RO
during execution. Here, indexing is done with respect to the cumulative numbering system, and so
. We let
if query
was never asked by
. We also define the following condition:
This definition allows us to build the following sets:
In other terms,
is the set of tuples
that yield a successful EFACM forgery when no collisions occur, and when
queried RO
on input
at some point during its execution such that condition
is met. This is none other than scenario 3 that was described earlier.
In other terms,
is the set of tuples
that yield a successful EFACM forgery when no collisions occur, and when the index of the
-query sent to RO
on input
is equal to
, and such that condition
is met.
Recall that:
, (non-negligible in
).
Clearly, partitions
. So:
This implies that such that
If this were not the case, then one would get the following contradiction:
So we introduce the set consisting of all indices that meet the
threshold, i.e.
We claim that .
Proof: By definition of the sets , we have:
The next step is to apply the splitting lemma to each . First note that:
Referring to the notation used in the splitting lemma (section 7 of part 1), we let:
is defined as the space of tuples of:
- All random tapes
- All random tapes
- All possibe RO
answers to the first (
) queries sent by
(note the usage of
-indexing since indexing is done with respect to RO
queries only)
- All RO
(this means all possible RO
answers to the
queries sent by
).
is defined as the space of all possible RO
answers to the last
queries sent by
. (Recall that
where
is the
query sent to RO
).
The splitting lemma guarantees the existence of a subset of tuples
such that:
, we have:
, and so
-
We would like to compute the probability of finding a successful tuple
given that
was a successful
tuple and such that
. That means finding the following probability:
From the splitting lemma results, we have a (non-negligible in ) lower-bound on:
Note however, that and
are generally distinct sets. And so we cannot conclude that:
and therefore we cannot conclude that the following quantity is non-negligible in
In order to show that the above quantity is non-negligible in , we proceed differently. Suppose we can show that the following probability is non-negligible in
:
This would imply that with non-negligible probability, we can find a tuple that belongs to (and hence corresponds to a successful forgery) and at the same time belongs to
. We can then invoke the splitting lemma result just mentioned, to find a second tuple coresponding to a second forgery and that has the desired properties.
To prove the above, we proceed as follows:
since the ‘s are disjoint.
( result of splitting lemma above)
(by the claim proven earlier)
.
And so we conclude that:
which is non-negligible in
So let be such an index and
such a tuple. From the result above, we know that finding such a
can be done with non-negligible probability. And since
, we must have
. We can then invoke the
consequence of the splitting lemma and write:
We still have one last constraint to impose and that is that . We show that the following quantity is non-negligible:
To prove this, we use the same technique employed in part 2 and part 4 of this series. Note that if and
are independent events, then we can write:
And so we get
This result allows us to write:
(because we chose )
, which is non-negligible in
Step 5: The final step uses the 2 forgeries obtained earlier to solve an instance of the Discrete Logarithm (DL) problem. Here is a recap of Step 4 results:
- With non-negligible probability of at least
we get a successful tuple
, s.t.
for some index
. By running
a number of times polynomial in
, we can find such a tuple.
- Once we find such a tuple, we’ve also shown that with non-negligible probability of at least
, we can find another successful tuple
such that
and
.
Let correspond to forgery
and correspond to forgery
Recall that is the index of the query
that
sends to RO
. Since the 2 experiments corresponding to the 2 successful tuples have:
- The same random tapes
and
- The same RO
- ROs
and
behave the same way on the first
queries,
we can be confident that the first queries sent to the 2 ROs
and
are identical.
In particular the two queries are the same. And so:
(by writing
Moreover, we have
(since is a valid forgery)
(by design of the forgery tuples)
(since is a valid forgery)
Since , then
such that
That means that we can solve for in polynomial time, contradicting the intractability of DL on elliptic curve groups. We conclude that the signature scheme is secure against EFACM in the RO model.
5. Security analysis – Exculpability
We encountered the notion of exculpability when we introduced the 2 anonymity definitions in part 3. In that context, we said that a signer is exculpable if her identity can not be established even if her private key gets compromised. In other terms, no one can prove that she was the actual signer under any circumstance. This ensures her exculpability. In this section, we introduce a different notion of exculpability described in [1]. It has to do with unforgeability as opposed to anonymity.
Suppose private keys have been compromised in an
-ring setting. Let
denote the index of the only non-compromised private key
, and let
denote the key-image (or tag) associated with the key pair
. We investigate whether it is likely to produce a valid forgery with key-image
. In what follows, we show that this can only happen with negligible probability. In essence, this means that a non-compromised honest ring member (by honest we mean a ring member that signs at most once using his private key) does not run the risk of encountering a forged signature that carries his key-image. In the context of Cryptonote, this implies that a non-compromised honest ring member cannot be accused of signing twice using the same key image or tag, and hence is exculpable.
Note that since the adversary has access to the
compromised private keys, it can easily calculate their corresponding public keys. Doing so will allow it to identify the public key
of the non-compromised ring member. That means that it can determine the index
of the non-compromised member in the ring
. In order to prove the exculpability of the Cryptonote scheme, we follow an almost identical proof to that of the previous section (i.e., unforgeability vis-a-vis EFACM) and apply the same 5-step approach.
Step 1: We proceed by contradiction and assume that there exists a PPT adversary such that:
for non-negligible in
.
We refer to “succeeds in creating a forgery” as
. We re-write the above equation as:
, for
non-negligible in
.
The notation used makes it explicit that can access the set of compromised keys
with
excluded. Success is defined as issuing a forged signature with key image or tag equal to
. (Recall that
is derived from
).
Step 2: The next step consists in building a simulator such that it:
- Does not have access to the private key of any signer.
- Has the same range as the original signing algorithm
(i.e., they output signatures taken from the same pool of potential signatures over all possible choices of RO functions and respective random tapes
and
).
- Has indistinguishable probability distribution from that of
over this range.
The simulator is the same as the one we built in the previous section. The only nuance is that
does not choose a random index
, since
already knows the index of the non-compromised ring member.
Step 3: The logical reasoning and procedure are identical to those of the previous section. We conclude that:
Step 4: Here too, the logical reasoning and procedure are identical to those of the previous section. In particular, we define the following sets in a similar way:
and conclude that:
, which is non-negligible in
Here , as before, is an appropriately defined index,
, and
for all
.(
denotes the
query sent to RO).
Step 5: The final step uses the 2 forgeries obtained earlier to solve an instance of the Discrete Logarithm (DL) problem. Here is a recap of Step 4 results:
- With non-negligible probability of at least
we get a successful tuple
, s.t.
for some index
. By running
a number of times polynomial in
, we can find such a tuple.
- Once we find such a tuple, we’ve also shown that with non-negligible probability of at least
, we can find another successful tuple
such that
and
.
Let correspond to forgery
and correspond to forgery
Recall that is the index of the query
that
sends to RO
. Since the 2 experiments corresponding to the 2 successful tuples have:
- The same random tapes
and
- The same RO
- ROs
and
behave the same way on the first
queries,
we can be confident that the first queries sent to the 2 ROs
and
are identical.
In particular the two queries are the same. And so:
, and
Let , and
. For each
, we get 2 identical systems of 2 equations dictated by
‘s verification computation:
, the first system is a linear system of 2 equations in variables
and
. Similarly, the second system is a linear system of 2 equations in variables
and
. The 2 systems are identical with different variable names. Hence, if
is a unique solution to the first system and
a unique solution to the second, we can be confident that
and
. (Note that when we previously proved resilience against EFACM in section 4, the 2 forged signatures did not necessarily share the same tag
and so the 2 systems of linear equations would have been different from each other). For either system to admit a unique solution, the 2 equations must be linearly independent. We re-write the 2 systems as follows:
If we multiply the second equation by (multiplication refers to
), we see that a sufficient condition for the system to be linearly independent is to have
. Next, we show that with overwhelming probability, the system of linear equations is indeed independent for all
:
- Recall that the range of
is
and that the order of
- Therefore,
such that
and
- We can then re-write the sufficient condition as
- Note that given
, and
, there is at most one value of
that satisfies
. Otherwise, we would have
,
, and
. This would imply that
, a contradiction.
- Noting that each
corresponds to a distinct
, we conclude that given
and
there is at most one
s.t.
.
- Since
is a RO outputing random values, the probability of getting the right value of
is
(negligible in
).
, we therefore conclude that with overwhelming probability we have
. In other terms, we can be confident that the linear system of 2 equations has a unique solution. Hence,
, we have
, and
Moreover, we have:
(since
is a valid forgery)
(by design of the forgery tuples)
(since is a valid forgery)
Since , we conclude that
such that
. But
, we showed earlier that with overwhelming probability we have
. We then conclude that with overwhelming probability
.
Going back to the system of 2 equations associated with , we write:
That means that we can solve for in polynomial time, contradicting the intractability of DL on elliptic curve groups. We conclude that the signature scheme is exculpable and hence secure against
in the RO model.
6. Security analysis – Anonymity
In this section, we show that Cryptonotes’ signature scheme satisfies the weaker anonymity definition #2 introduced in part 3. The reason it cannot satisfy the stronger definition #1 has to do with the inclusion of the key image or tag. Linkable signatures in general cannot satisfy anonymity definition #1. By introducing a tag that is fully determined by a ring member’s private key, anyone with access to the signature can rule-out or confirm whether a member was the author if the member releases her private key. To see this, suppose that ring member releases her private key
, and let
be a valid signature. Anyone can calculate
and compare it to
. If the 2 values differ, then the signature could not have originated from member
who can now be ruled out. The identity of the signer is then confined to the remaining non-compromised members. On the other hand, if the 2 values match, then one can confidently assume that member
was the author. The confidence is derived from the exculpability property of the scheme that we demonstrated earlier (i.e., with overwhelming probability, no one can forge a signature with a key image or tag corresponding to a non-compromised signer). This shows that a linkable signature scheme is not exculpable in the anonymity-metric sense, although it can be exculpable in the unforgeability-metric sense.
More formally, we let be a PPT adversary with random tape
that takes 4 inputs:
- Any message
.
- A ring
of the
public keys
of the ring members.
includes the public key
of the actual signer.
- A list
of compromised private keys of ring members
. Note that
can be empty. Also note that
may be different than
but we always have
- A valid signature
on message
, with ring
and actual signer private key
.
outputs an index corresponding to the ring member in
that it thinks is the actual signer. Definition # 2 mandates that for any polynomial
in security parameter
, we have:
, if
and
, if
or
In the RO model, we allow to send a number of queries (polynomial in
) to RO
and RO
. The probability of
‘s success is then computed over the distributions of
and
. Making explicit the dependence on the ROs, definition # 2’s condition becomes:
, if
and
, if
or
In order to prove that anonymity holds in the above sense, we proceed by contradiction and rely on the intractability of another hard problem over cyclic groups known as the Decisional Diffie Hellman problem (DDH for short). A proof that relies on an intractable problem is referred to as a conditional proof. In what follows, we first introduce the DDH problem and then prove anonymity of the Cryptonote’s scheme.
DDH formulation: Let be a generator of a cyclic group of order
. DDH states that if
and
are uniformly and independently chosen in
, then the value of
looks like a random element in the group. Intuitively, this means that the following 2 distributions of tuples are indistinguishable:
, where
and
are randomly and independently chosen in
.
, where
,
, and
are randomly and independently chosen in
.
In other terms, we don’t know of any PPT algorithm that when given a tuple
as input, can do better than random guessing as to whether
or not. Formally, we describe the DDH problem as follows:
- Let
be uniformly drawn from
- Let
be uniformly drawn from
, and let
- Let
be uniformly drawn from
- Let
. We don’t know any
(probabilistic polynomial time in
) such that
fornon-negligible in
The previous formalization means that if we randomly decide whether a tuple that we send to is definitely a DDH instance or not, there is no known PPT(
)
that can tell what was decided with probability better than random guessing. Concretely:
- We flip a coin and assign the value 0 or 1 to variable
- If
, we feed
a random tuple
. There is a negligible probability that
, but most likely
and hence the tuple is not DDH.
- If on the other hand
, we feed
a tuple
that we know for sure to be a DDH instance.
The objective is for to devise a mechanism to guess, depending on the input tuple, whether
or
. The intractability of DDH means that there does not exist
in the set of PPT(
) that can outperform random guessing in that endeavour.
Note that if is a DDH instance, we write
to make explicit the group generator
. Clearly, a DDH instance by definition satisfies
and so will satisfy
. In general, if
is a tuple with elements chosen from a group with generator
, saying that
is a DDH instance is equivalent to saying that
. This is helpful when we are not given the powers
, but rather the group elements
and
. We are now in a position to prove the anonymity of Cryptonote’s signature scheme.
Anonymity proof: We consider 3 separate cases.
Case 1: and
We proceed by contradiction. Suppose that in PPT(
) and
non-negligible in
such that
, if
and
Recall that since , one can automatically rule out all the compromised ring members as possible signers (the logic was described in the introductory paragraph of this section). One can then limit the guessing range of the identity of the signer to the uncompromised batch of
remaining members.
We now build PPT(
) that colludes with
to solve the DDH problem. The input of
consists of 1) The tuple
being tested for DDH, 2) A certain ring size
( randomly chosen), 3) A number
of compromised members (randomly chosen), and 4) A message
(randomly chosen).
will output a tuple consisting of 1) The message
, 2) A randomly generated ring
of size
, 3) A randomly chosen set
of
compromised secret keys, and 4) A not-necessarily valid signature
assigned to ring member
such that
.
feeds its output
to
. In order for
to use its advantage in guessing the signer’s identity, it must be given a valid signature. For
to be a valid signature,
must be a DDH instance. Indeed, we have the following implications:
is DDH instance
By design of , we also have
,
. We then conclude that
is a valid signature.
On the other hand, if is not a DDH instance, then
and with overwhelming probability
and is not a valid signature.
Recall that can send queries to
and
during execution. It is important to enforce consistency between
and
‘s query results obtained from RO
and RO
on the same input. There are no risks of faulty collisions in so far as
is concerned (by design of
). However,
bypasses RO
and conducts its own backpatching to
. If
queries
on input
, then with overwhelming probability, it will conflict with
‘s backpatched value causing the execution to halt. The aforementioned collision must be avoided. In order to do so, we first calculate the probability of its occurence. We assume that during execution,
can make a maximum of
queries to RO
.
is assumed to be polynomial in the security parameter
, since the adversary is modeled as a PPT Turing machine.
and so we conclude that:
whenever and
. Here,
is non-negligibale in
After execution, returns to
an integer
.
then outputs 1 if
, or outputs 0/1 with equal probability otherwise. The following diagram summarizes the process:
Using the setting described above, we now calculate the probability of guessing whether
is DDH or not. In what follows we make use of the following notational simplifications:
- We refer to
simply as
- We refer to
simply as
We start by noticing that
- Case (
): In this case,
is a DDH instance and so as we saw earlier,
will be a valid signature.
would then use its hypothetical advantage to guess the index of the signer among the
non-compromised ring members. We get:
(by design of).
Since
is a valid signature, we have:
fornon-negligible in
Let
for some
Hence
. We get:
- Case (
): In this case, we do not know if
is a DDH instance or not, and hence can not be sure whether
is a valid signature. Consequently,
can no longer use its advantage in guessing the index of the signer, because this advantage works only when it is fed a valid signature. We get:
(by design of).
and since
can no longer use its advantage to guess the index of the signer, the best thing it can do is random guessing among non-compromised members. Hence:
and
We get:
Putting it altogether, we conclude that:
Since is non-negligible in
, the above probability outperforms random guessing. This contradicts the intractability of DDH. Similarly, we can show
is also bounded from below. We finally conclude that for any polynomial :
, if
and
Case 2: and
In this case, can check if
(the key-image or tag of
) matches any of the compromised tags
, for
. With overwhelming probability, none of them will match since we proved that the scheme is exculpable and so no one can forge a signature with a tag of a non-compromised member. Proceeding by elimination,
can then conclude that the signer is
Case 3:
In this case, can check which of the compromised tags
matches
(the key-image or tag of
). Only one of them will match (due to exculpability), subsequently revealing the identity of the signer.
7. Security analysis – Linkability
In essence, the linkability property means that if a secret key is used to issue more than one signature, then the resulting signatures will be linked and flagged by (the linkability algorithm). We claim that:
A signature scheme is linkable
a ring of
members, it is not possible to produce
valid signatures with pairwise different key-images such that all of them get labeled independent by
Proof of : Consider the case
with
. Then it is not possible to use
‘s secret key
to produce 2 valid signatures such that they have different key-images and both are labeled independent by
. In other words, the signature scheme is linkable.
Proof of : We prove the contrapositive. Assume that
a ring of
members
such that it can produce
valid signatures with pairwise different key-images, and such that all of them get labeled independent by
. This implies that
such that the ring member with public key
produced at least 2 valid signatures with different key-images, both labeled independent by
. This means that the scheme is not linkable.
To prove that Cryptonote’s scheme is linkable we follow a reductio ad absurdum approach:
- Assume that the scheme is not linkable.
- The equivalence above would imply that
such that it can produce
valid signatures with pairwise different key-images (i.e.,
), and such that all of them get labeled independent by
.
- This means that there must exist a signature (from the set of
valid signatures) with key-image
such that
. Denote this signature by
.
- When verifying the validity of
will first compute the following for all
:
, the system of 2 equations above can be equivalently written as:
For a given
,
, and
, this constitutes a system of 2 equations in variables
and
.
- Since
, the system of 2 equations corresponding to each
is independent and admits a unique solution
for any given
, and
. That means that for given
, and
, the value
is well defined.
- By virtue of being a valid signature,
must satisfy
‘s verification equation. More specifically, it must be that
. But RO
is random by definition. The probability that it outputs a specific value is eqal to
(recall that the range of
). And since by design we have
, we conclude that the probability that
is upper-bounded by
and is hence negligible. In other terms, the probability that
is a valid signature is negligible.
We can then conclude that with overwhelming probability, the ring can not produce
valid signatures with pairwise different key-images and such that all of them get labeled independent by
. Cryptonote’s scheme is hence linkable.
References
[1] E. Fujisaki and K. Suzuki. Traceable ring signatures. Public Key Cryptography, pages 181 (200, 2007).
[2] N. Van Saberhagen. Cryptonote 2.0., 2013.
Tags: anonymity, Cryptonote, Monero, Privacy, ring signature
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