1. Introduction
For a given ring size , Cryptonote’s original scheme (as introduced in part 5), generates signatures of the form consisting of arguments. It turns out that a more efficient scheme initially introduce in [3] and later adapted by Adam Back in [1] can achieve the same security properties as Cryptonote’s with arguments instead (a reduction factor that tends to 2 as tends to ). The scheme introduced in [3] is known as Linkable Spontaneous Anonymous Group signature or LSAG signature scheme for short. In part 7 of this series, we will see how [4] generalizes the LSAG construct to build the foundation of Monero’s current ringCT signature scheme.
2. The LSAG scheme
The LSAG signature introduced in [3] is built on a group of prime order and generator . Moreover, it uses 2 statistically independent ROs:
In what follows we introduce a slightly modified LSAG scheme that will allow an easier comparison to Cryptonote’s original scheme. We carry forward all the notation used in the Cryptonote scheme to the current LSAG definition. In particular, we let be a large finite group generated by the same elliptic curve introduced in part 5 (refer to the post entitled Elliptic Curve Groups for an introduction to this topic). We also consider the same base point . Recall that the base point is chosen in such a way to ensure that it has a large prime order All arithmetic is done in the subgroup of the elliptic curve group As a matter of convention, we write
With a slight divergence from [3], 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 LSAG’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 [3] defines as a map from to . Here we restricted the domain and the range to instead. This is because in this version of LSAG, is strictly applied to public keys as opposed to any element of . 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 ). 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 LSAG scheme is defined by a set of 4 algorithms:
 The key generation algorithm . On input ( is the security parameter that by design we request 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 signerspecific 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. We observe that in [3], the keyimage or tag is computed as part of the ring signing algorithm as opposed to We include it in to ensure consistency with Cryptonote’s original construct.
 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 keyimage (or tag) given by . does the following:
 Choose random . Assign:
 , pick random . Assign:
 , where
 Set . Here denotes regular scalar multiplication in modulo arithmetic.
outputs a signature . is a PPT algorithm.
 Choose random . Assign:
 The ring verification algorithm . Given a ring signature , a message , and the set of public keys of the ring members:
 (Verification equations #1 to ): let . assigns
 (Verification equation ): checks whether
 , where
If equality holds, the signature is valid and outputs True. Otherwise, it outputs False. is a deterministic algorithm.
 (Verification equations #1 to ): let . assigns
 The ring linkability algorithm . It takes a verified valid signature . It checks if the keyimage was used in the past by comparing it to previous keyimages 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 when we prove the exculpability of LSAG in section 5 below), and outputs Linked. Otherwise, its keyimage is added to and outputs Independent.
3. Security analysis – Correctness
Let be an LSAG generated signature. Without loss of generality, assume . Then , we have the following implication:
If , then:
Recall that (by design of ) and so and . We therefore conclude by induction on that , . In particular, . This implies:
We can then invoke a similar induction argument on as the one stated earlier, but this time for . We therefore conclude that:
(by design of )
(by induction proof showing that and )
Subsequently, any generated LSAG signature will satisfy ‘s verification test.
4. Security analysis – Unforgeability visavis EFACM
For unforgeability proofs, we follow the 5step 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 the LSAG scheme is secure against EFACM in the RO model, we proceed by contradiction and assume that there exists a PPT adversary such that:
for nonnegligible 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 such that and where and are calculated as follows:
 Let
 , compute:
 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 and the ‘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 and the ‘s
 For
With adequately built, we conclude that (refer to section 6 of part 1 for a justification):
, for nonnegligible 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:
 : there exists such that 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.
 : there exists such that 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 nonnegligible 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 nonnegligible in
Here is an appropriate index that we will define in the proof. Moreover, 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 where we let , and :
We distinguish between 3 scenarios (without loss of generality, we assume that all queries sent to RO are distinct from eachother. Similarly, all queries sent to RO are distinct from eachother. 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
 such that 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 upperbounded by the probability that picks such that it matches the value of . If the 2 values don’t match, then will be different than (by the verification algorithm ). It is upperbounded because at the very least, this constraint must be observed to pass the verification test. 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 upperbounded 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 nonnegligible 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,
By definition of scenario 3, we know for a fact that , there exists an integer such that is the index of the query . We define to be the vector of indices corresponding to the queries that sends to RO during execution. Here, indexing is with respect to the cumulative numbering system. 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.

where
We let denote that the cardinality of . We have
We can see that represents the set of tuples that yield a successful EFACM forgery when no collisions occur, and when queried RO on all such that the index of the input query is equal to (i.e., the component of ), and such that condition is met.
Recall that
(nonnegligible in )
Clearly, partitions . So
This implies that
If this were not the case, then one would get the following contradiction:
So we introduce the set consisting of all vectors 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:
Let . 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 (nonnegligible in ) lowerbound 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 nonnegligible in
In order to show that the above quantity is nonnegligible in , we proceed differently. Suppose we can show that the following probability is nonnegligible in
This would imply that with nonnegligible 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 nonnegligible 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 nonnegligible 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 nonnegligible:
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 nonnegligible 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 nonnegligible probability of at least we get a successful tuple , s.t. for some vector of indices . 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 nonnegligible probability of at least , we can find another successful tuple s.t. and
Let correspond to forgery
and correspond to forgery
Recall that is the index of the last query of the form 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 other words, . Without loss of generality, let , (where ), correspond to the last query of this type sent to RO . That means that is the query sent to RO . We have:
(where (, whenever )
(by writing
Moreover, we have
(by definition of in )
(by design of the forgery tuples)
(by definition of in )
That means that we can solve for in polynomial time, contradicting the intractability of DL on elliptic curve groups. We conclude that the LSAG signature scheme is secure against EFACM in the RO model.
5. Security analysis – Exculpability
In part 5 of this series we discussed 2 different notions of exculpability. One of them had to do with the security property of anonymity and the other with the security property of unforgeability. Exculpability in the anonymity sense roughly meant that a signer’s identity can not be established even if her private key gets compromised (i.e., no one can prove that she was the actual signer under any circumstance). This section is concerned with the notion of exculpability from an unforgeability standpoint as described in [2].
The setting is similar to the one previously described in part 5. Suppose private keys have been compromised in an ring setting. Let denote the index of the only noncompromised private key , and let denote the keyimage (or tag) associated with the key pair . We investigate whether it is likely to produce a valid forgery with keyimage . In what follows, we show that this can only happen with negligible probability. In essence, this means that a noncompromised 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 keyimage. In the context of Cryptonote, this implies that a noncompromised 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 noncompromised ring member. That means that it can determine the index of the noncompromised member in the ring . In order to prove the exculpability of LSAG, we follow an almost identical proof to that of the previous section (i.e., unforgeability visavis EFACM) and apply the same 5step approach. The objective is to show that this particular type of forgery would imply the ability to solve the DL of . The nuance resides in the specific index for which the DL will be solved, as opposed to any other index. This is because we assume that all the other members are compromised and hence their DLs (i.e., private keys) are commonknowledge.
Step 1: We proceed by contradiction and assume that there exists a PPT adversary such that:
, for nonnegligible in
We refer to the event succeeds in creating a forgery as . We rewrite the above equation as:
, for nonnegligible 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 noncompromised 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 nonnegligible 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 nonnegligible probability of at least we get a successful tuple , s.t. for some vector of indices . 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 nonnegligible probability of at least , we can find another successful tuple s.t. and
Let correspond to forgery
and correspond to forgery
Recall that is the index of the last query of the form 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 other words, we have
, 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 rewrite 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 rewrite 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 . We can then be confident that the linear system of 2 equations has a unique solution. Hence, , we have , and
Moreover, by design of the 2 forgeries, we know that there exists one and only one (corresponding to the query sent to RO ) that satisfies
(by definition of in )
(by design of the forgery tuples)
(by definition of in )
(where whenever )
But , we showed that with overwhelming probability . Therefore, it must be that and so
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 secure against in the RO model.
6. Security analysis – Anonymity
In this section, we show that the LSAG scheme satisfies the weaker anonymity definition #2 introduced in part 3 of this series. Note that as we previously observed in part 5, linkable signatures cannot satisfy anonymity definition #1.
More formally, let be a PPT adversary with random tape that takes 4 inputs:
 Any message
 A ring that includes the public key of the actual signer.
 A list of compromised private keys of ring members (). can be empty, and may be different than , but
 A valid signature on message , ring and actual signer private key
outputs an index 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, can send a number of queries (polynomial in ) to RO and RO . The probability of ‘s success is 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 the Decisional Diffie Hellman problem (DDH for short). (Refer to part 5 for a discussion of DDH). We consider 3 separate cases:
Case 1: and
Suppose that in PPT() and nonnegligible 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 anonymity section of part 5). 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. ‘s input 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).
outputs 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 notnecessarily valid signature assigned to ring member s.t.
We let run the following algorithm:
feeds its output to . In order for to use its advantage in guessing the signer’s identity, it must be given a valid signature (i.e., a signature that is an element of the range of acceptable signatures over all RO . For to be a valid signature, must be a DDH instance. Indeed, let be partially defined as per the design of . We show that for this particular , the signature obtained is an element of the range of acceptable signatures. First note that:
If then we get:
Since is a DDH instance then we necessarily have
Moreover, recall that (by design of ). And so and . We therefore conclude by induction on that , . In particular, . This in turn implies that is a valid signature.
On the other hand, if is not a DDH instance, then and with overwhelming probability 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 such that 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 nonnegligibale 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. The calculation is the same as the one previously conducted in part 5. 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 noncompromised ring members. We get:
(by design of ).Since is a valid signature, we have:
for nonnegligible inLet 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 noncompromised members. Hence:
and
We get:
Putting it altogether, we conclude that:
Since is nonnegligible 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 keyimage 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 noncompromised 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 keyimage or tag of ). Only one of them will match (due to exculpability), subsequently revealing the identity of the signer.
7. Security analysis – Linkability
Recall that 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 proved in part 5 of this series that a signature scheme is linkable if and only if a ring of members, it is not possible to produce valid signatures with pairwise different keyimages such that all of them get labeled independent by
To prove that the LSAG scheme is linkable we follow a reductio ad absurdum approach, similar to the one described in part 5:
 Assume that the LSAG scheme is not linkable.
 The equivalence above would imply that such that it can produce valid signatures with pairwise different keyimages (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 keyimage such that . Denote this signature by
 When verifying the validity of will first compute the following:
 for all :
 for all :
 , the system of 2 equations given by and 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 . In particular, that means that the value is well defined and equal to

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 ). Since by design we have , we conclude that the probability that is upperbounded 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 keyimages and such that all of them get labeled independent by . The LSAG scheme as we introdued it is hence linkable.
References
[1] A. Back. Ring signature efficiency. 2015.
[2] E. Fujisaki and K. Suzuki. Traceable ring signatures. Public Key Cryptography, pages 181200, 2007.
[3] J. K. Liu, V. K. Wei, and D. S. Wong. Linkable spontaneous anonymous group signature for ad hoc groups. ACISP, Lecture Notes in Computer Science(3108):325335, 2004.
[4] S. Noether and A. Mackenzie. Ring condential transactions. Monero Research Lab, 2016.
Tags: anonymity, Group, Linkable, LSAG, Monero, Privacy, ring signature
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