Chaum's "The Dining Cryptographers Problem" (VERY LONG)

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J. Cryptology (1988) 1:65-75

The Dining Cryptographers Problem:

Unconditional Sender and Recipient Untraceability

David Chaum
Centre for Mathematics and Computer Science, Kruislan 413, 1098 SJ 
Amsterdam, The Netherlands

Abstract.  Keeping confidential who sends which messages, in a 
world where any physical transmission can be traced to its 
origin, seems impossible. The solution presented here is 
unconditionally or cryptographically secure, depending on whether 
it is based on one-time-use keys or on public keys, respectively. 
It can be adapted to address efficiently a wide variety of 
practical considerations.

Key words.  Untraceability, Unconditional Security, Pseudonymity.

Introduction

Three cryptographers are sitting down to dinner at their favorite 
three-star restaurant. Their waiter informs them that arrangements 
have been made with the maitre d'hotel for the bill to be paid 
anonymously. One of the cryptographers might be paying for the dinner, 
or it might have been NSA (U.S. National Security Agency). The three 
cryptographers respect each other's right to make an anonymous 
payment, but they wonder if NSA is paying. They resolve their 
uncertainty fairly by carrying out the following protocol:

Each cryptographer flips an unbiased coin behind his menu, between 
him and the cryptographer on his right, so that only the two of them can 
see the outcome. Each cryptographer then states aloud whether the two 
coins he can see--the one he flipped and the one his left-hand neighbor 
flipped--fell on the same side or on different sides. If one of the 
cryptographers is the payer, he states the opposite of what he sees. An 
odd number of differences uttered at the table indicates that a 
cryptographer is paying; an even number indicates that NSA is paying 
(assuming that the dinner was paid for only once). Yet if a 
cryptographer is paying, neither of the other two learns anything from 
the utterances about which cryptographer it is.

To see why the protocol is unconditionally secure if carried out 
faithfully, consider the dilemma of a cryptographer who is not the 
payer and wishes to find out which cryptographer is. (If NSA pays, there 
is no anonymity problem.) There are two cases. In case (1) the two 
coins he sees are the same, one of the other cryptographers said 
"different," and the other one said "same." If the hidden outcome was 
the same as the two outcomes he sees, the cryptographer who said 
"different" is the payer; if the outcome was different, the one who said 
"same" is the payer. But since the hidden coin is fair, both possibilities 
are equally likely. In case (2) the coins he sees are different; if both 
other cryptographers said "different," then the payer is closest to the 
coin that is the same as the hidden coin; if both said "same," then the 
payer is closest to the coin that differs from the hidden coin. Thus, in 
each subcase, a nonpaying cryptographer learns nothing about which of 
the other two is paying.

The cryptographers become intrigued with the ability to make 
messages public untraceably. They devise a way to do this at the table 
for a statement of arbitrary length: the basic protocol is repeated over 
and over; when one cryptographer wishes to make a message public, he 
merely begins inverting his statements in those rounds corresponding 
to 1 's in a binary coded version of his message. If he notices that his 
message would collide with some other message, he may for example 
wait a number of rounds chosen at random from a suitable distribution 
before trying to transmit again.

1. Generalizing the Approach

During dinner, the cryptographers also consider how any number of 
participants greater than one can carry out a version of the protocol. 
(With two participants, only nonparticipant listeners are unable to 
distinguish between the two potential senders.) Each participant has a 
secret key bit in common with, say, every other participant. Each 
participant outputs the sum, modulo two, of all the key bits he shares, 
and if he wishes to transmit, he inverts his output. If no participant 
transmits, the modulo two sum of the outputs must be zero, since every 
key bit enters exactly twice; if one participant transmits, the sum 
must be one. (In fact, any even number of transmitting participants 
yields zero, and any odd number yields one.) For j rounds, each 
participant could have a j-bit key in common with every other 
participant, and the ith bit of each such key would be used only in the 
ith round. Detected collision of messages leads to attempted 
retransmission as described above; undetected collision results only 
from an odd number of synchronized identical message segments. 
(Generalization to fields other than GF(2) is possible, but seems to 
offer little practical advantage.)

Other generalizations are also considered during dinner. The 
underlying assumptions are first made explicit, including modeling 
key-sharing arrangements as graphs. Next, the model is illustrated 
with some simple examples. The potential for cooperations of 
participants to violate the security of others is then looked at. Finally, 
a proof of security based on systems of linear equations is given.

1.1. Model

Each participant is assumed to have two kinds of secret: (a) the keys 
shared with other participants for each round; and (b) the inversion 
used in each round (i.e., a 1 if the participant inverts in that round and a 
0 if not). Some or all of a participant's secrets may be given to other 
participants in various forms of collusion, discussion of which is 
postponed until Section 1.3. (For simplicity in exposition, the 
possibility of secrets being stolen is ignored throughout.)

The remaining information about the system may be described as: (a)
who shares keys with whom; and (b) what each participant outputs
during each round (the modulo two sum of that participant's keys and
inversion). This information need not be secret to ensure
untraceability. If it is publicly known and agreed, it allows various
extensions discussed in Sections 2.5 and 2.6. The sum of all the
outputs will, of course, usually become known to all participants.

In the terminology of graphs, each participant corresponds to a 
vertex and each key corresponds to an edge. An edge is incident on the 
vertices corresponding to the pair of participants that shares the 
corresponding key. From here on, the graph and dinner-table 
terminologies will be used interchangeably. Also, without loss of 
generality, it will be assumed that the graph is connected (i.e., that a 
path exists between every pair of vertices), since each connected 
component (i.e., each maximal connected subgraph) could be considered 
a separate untraceable-sender system.

An anonymity set seen by a set of keys is the set of vertices in a 
connected component of the graph formed from the original graph by 
removing the edges concerned. Thus a set of keys sees one anonymity 
set for each connected partition induced by removing the keys. The main 
theorem of Section 1.4 is essentially that those having only the public 
information and a set of keys seeing some anonymity set can learn 
nothing about the members of that anonymity set except the overall 
parity of their inversions. Thus, for example, any two participants 
connected by at least one chain of keys unknown to an observer are both 
in the same anonymity set seen by the observer's keys, and the observer 
gains nothing that would help distinguish between their messages.

1.2. Some Examples

A few simple consequences of the above model may be illustrative. The 
anonymity set seen by the empty set (i.e., by a nonparticipant observer) 
is the set of all vertices, since the graph is assumed connected and 
remains so after zero edges are removed. Also, the anonymity sets seen 
by the full set of edges are all singleton sets, since each vertex's 
inversion is just the sum of its output and the corresponding key bits.

If all other participants cooperate fully against one, of course no 
protocol can keep that singleton's messages untraceable, since 
untraceability exists only among a set of possible actors, and if the set 
has only one member, its messages are traceable. For similar reasons, 
if a participant believes that some subset of other participants will 
fully cooperate against him, there is no need for him to have keys in 
common with them.

A biconnected graph (i.e., a graph with at least two vertex-disjoint 
paths between every pair of vertices) has no cut-vertices (i.e., a single 
vertex whose removal partitions the graph into disjoint subgraphs). In 
such a graph, the set of edges incident on a vertex v sees (apart from v) 
one anonymity set containing all other vertices, since there is a path 
not containing v between every pair of vertices, and thus they form a 
connected subgraph excluding v; each participant acting alone learns 
nothing about the contribution of other participants.

1.3. Collusion of Participants

Some participants may cooperate by pooling their keys in efforts to 
trace the messages of others; such cooperation will be called collusion. 
For simplicity, the possibilities for multiple collusions or for pooling 
of information other than full edges will be ignored. Colluders who lie 
to each other are only touched on briefly, in Section 2.6.

Consider collusion in a complete graph. A vertex is only seen as a 
singleton anonymity set by the collection of all edges incident on it; all 
other participants must supply the key they share with a participant in 
order to determine that participant's inversions. But since a collusion 
of all but one participant can always trace that participant merely by 
pooling its members' inversions as already mentioned, it gains nothing 
more by pooling its keys. The nonsingleton anonymity set seen by all 
edges incident on a colluding set of vertices in a complete graph is the 
set of all other vertices; again, a collusion yields nothing more from 
pooling all its keys than from pooling all its inversions.

Now consider noncomplete graphs. A full collusion is a subset of 
participants pooling all of their keys. The pooled keys see each colluder 
as a singleton anonymity set; the colluders completely sacrifice the 
untraceability of their own messages. If a full collusion includes a cut-
set of vertices (i.e., one whose removal partitions the graph), the 
collusion becomes nontrivial because it can learn something about the 
origin of messages originating outside the collusion; the noncolluding 
vertices are partitioned into disjoint subgraphs, which are the 
anonymity sets seen by the pooled keys.

Members of a partial collusion pool some but not all of their keys. 
Unlike the members of a full collusion, each member of a partial 
collusion in general has a different set of keys. For it to be nontrivial, 
a partial collusion's pooled keys must include the bridges or separating 
edges of a segregation or splitting of the graph (i.e., those edges whose 
removal would partition the graph). Settings are easily constructed in 
which the pooled keys see anonymity sets that partition the graph and 
yet leave each colluder in a nonsingleton partition seen by any other 
participant. Thus, colluders can join a collusion without having to make 
themselves completely traceable to the collusion's other members.

1.4. Proof of Security

Consider, without loss of generality, a single round in which say some 
full collusion knows some set of keys. Remove the edges known to the 
collusion from the key-sharing graph and consider any particular 
connected component C of the remaining graph. The vertices of C thus 
form an anonymity set seen by the pooled keys.

Informally, what remains to be shown is that the only thing the 
collusion learns about the members of C is the parity sum of their 
inversions. This is intuitively apparent, since the inversions of the 
members of C are each in effect hidden from the collusion by one or 
more unknown key bits, and only the parity of the sum of these key bits 
is known (to be zero). Thus the inversions are hidden by a one-time pad, 
and only their parity is revealed, because only the parity of the pad is 
known.

The setting is formalized as follows: the connected component C is 
comprised of rn vertices and n edges. The incidence matrix M of C is 
defined as usual, with the vertices labeling the rows and the edges 
labeling the columns. Let K, I, and A be stochastic variables defined on 
GF(2)^n, GF(2)^m, and GF(2)^m, respectively, such that
K is uniformly distributed over GF(2)^n, K and I are mutually 
independent, and A = (MK) cross I. In terms of the protocol, K comprises 
the keys corresponding to the edges, I consists of the inversions 
corresponding to the vertices, and A is formed by the outputs of the 
vertices. Notice that the parity of A (i.e., the modulo two sum of its 
components) is always equal to the parity of I, since the columns of M 
each have zero parity. The desired result is essentially that A reveals 
no more information about I than the parity of 1. More formally:

Theorem.  Let a be in GF(2)^n. For each i in GF(2)^n, which is assumed by 
I with nonzero probability and which has the same parity as a, the 
conditional probability that A = a given that I = i is 2^(1 - m). Hence, 
the conditional probability that I = i given that A = a is the a priori 
probability that I = i.

Proof.  Let i be an element of GF(2)^n have the same parity as a. 
Consider the system of linear equations (MK) cross i = a, in k an 
element of GF(2)^n. Since the columns of M each have even parity, as 
mentioned above, its rows are linearly dependent over GF(2)^m. But as a 
consequence of the connectedness of the graph, every proper subset of 
rows of M is linearly independent. Thus, the rank of M is m - 1, and so 
each vector with zero parity can be written as a linear combination of 
the columns of M. This implies that the system is solvable because i 
cross a has even parity. Since the set of n column vectors of M has rank 
m - 1, the system has exactly 2^(n - m + 1) solutions.

Together with the fact that K and I are mutually independent and 
that K is uniformly distributed, the theorem follows easily.                           

2. Some Practical Considerations

After dinner, while discussing how they can continue to make 
untraceable statements from this respective homes, the cryptographers 
take up a variety of other topics. In particular, they consider different 
ways to establish the needed keys; debate adapting the approach to 
various kinds of communication networks; examine the traditional 
problems of secrecy and authentication in the context of a system that 
can provide essentially optimal untraceability; address denial of 
service caused by malicious and devious participants; and propose 
means to discourage socially undesirable messages from being sent.

2.1. Establishing Keys

One way to provide the keys needed for longer messages is for one 
member of each pair to toss many coins in advance. Two identical 
copies of the resulting bits are made, say each on a separate optical 
disk. Supplying one such disk (which today can hold on the order of 
10^10 bits) to a partner provides enough key bits to allow people to 
type messages at full speed for years. If participants are not 
transmitting all the time, the keys can be made to last even longer by 
using a substantially slower rate when no message is being sent; the 
full rate would be invoked automatically only when a 1 bit indicated 
the beginning of a message. (This can also reduce the bandwidth 
requirements discussed in Section 2.2.)

Another possibility is for a pair to establish a short key and use a 
cryptographic pseudorandom-sequence generator to expand it as needed. 
Of course this system might be broken if the generator were broken. 
Cryptanalysis may be made more difficult, however, by lack of access 
to the output of individual generators. Even when the cryptographers do 
not exchange keys at dinner, they can safely do so later using a public-
key distribution system (first proposed by [4] and [3]).

2.2 Underlying Communication Techniques

A variety of underlying communication networks can be used, and their 
topology need not be related to that of the key-sharing graph.

Communication systems based on simple cycles, called rings, are 
common in local area networks. In a typical ring, each node receives 
each bit and passes it round-robin to the next node. This technology is 
readily adapted to the present protocols. Consider a single-bit message 
like the "I paid" message originally sent at the dinner table. Each 
participant exclusive-or's the bit he receives with his own output 
before forwarding it to the next participant. When the bit has traveled 
full circle, it is the exclusive-or sum of all the participants' outputs, 
which is the desired result of the protocol. To provide these messages 
to all participants, each bit is sent around a second time by the 
participant at the end of the loop.

Such an adapted ring requires, on average, a fourfold increase in 
bandwidth over the obvious traceable protocols in which messages 
travel only halfway around on average before being taken off the ring by 
their recipients. Rings differ from the dinner table in that several bit-
transmission delays may be required before all the outputs of a 
particular round are known to all participants; collisions are detected 
only after such delays.

Efficient use of many other practical communication techniques 
requires participants to group output bits into blocks. For example, in 
high-capacity broadcast systems, such as those based on coaxial cable, 
surface radio, or satellites, more efficient use of channel capacity is 
obtained by grouping a participant's contribution into a block about the 
size of a single message (see, e.g., [5]). Use of such communication 
techniques could require an increase in bandwidth on the order of the 
number of participants.

In a network with one message per block, the well-known contention 
protocols can be used: time is divided evenly into frames; a participant 
transmits a block during one frame; if the block was garbled by 
collision (presumably with another transmitted block), the participant 
waits a number of frames chosen at random from some distribution 
before attempting to retransmit; the participants' waiting intervals 
may be adjusted on the basis of the collision rate and possibly of other 
heuristics [5].

In a network with many messages per block, a first block may be 
used by various anonymous senders to request a "slot reservation" in a 
second block. A simple scheme would be for each anonymous sender to 
invert one randomly selected bit in the first block for each slot they 
wish to reserve in the second block. After the result of the first block 
becomes known, the participant who caused the ith 1 bit in the first 
block sends in the ith slot of the second block.

2.3. Example Key-Sharing Graphs

In large systems it may be desirable to use fewer than the m(m - 1)/2 
keys required by a complete graph. If the graph is merely a cycle, then 
individuals acting alone learn nothing, but any two colluders can 
partition the graph, perhaps fully compromising a participant 
immediately between them. Such a topology might nevertheless be 
adequate in an application in which nearby participants are not likely 
to collude against one another.

A different topology assumes the existence of a subset of 
participants who each participant believes are sufficiently unlikely to 
collude, such as participants with conflicting interests. This subset 
constitutes a fully connected subgraph, and the other participants each 
share a key with every member of it. Every participant is then 
untraceable among all the others, unless all members of the completely 
connected subset cooperate. (Such a situation is mentioned again in 
Section 3.)

If many people wish to participate in an untraceable communication 
system, hierarchical arrangements may offer further economy of keys. 
Consider an example in which a representative from each local fully 
connected subgraph is also a member of the fully connected central 
subgraph. The nonrepresentative members of a local subgraph provide 
the sum of their outputs to their representative. Representatives would 
then add their own contributions before providing the sum to the 
central subgraph. Only a local subgraph's representative, or a collusion 
of representatives from all other local subgraphs, can recognize 
messages as coming from the local subgraph. A collusion comprising 
the representative and all but one nonrepresentative member of a local 
subgraph is needed for messages to be recognized as coming from the 
remaining member.

2.4. Secrecy and Authentication

What about the usual cryptologic problems of secrecy and 
authentication?

A cryptographer can ensure the secrecy of an anonymous message by 
encrypting the message with the intended recipient's public key. (The 
message should include a hundred or so random bits to foil attempts to 
confirm a guess at its content [1].) The sender can even keep the 
identity of the intended recipient secret by leaving it to each recipient 
to try to decrypt every message. Alternatively, a prearranged prefix 
could be attached to each message so that the recipient need only 
decrypt messages with recognized prefixes. To keep even the 
multiplicity of a prefix's use from being revealed, a different prefix 
might be used each time. New prefixes could be agreed in advance, 
generated cryptographically as needed, or supplied in earlier messages.

Authentication is also quite useful in systems without identification.
Even though the messages are untraceable, they might still bear
digital signatures corresponding to public-key "digital pseudonyms"
[1]; only the untraceable owner of such a pseudonym would be able to
sign subsequent messages with it. Secure payment protocols have
elsewhere been proposed in which the payer and/or the payee might be
untraceable [2]. Other protocols have been proposed that allow
individuals known only by pseudonyms to transfer securely information
about themselves between organizations [2]. All these systems require
solutions to the sender untraceability problem, such as the solution
presented here, if they are to protect the unlinkability of pseudonyms
used to conduct transactions from home.

2.5. Disruption

Another question is how to stop participants who, accidentally or even 
intentionally, disrupt the system by preventing others from sending 
messages. In a sense, this problem has no solution, since any 
participant can send messages continuously, thereby clogging the 
channel. But nondisupters can ultimately stop disruption in a system 
meeting the following requirements: (1) the key-sharing graph is 
publicly agreed on; (2) each participant's outputs are publicly agreed on 
in such a way that participants cannot change their output for a round 
on the basis of other participants' outputs for that round; and (3) some 
rounds contain inversions that would not compromise the 
untraceability of any nondisrupter.

The first requirement has already been mentioned in Section 1.1, 
where it was said that this information need not be secret; now it is 
required that this information actually be made known to all 
participants and that the participants agree on it.

The second requirement is in part that disrupters be unable (at least 
with some significant probability) to change their output after hearing 
other participants' outputs. Some actual channels would automatically 
ensure this, such as broadcast systems in which all broadcasts are 
made simultaneously on different frequencies. The remainder of the 
second requirement, that the outputs be publicly agreed on, might also 
be met by broadcasting. Having only channels that do not provide it 
automatically, an effective way to meet the full second requirement 
would be for participants to "commit" to their outputs before making 
them. One way to do this is for participants to make public and agree on 
some (possibly compressing and hierarchical, see Section 2.6) one-way 
function of their outputs, before the outputs are made public.

The third requirement is that at least some rounds can be contested 
(i.e., that all inversions can be made public) without compromising the 
untraceability of non-disrupting senders. The feasibility of this will be 
demonstrated here by a simple example protocol based on the slot 
reservation technique already described in Section 2.2.

Suppose that each participant is always to make a single reservation 
in each reserving block, whether or not he actually intends to send a 
message. (Notice that, because of the "birthday paradox," the number of 
bits per reserving block must be quadratic in the number of 
participants.) A disrupted reserving block would then with very high 
probability have Hamming weight unequal to the number of participants. 
All bits of such a disrupted reserving block could be contested without 
loss of untraceability for nondisrupters.

The reserved blocks can also be made to have such safely contestable
bits if participants send trap messages. To lay a trap, a participant
first chooses the index of a bit in some reserving block, a random
message, and a secret key. Then the trapper makes public an
encryption, using the secret key, of both the bit index and the random
message. Later, the trapper reserves by inverting in the round
corresponding to the bit index, and sends the random message in the
resulting reserved slot. If a disrupter is unlucky enough to have
damaged a trap message, then release of the secret key by the trapper
would cause at least one bit of the reserved slot to be contested.

With the three requirements satisfied, it remains to be shown how 
if enough disrupted rounds are contested, the disrupters will be 
excluded from the network.

Consider first the case of a single participant's mail computer 
disrupting the network. If it tells the truth about contested key bits it 
shares (or lies about an even number of bits), the disrupter implicates 
itself, because its contribution to the sum is unequal to the sum of 
these bits (apart from any allowed inversion). If, on the other hand, the 
single disrupter lies about some odd number of shared bits, the values 
it claims will differ from those claimed for the same shared bits by 
the other participants sharing them. The disrupter thereby casts 
suspicion on all participants, including itself, that share the disputed 
bits. (It may be difficult for a disrupter to cast substantial suspicion 
on a large set of participants, since all the disputed bits will be in 
common with the disrupter.) Notice, however, that participants who 
have been falsely accused will know that they have been--and by 
whom--and should at least refuse to share bits with the disrupter in 
the future.

Even with colluding multiple disrupters, at least one inversion must 
be revealed as illegitimate or at least one key bit disputed, since the 
parity of the outputs does not correspond to the number of legitimate 
inversions. The result of such a contested round will be the removal of 
at least one edge or at least one vertex from the agreed graph. Thus, if 
every disruptive action has a nonzero probability of being contested, 
only a bounded amount of disruption is possible before the disrupters 
share no keys with anyone in the network, or before they are revealed, 
and are in either case excluded from the network.

The extension presented next can demonstrate the true value of 
disputed bits, and hence allows direct incrimination of disrupters.

2.6. Tracing by Consent

Antisocial use of a network can be deterred if the cooperation of most 
participants makes it possible, albeit expensive, to trace any message. 
If, for example, a threatening message is sent, a court might order all 
participants to reveal their shared key bits for a round of the message. 
The sender of the offending message might try to spread the blame, 
however, by lying about some odd number of shared bits. Digital 
signatures can be used to stop such blame-spreading altogether. In 
principle, each party sharing a key could insist on a signature, made by 
the other party sharing, for the value. of each shared bit.

Such signatures would allow for contested rounds to be fully resolved,
for accused senders to exonerate themselves, and even for colluders to
convince each other that they are pooling true keys.  Unfortunately,
cooperating participants able to trace a message to its sender could
convince others of the message's origin by revealing the sender's own
signatures. A variation can prevent a participant's signatures from
being used against him in this way: instead of each member of a pair
of participants signing the same shared key bit, each signs a separate
bit, such that the sum of the signed bits is the actual shared key
bit. Signatures on such "split" key bits would still be useful in
resolving contested rounds, since if one contester of a bit shows a
signature made by the second contester, then the second would have to
reveal the corresponding signature made by the first or be thought to
be a disrupter.

In many applications it may be impractical to obtain a separate 
signature on every key bit or split key bit. The overhead involved could 
be greatly reduced, however, by digitally signing cryptographic 
compressions of large numbers of key bits. This might of course require 
that a whole block of key bits be exposed in showing a signature, but 
such blocks could be padded with cryptographically generated 
pseudorandom (or truly random) bits, to allow the exposure of fewer 
bits per signature. The number of bits and amount of time required to 
verify a signature for a single bit can be reduced further by using a 
rooted tree in which each node is the one-way compression function of 
all its direct descendants; only a digital signature of each participant's 
root need be agreed on before use of the keys comprising the leaves.

3. Relation to Previous Work

There is another multiparty-secure sender-untraceability protocol in 
the literature [1]. To facilitate comparison, it will be called a mix-net 
here, while the protocol of the present work is called a dc-net. The 
mix-net approach relies on the security of a true public-key system 
(and possibly also of a conventional cryptosystem), and is thus at best 
computationally secure; the dc-net approach can use unconditional 
secrecy channels to provide an unconditionally secure untraceable-
sender system, or can use public-key distribution to provide a 
computationally secure system (as described in Section 2.1).

Under some trust assumptions and channel limitations, however, 
mix-nets can operate where dc-nets cannot. Suppose that a subset of 
participants is trusted by every other participant not to collude and 
that the bandwidth of at least some participants' channels to the 
trusted subset is incapable of handling the total message traffic. Then 
mix-nets may operate quite satisfactorily, but dc-nets will be unable 
to protect fully each participant's untraceability. Mix-nets can also 
provide recipient untraceability in this communication environment, 
even though there is insufficient bandwidth for use of the broadcast 
approach (mentioned in Section 2.4).

If optimal protection against collusion is to be provided and the 
crypto-security of mix-nets is acceptable, a choice between mix-nets 
and dc-nets may depend on the nature of the traffic. With a mail-like 
system that requires only periodic deliveries, and where the average 
number of messages per interval is relatively large, mix-nets may be 
suitable. When messages must be delivered continually and there is no 
time for batching large numbers of them, dc-nets appear preferable.

4. Conclusion

This solution to the dining cryptographers problem demonstrates that
unconditional secrecy channels can be used to construct an
unconditional sender-untraceability channel. It also shows that a
public-key distribution system can be used to construct a
computationally secure sender-untraceability channel. The approach
appears able to satisfy a wide range of practical concerns.

Acknowledgments

I am pleased to thank Jurjen Bos, Gilles Brassard, Jan-Hendrik Evertse, 
and the untraceable referees for all their help in revising this article. 
It is also a pleasure to thank, as in the original version that was 
distributed at Crypto 84, Whitfield Diffie, Ron Rivest, and Gus Simmons 
for some stimulating dinner-table conversations.

References

[1]	Chaum, D., Untraceable Electronic Mail, Return Addresses, and 
Digital Pseudonyms, Communications of the  ACM, vol. 24, no. 2, 
February 1981, pp. 84-88.
[2]	Chaum, D., Security Without Identification: Transaction Systems 
to Make Big Brother Obsolete, Communications of the ACM, vol. 28, 
no. 10, October 1985, pp. 1030-1044.
[3]	Diffie, W., and Hellman, M.E., New Directions in Cryptography, IEEE 
Transactions on Information Theory, vol. 22, no. 6, November 1976, 
pp. 644-654.
[4]	Merkle, R.C., Secure Communication over Insecure Channels, 
Communications of the ACM, vol. 21, no. 4, 1978, pp. 294-299.
[5]	Tanenbaum, A.S., Computer Networks, Prentice Hall, Englewood 
Cliffs, New Jersey, 1981.


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