Anonymous Communications Systems
Michael J. Freedman
< email@example.com >
We earlier described several major implementations of anonymous
communications channels. This appendix serves to give a more detailed
survey of research and development in the area of anonymous
communications. Some of these projects are not implemented; some exist
more as a proof-of-concept by their respective designers; and still
others repeat design and functionality provided by like systems.
We review three main types of design: proxy-servers, mix-nets, and other
anonymous communications channels.
Proxy services provide one of the most basic forms of anonymity,
inserting a third party between the sender and recipient of a given
message. Proxy services are characterized as having only one centralized
layer of separation between message sender and recipient. The proxy
serves as a ``trusted third party,'' responsible for sufficiently
stripping headers and other distinguishing information from sender
Proxies only provide unlinkability between sender and receiver, given
that the proxy itself remains uncompromised. This unlinkability does
not have the quality of perfect forward anonymity, as proxy users often
connect from the same IP address. Therefore, any future information
used to gain linkability between sender and receiver (i.e., intersection
attacks, traffic analysis) can be used against previously recorded
Sender and receiver anonymity is lost to an adversary that may monitor
incoming traffic to the proxy. While the actual contents of the message
might still be computationally secure via encryption, the adversary can
correlate the message to a sender/receiver agent.
This loss of sender/receiver anonymity plagues all systems which include
external clients which interact through a separate communications
channel - that is, we can define some distinct edge of the channel. If
an adversary can monitor this edge link or the first-hop node within the
channel, this observer gains agent-message correlation. Obviously, the
ability to monitor this link or node depends on the adversary's
resources and the number of links and nodes which exist. In a proxy
system, this number is small. In a globally-distributed mixnet, this
number could be very large. The adversary's ability also depends on her
focus: whether she is observing messages and agents at random, or if she
is monitored specific senders/receivers on purpose.
The Anonymizer was one of the first examples of a form-based web
proxy [#!anonymizer!#]. Users point their browsers at the
Anonymizer page at www.anonymizer.com. Once there, they enter
their destination URL into a form displayed on that page. The Anonymizer
then acts as an http proxy for these users, stripping off all
identifying information from http requests and forwarding them
on to the destination URL.
The functionality is limited. Only http requests are proxied,
and the Anonymizer does not handle cgi scripts. In addition, unless the
user chains several proxies together, he or she may be vulnerable to an
adversary which tries to correlate incoming and outgoing http
requests. Only the data stream is anonymized, not the connection
itself. Therefore, the proxy does not prevent traffic analysis attacks
like tracking data as it moves through the network.
Chaining multiple proxies together by hand is a tedious business,
requiring many preliminaries before the first web page is reached.
Lucent's Proxymate software automates the process[#!lucent!#]. The
software looks like a proxy sitting on the user's computer. By setting
software to use the Proxymate proxy, the user causes the software's
requests and traffic to go to the software, which then automatically
negotiates a chain of proxies for each connection.
Another piece of software which helps manage many distinct proxies in a
transparent manner is Proxomitron[#!proxomitron!#]. In addition to basic
listing and chaining of proxies, Proxomitron allows users to write
filter scripts. These filters can then be applied to incoming and
outgoing traffic to do everything from detecting a request for the
user's e-mail address by a web site to automatically changing colors on
incoming web pages.
The project of anonymity on the Internet was kicked off by David Chaum
in 1981 with a paper in Communications of the ACM describing a system
called a ``Mix-net.'' This system uses a very simple technique to
provide anonymity: a sender and receiver are linked by a chain of
servers called Mixes. Each Mix in the chain strips off the identifying
marks on incoming messages and then sends the message to the next Mix,
based on routing instructions which encrypted with its public key.
Comparatively simple to understand and implement, this Mix-net (or
``mix-net'' or ``mixnet'') design is used in almost all of today's
practical anonymous channels.
Chaum's original paper introduced the basic concept of a Mix as a sort
of ``permutation box.'' On the incoming side is a list of messages
representing the messages which have arrived at the Mix server, each of
which is identified with a particular sender. On the outgoing side is a
randomly permuted list of messages, which have lost their identification
with the sender. The assumption is that if the Mix works correctly, no
adversary can do better than guessing to link an incoming message with
an outgoing message.
Chaum's original Digital Mix was described in terms of a series of Mix
nodes which passed idealized messages over a network. The first proposal
for the practical application of mixes came from Pfitzmann et. al.
[#!ISDN-mix!#], who showed how a mix-net could be used with ISDN lines
to anonymize a telephone user's real location. Their motivation was to
protect the privacy of the user in the face of a telephone network owned
by a state telephone monopoly.
Their paper introduced a distinction between explicit and
implicit addresses. An explicit address is something about a
message which clearly and unambiguously links it to a recipient and can be
read by everyone, such as a To: header. An implicit address is an
attribute of a message which links it to a recipient and can only be
determined by that recipient. For example, being encrypted with the
recipient's public key in a recipient-hiding public key is an implicit
Until the rise of proxy-based and TCP/IP-based systems, the most popular
form of anonymous communication was the anonymous remailer: a
form of mix which works for e-mail sent over SMTP. Remailers are
informally divided into three categories, called Type 0, Type 1, and
One of the first and most popular remailers was anon.penet.fi,
run by Johan Helsingius. This remailer was very simple to use. A user
simply added an extra header to e-mail indicating the final destination,
which could be either an e-mail address or a Usenet newsgroup. This
e-mail was sent to the anon.penet.fi server, which stripped off
the return address and forwarded it along. In addition, the server
provided for return addresses of the form ``anXXXX@anon.penet.fi''; mail
sent to such an address would automatically be forwarded to another
e-mail address. These pseudonyms could be set up with a single e-mail to
the remailer; the machine simply sent back a reply with the user's new
The anon.penet.fi remailer is referred to as a Type 0 remailer
for two reasons. First, it was the original ``anonymous remailer.'' More
people used anon.penet.fi than are known to have used any
following type of remailer. Exact statistics are hard to come by, but X
number of accounts were registered at penet.fi, and only Y are
currently registered at nym.alias.net.
Second, anon.penet.fi did not provide some of the features
which motivated the development of ``Type I'' and ``Type II''
remailers. In particular, it provided a single point of failure and the
remailer administrator had access to each user's ``real'' e-mail
address. In general, any remailer system which consists of a single hop
is considered Type 0.
This last feature proved to be the service's undoing. The Church of
Scientology, a group founded by the science fiction writer L. Ron
Hubbard, sued a penet.fi pseudonym for distributing materials
reserved for high initiates to a Usenet newsgroup. Scientology claimed
that the material was copyrighted ``technology.'' The poster claimed it
was a fraud used to extort money from gullible and desperate
fools. Scientology won a court judgment requiring the
anon.penet.fi remailer to give up the true name of the
pseudonymous poster, which the operator eventually did. This incident,
plus several allegations of traffic in child pornography, eventually
convinced Johan Helsingius to close the service in 1995[#!helsingius!#].
Services similar to Type 0 remailers now exist in the form of ``free
e-mail'' services such as Hotmail, Hushmail, and ZipLip, which allow
anyone to set up an account via a web form. Hushmail and ZipLip even
keep e-mail in encrypted form on their server. Unfortunately, these
services are not sufficient by themselves, as an eavesdropping adversary
can determine which account corresponds to a user simply by watching him
or her login.
The drawbacks of anon.penet.fi spurred the development of
``cypherpunks'' or ``Type 1'' remailers, so named because their design
took place on the cypherpunks mailing list. This generation of remailers
addressed the the two major problems with anon.penet.fi: first,
the single point of failure, and second, the vast amount of information
about users of the service collected at that point of failure. Several
remailers exist; a current list can be found at the Electronic Frontiers
Georgia site [#!mixmaster!#] or on the newsgroup
Each cypherpunk remailer has a public key and uses PGP for
encryption. Mail can be sent to each remailer encrypted with its key,
preventing an eavesdropper from seeing it in transit. A message sent to
a remailer can consist of a request to remail to another remailer and a
message encrypted with the second remailer's public key. In this way a
chain of remailers can be built, such that the first remailer in the
chain knows the sender, the last remailer knows the recipient, and the
middle remailers know neither.
Cypherpunk remailers also allow for reply blocks. These consist
of a series of routing instructions for a chain of remailers which
define a route through the remailer net to an address. Reply blocks
allow users to create and maintain pseudonyms which receive e-mail. By
prepending the reply block to a message and sending the two together to
the first remailer in the chain, a message can be sent to a party
without knowing his or her real e-mail address.
While Cypherpunk remailers represented a major advance over
anon.penet.fi, they fell short of the anonymity provided by the ideal
mix. In 1995, Lance Cottrell outlined some of the problems with ``Type
I'' remailers [#!mixmaster!#]:
- Traffic Analysis: Cypherpunk remailers tend to send messages
as soon as they arrive, or after some specified amount of delay. The
first option makes it easy for an adversary to correlate messages across
the mix-net. It's not clear how much delay helps protect against this
- Does Not Hide Length: The length of messages is not hidden by
the encryption used by cypherpunk remailers. This allows an adversary to
track a message as it passes through the mixnet by looking for messages
of approximately the same length. [Note that the definitions of
semantic security and non-malleability do not seem to imply
Cottrell wrote the Mixmaster, or ``Type II'', remailer to address these
problems. Instead of using PGP, Mixmaster uses its own client software
(which is also the server software), which understands a special
Mixmaster packet format. All Mixmaster packets are the same
length. Every message is encrypted with a separate 3DES key for each mix
node in a chain between the sender and receiver; these 3DES keys are in
turn encrypted with the RSA public keys of each mix node.
When a message reaches a mix node, it decrypts the header, decrypts the
body of the message, and then places the message in a ``message pool.''
Once enough messages have been placed in the pool, the node picks a
random message to forward.
As of this writing, Mixmaster is in version 2.9b22[#!mixmaster-code!#].
Discussion of the project can be found on the mix-l mailing
list[#!mix-l!#]. A Mixmaster version 3 is planned in which nodes will
communicate with each other via TCP/IP connections. All traffic will be
encrypted with a key derived by a Diffie-Hellman key exchange and then
destroyed immediately after the transaction is ended, thereby providing
perfect forward secrecy. Unfortunately, the prototype specification for
this protocol is only available in German and is not finished.
The reply blocks used by cypherpunks remailers are important for
providing for return traffic, but they must be sent to every
correspondent individually. In addition, using a reply block requires
that a correspondent be familiar with the use of specialized
software. This problem is addressed by nymservers, which act as
holding and processing centers for reply blocks.
To use a nymserver, a user simply registers an e-mail address of the
form ``firstname.lastname@example.org'' and associates a reply block with it. This
association can be carried out via anonymous e-mail. Then whenever a
message is sent to ``email@example.com,'' the nymserver automatically
prepends the associated reply block, encrypts the aggregate, and sends
it off to the appropriate anonymous remailer.
The most popular nymserver may be the one run at nym.alias.net, which is
hosted at MIT's Lab for Computer Science. A recent report by Mazieres
and Kaashoek details the technical and social details of running the
nymserver, including problems of abuse[#!nymserver!#].
The major reason for the massive popularity of anon.penet.fi
was that it was extremely easy to use. Anyone who could type
``Request-Remailing-To:'' at the top of an e-mail message could send
anonymous e-mail. With the advent of remailers which required the use of
PGP or the Mixmaster software, the difficulty of using remailers
increased. This difficulty was aggravated by the fact that for years,
both PGP and Mixmaster were only available as command-line applications
with a bewildering array of options.
Goldberg and Wagner applied Mixes to the task of designing an anonymous
publishing network called Rewebber[#!taz-rewebber!#]. Rewebber uses
URLs which contain the name of a Rewebber server and a packet of
encrypted information. When typed into a web browser, the URL sends the
browser to the Rewebber server, whch decrypts the associated packet to
find the address of either another Rewebber server or a legitimate web
site. In this way, web sites can publish content without revealing
Mapping between intelligible names and Rewebber URLs is performed by a
name server called the Temporary Autonomous Zone(TAZ), named after a
novel by Hakim Bey. The point of the ``Temporary'' in the name of the
nameserver (and the novel) is that static structures are vulnerable to
attack. Continually refreshing the Rewebber URL makes it harder for an
adversary to gain information about the server to which it refers.
Contemporary with Cotrell's Mixmaster is an effort by Gulcu and Tsudik
called ``Babel''[#!babel!#]. Babel uses a modified version of PGP as its
underlying encryption engine. This modified version does not include
normal headers, which would include the identity of the receiver, the
PGP version number, and other identifying information.
The Babel paper defines quantities called the ``guess factor'' and the
``mix factor'' which model the ability of an adversary to match messages
passing through the mix with their original senders. Then several
attacks are presented, including the trickle and flooding attack, along
with some countermeasures. The paper is noteworthy in that it attempts
to give an analysis of just how much the practice of batching messages
helps the untraceability of a mix-net node.
The next step in probabilistic analysis for mixnets comes in the work of
Kesdogan, Egner, and Buschkes [#!sg-mix!#], who proposed the ``Stop and
Go Mix.'' They divide networks into two kinds: ``closed'' networks, in
which the number of users is small, known in advance, and all users can
be made distinct, and ``open'' networks like the Internet with extremely
large numbers of users. They claim that perfect anonymity cannot be
achieved in these open networks, because there is no guarantee that
every single client of the mix node is not the same person coming under
Instead, they define and consider a notion of probabilistic
anonymity: given that the adversary controls some percentage of the
clients, some other set of mix servers, and is watching a Mix, can the
probability of correlating messages be quantified in terms of some
security parameter? They consider queueing theory as an inspiration for
a statistical model and manage to prove theorems about the adversary's
knowledge in this model.
Later, Kesdogan et. al. applied Mixes to the GSM mobile telephone
setting[#!kesdogan-vil!#]. Here, the point is to allow for GSM roaming
from cell to cell while still protecting the user's real location from
discovery by the phone company or an outside intruder. This is done by
the use of variable implicit addresses, which work as follows :
each roaming area has a publically known and static explicit
address. When the client GSM phone comes online or crosses the
boundaries of a cell, it queries the surrounding cells and downloads
these addresses. Then it creates a new address for itself which combines
the addresses of its surrounding cells.
Then, instead of sending the entirety of the new address, the phone
sends only some characters, say log n, of the address to the
network to identify itself. The network then directs traffic intended
for the phone to any cell which has those log n characters in its
address. A refinement process then takes place in which the phone gives
out slightly more information to the system to improve performance by
sending information to fewer cells, but not so much as to allow its
location to be restricted to only one cell.
At EUROCRYPT '98, Jakobsson proposed a mixnet which was both practical
and could be proved to mix correctly as long as less than 1/2 of the
servers were corrupted[#!jakobsson-practical-mix!#]. The crucial idea is
to treat the mixing as a secure multiparty computation in which each
party is collaborating to make the collective mix look like a ``random
enough// permutation on a batch of messages. Then techniques of
zero-knowledge proof are used by which each server can prove to all
other servers that they are in fact conforming to the mix
protocol. Deviating servers cannot produce valid proofs, and so can be
caught and excluded from future mixing. Jakobsson's original protocol
requires in the neighborhood of 160 modular exponentiations per message
At PODC '99, Jakobsson showed how the use of precomputation could reduce
the cost even further[#!jakobsson-flash-mix!#]. This new ``flash mix''
required only around 160 modular multiplications per message
per server. This level of efficiency makes flash mixing competitive with
the encryption used in anonymous remailers, and a serious candidate for
At Eurocrypt '00, ``How to Break a practical mix, and fix it.''
With Jakobsson's design, the correctness of a mix-net can only be
verified by the mix servers themselves. When more than a threshold of
servers is corrupt, the verification fails. Because a user of the
mix-net may not be aware of the corruption, this failure may be silent
and therefore dangerous. One solution to this problem is a
universally verifiable mix-net - a mix-net whose correctness
can be verified by anyone, regardless of their status as server or user.
The concept was introduced by Killian [#!universal-verifiable-mix-1!#],
and recently a design of this type was proposed at EUROCRYPT '98 by Abe
[#!abe-mix!#]. This design works along the similar broad lines as the
Jakobsson design; each mix server uses zero-knowledge proofs to prove
that it is acting in accordance with some protocol to randomly mix
messages. The difference here is that these proofs are posted
publically by the mix nodes instead of being multicast only to other mix
nodes. The novel feature of Abe's design is that the work necessary to
verify these proofs grows in a fashion independent of the number of
servers. Unfortunately, verifying these proofs requires on the order of
1600 modular exponentiations per message.
The Onion Routing system designed by Syverson, et. al. creates a mix-net
for TCP/IP connections [#!onion-routing-paper!#,#!onion-router!#]. In
the Onion Routing system, a mixnet packet, or ``onion'', is created by
successively encrypting a packet with the public keys of several mix
servers, or ``onion routers.''
When a user places a message into the system, an ``onion proxy''
determines a route through the anonymous network and onion encrypts the
message accordingly. Each onion router which receives the message peels
the topmost layer, as normal, then adds some key seed material to be
used to generate keys for the anonymous communication. As usual, the
changing nature of the onion - the ``peeling'' process - stops message
coding attacks. Onions are numbered and have expire times, to stop
replay attacks. Onion routers maintain network topology by
communicating with neighbors, using this information to initially build
routes when messages are funneled into the system. By this process,
routers also establish shared DES keys for link encryption.
The routing is performed on the application layer of onion proxies, the
path between proxies dependent upon the underlying IP network.
Therefore, this type of system is comparable to loose source routing.
Onion Routing is mainly used for sender-anonymous communications with
non-anonymous receivers. Users may wish to Web browse, send email, or
use applications such as rlogin. In most of these real-time
applications, the user supplies the destination hostname/port or IP
address/port. Therefore, this system only provides receiver-anonymity
from a third-party, not from the sender.
Furthermore, Onion Routing makes no attempt to stop timing attacks using
traffic analysis at the network endpoints. They assume that the routing
infrastructure is uniformly busy, thus making passive intra-network
timing difficult. However, the network might not be statistically
uniformly busy, and attackers can tell if two parties are communicating
via increased traffic at their respective endpoints. This
endpoint-linkable timing attack remains a difficulty for all low-latency
Recently, the Canadian company Zero Knowledge Systems has begun the
process of building the first mix-net operated for profit, known as
Freedom [#!zks!#]. They have deployed two major systems, one for
e-mail and another for TCP/IP. The e-mail system is broadly similar to
Mixmaster, and the TCP/IP system similar to Onion Routing.
ZKS's ``Freedom 1.0'' application is designed to allow users to use a
nym to anonymously access web pages, use IRC, etc. The anonymity comes
from two aspects: first of all, ZKS maintains what it calls the Freedom
Network, which is a series of nodes which route traffic amongst
themselves in order to hide the origin and destination of packets, using
the normal layered encryption mixnet mechanism. All packets are of the
same size. The second aspect of anonymity comes from the fact that
clients purchase ``tokens'' from ZKS, and exchange these token for nyms
- supposedly even ZKS isn't able to correlate identities with their use
of their nyms.
The Freedom Network looks like it does a good job of actually
demonstrating an anonymous mixnet that functions in real-time. The
system differs from Onion Routing in several ways.
First of all, the system maintains Network Information Query and Status
Servers, which are databases which provide network topology, status, and
ratings information. Nodes also query the key servers every hour to
maintain fresh public keys for other nodes, then undergo authenticated
Diffie-Hellman key exchange to allow link encryption. This system
differs from online inter-node querying that occurs with Onion Routing.
Combined with centralized nym servers, time synchronization, and key
update/query servers, the Freedom Network is not fully decentralized
Second, the system does not assume uniform traffic distribution, but
instead uses a basic ``heartbeat'' function that limits the amount of
inter-node communication. Link padding, cover traffic, and a more
robust traffic-shaping algorithm have been planned and discussed, but
are currently disabled due to engineering difficulty and load on the
servers. ZKS recognizes that statistical traffic analysis is possible
Third, Freedom loses anonymity for the primary reason that it is a
commercial network operated for profit. Users must purchase the nyms
used in pseudonymous communications. Purchasing is performed
out-of-band via an online Web store, through credit-card or cash
payments. ZKS uses a protocol of issuing serial numbers, which are
reclaimed for nym tokens, which in turn are used to anonymously purchase
nyms. However, this system relies on ``trusted third party'' security:
the user must trust that ZKS is not logging IP information or recording
serial-token exchanges that would allow them to correlate nyms to users
[#!freedom-nyms!#]. The future adoption of anonymous ecash purchasing
should remove this weakness, and allow truely anonymous nym issuing.
Another more recent effort to apply a Mix network to web browsing is due
to Federrath et. al.[#!web-mix!#] who call their system, appropriately
enough, ``Web Mixes.'' From Chaum's mix model, similar to other
real-time systems, they use: layered public-key encryption, prevention
of replay, constant message length within a certain time period, and
reordering outgoing messages.
The Web Mixes system incorporates several new concepts. First, they use
an adaptive ``chop-and-slice'' algorithm that adjusts the length used
for all messages between time periods according to the amount of network
traffic. Second, dummy messages are sent from user clients as long as
the clients are connected to the Mix network. This cover traffic makes
it harder for an adversary to perform traffic analysis and determine
when a user sends an anonymous message, although the adversary can still
tell when a client is connected to the mixnet. Third, Web Mixes attempt
to restrict insider and outsider flooding attacks by limited either
available bandwidth or the number of used time slices for each user. To
do this, users are issued a set number of blind signature tickets for
each time slice, which are spent to send anonymous messages. Lastly,
this effort includes an attempt to build a statistical model which
characterizes the knowledge of an adversary attempting to perform
The Dining Cryptographers protocol was introduced by David
Chaum[#!chaum-dc!#] and later improved by Pfitzmann and Waidner as a
means of guaranteeing untraceability for the sender and receiver of a
message, even against a computationally all-powerful adversary. The
protocol converts any broadcast channel into an anonymous broadcast
channel. In the context of Free Haven, however, we have a problem : the
participants in the protocol are identified, even though the sender and
receiver of any given message is not. If the only long-term participants
in the protocol are likely to be Free Haven servnet nodes, then we do
not achieve the server-anonymity we desire. Less serious, but still
important, problems are the efficiency of the protocol and the
difficulty of correct implementation.
Therefore we have not seriously considered using the dining
cryptographers protocol to provide Free Haven's anonymous channel. If we
were to do so, we might consider running a dining cryptographer protocol
using Mixes to hide the legal identity of each participant. In that
case, while a failure of the Mix would reveal a participant's identity,
the anonymous broadcast would prevent him or her from being linked to
any particular message.
The Crowds system was proposed and implemented by AT&T Research,
named for collections of users that are used to achieve partial
anonymity for Web browsing [#!crowds!#]. A user initially joins some
crowd and her system begins acting as a node, or anonymous jondo,
within that crowd. In order to instantiate communications, the user
creates some path through the crowd by a random-walk of jondos,
in which each jondo has some small probability of sending the
actual http request to the end server. Once established, this
path remains static as long as the user remains a member of that crowd.
The Crowds system does not use dynamic path creation so that colluding
crowd eavesdroppers are not able to probabilistically determine the
initiator (i.e., the actual sender) of requests, given repeated requests
through a crowd. The jondos in a given path also share a secret
path key, such that local listeners, not part of the path, only
see an encrypted end server address until the request is finally sent
off. The Crowds system also includes some optimizations to handle timing
attacks against repeated requests, as certain HTML tags cause browsers
to automatically issue re-requests.
Similar to other real-time anonymous communication channels (Onion
Routing, the Freedom Network, Web Mixes), Crowds is used for senders to
communicate with a known destination. The system attempts to achieve
sender-anonymity from the receiver and a third-party adversary.
Receiver-anonymity is only meant to be kept from adversaries, not from
the sender herself.
The Crowds system serves primarily to achieve sender and receiver
anonymity from an attacker, not provide unlinkability between the two
agents. Due to high availibility of data - real-time access is faster
that mix-nets as Crowds does not use public key encryption - an
adversary can more easily use traffic analysis or timing attacks.
However, Crowds differs from all other systems we have discussed, as
users are members of the communications channel, rather than
merely communicating through it. Sender-anonymity is still lost
to a local eavesdropper that can observe all communications to and from
a node. However, other colluding jondos along the sender's path
- even the first-hop - cannot expose the sender as originated the
message. Reiter and Rubin show that as the number of crowd members goes
to infinity, the probable innocence of the last-hop being the sender
In CRYPTO '97, Ostrovsky considered a slightly different model of
anonymous broadcast. In this model, there are n servers
broadcasting into a shared broadcast channel. One of the servers is a
special ``Command and Control'' server; the rest are broadcasting dummy
traffic. Then there is an adversary who has control of some of the
servers and wants to know which server is the ``Command and Control.''
Ostrovsky shows how to use correlated pseudo-random number generators
whose output reveals a certain message when XORed together to create a
protocol which prevents the adversary from discovering which server is
the correct one, even if he can eavesdrop on all communications and
corrupt up to k servers, where k is a security parameter
which affects the efficiency of the protocol.
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The translation was initiated by Michael J Freedman on 2000-07-27