Communications of the ACM February 1981 Volume 24 Number 2 --------------------------------------------------------------------------- Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. This work was partially supported by the National Science Foundation under Grant MCS 75-23739 and by teh Air Force Office of Scientific Research under Contract F49620-9-CO173. Author's present address: Computer Science Division, Electrical Engineering and Computer Sciences Department, University of California, Berkeley, California 94720. (415)42-1024. Copyright (C) 1981 ACM 0001-082/81/0200-0084 $00.75. --------------------------------------------------------------------------- Technical Note Programming Techniques and Data Structures R. Rivest, Editor --------------------------------------------------------------------------- Untraceable Electronic Mail, Return addresses, and Digital Pseudonyms David L. Chaum University of California, Berkeley Abstract A technique based on public key cryptography is presented that allows an electronic mail system to hide who a participant communicates with as well as the content of the communication - in spite of an unsecured underlying telecommunication system. The technique does not require a universally trusted authority. One correspondent can remain anonymous to a second, while allowing the second to respond via an untraceable return address. The technique can also be used to form rosters of untraceable digital pseudonyms from selected applications. Applicants retain the exclusive ability to form digital signatures corresponding to their pseudonyms. Elections in which any interested party can verify that the ballots have been properly counted are possible if anonymously mailed ballots are signed with pseudonyms from a roster of registered voters. Another use allows an individual to correspond witha record-keeping organization under a unique pseudonym which appears in a roster of acceptable clients. Key Words and Phrases: electronic mail, public key cryptosystems, digital signatures, traffic analysis, security, privacy CR Categories: 2.12, 3.81 Introduction Cryptology is the science of secret communication. Cryptographic techniques have been providing secrecy of message content for thousands of years [3]. Recently some new solutions to the "key distribution problem" (the problem of providing each communicant with a secret key) have been suggested [2,4], under the name of public key cryptography. Another cryptographic problem, "the traffic analysis problem" (the problem of keeping confidential who converses with whom, and when they converse), will become increasingly important with the growth of electronic mail. This paper presents a solution to the traffic analysis problem that is based on public key cryptography. Baran has solved the traffic analysis problem for networks [1], but requires each participant to trust a common authority. In contrast, systems based on the solution advanced here can be compromised only by subversion or conspiracy of all of a set of authorities. Ideally, each participant is an authority. The following two sections introduce the notation and assumptions. Then the basic concepts are introduced for some special cases involving a series of one or more authorities. The final section covers general purpose mail networks. Notation Someone becomes a user of a public key cryptosystem (like that of Rivest, Shamir, and Adleman [5]) by creating a pair of keys K and Inv(K) from a suitable randomly generated seed. The public key K is made known to the other users or anyone else who cares to know it; the private key Inv(K) is never divulged. The encryption of X with key K will be denoted K( X ), and is just the image of X under the mapping implemented by the cryptographic algorithm using key K. The increased utility of these algorithms over conventional algorithms results because the two keys are inverses of each other, in the sense that Inv(K)( K( X ) ) = K( Inv(K)( X ) ) = X. A message X is "sealed" with a public key K so that only the holder of the private key Inv(K) can discover its content. If X is simply encrypted with K, then anyone could verify a guess that Y = X by checking whether K( Y ) = K( X ). This threat can be eliminated by attaching a large string of random bits R to X before encrypting. The sealing of X with K is then denoted K( R, X ). A user "signs" some material X by prepending a large constant C (all zeros, for example) and then encrypting with its private key, denoted Inv(K)( C, X ) = Y. Anyone can verify that Y has been signed by the holder of Inv(K) and determine the signed matter X, by forming K( Y ) = C, X and checking for C. Assumptions The approach taken here is based on two important assumptions: (1) No one can determine anything about the correspondences between a set of sealed items and the corresponding set of unsealed items, or create forgeries without the appropriate random string or private key. (2) Anyone may learn the origin, destination(s), and representation of all messages in the underlying telecommunication system and anyone may inject, remove, or modify messages. Mail System The users of the cryptosystem will include not only the correspondents but a computer called a "mix" that will process each item of mail before it is delivered. A participant prepares a message M for delivery to a participant at address A by sealing it with the addressee's public key Ka, appending the address A, and then sealing the result with the mix's public key K1. The left-hand side of the following expression denotes this item which is input to the mix: K1( R1, Ka( R0, M ), A ) --> Ka( R0, M ), A. The --> denotes the transformation of the input by the mix into the output shown on the right-hand side. The mix decrypts its input with its private key, throws away the random string R1, and outputs the remainder. One might imagine a mechanism that forwards the sealed messages Ka( R0, M ) of the output to the addressees who then decrypt them with their own private keys. The purpose of a mix is to hide the correspondences between the items in its input and those in its output. The order of arrival is hidden by outputting the uniformly sized items in lexicographically ordered batches. By assumption (1) above, there need be no concern about a cryptanalytic attack yielding the correspondence between the sealed items of a mix's input and its unsealed output - if items are not repeated. However, if just one item is repeated in the input and allowed to be repeated in the output, then the correspondence is revealed for that item. Thus, an important function of a mix is to ensure that no item is processed more than once. This function can be readily achieved by a mix for a particular batch by removing redundant copies before outputting the batch. If a single mix is used for multiple batches, then one way that repeats across batches can be detected is for the mix to maintain a record of items used in previous batches. (Records can be discarded once a mix changes its public key by, for example, announcing the new key in a statement signed with its old private key.) A mix need not retain previous batches if part of each random string R1 constains something - such as a time-stamp - that is only valid for a particular batch. If a participant gets signed receipts for messages it submits to a mix, then the participant can provide substantial evidence that the mix failed to output an item properly. Only a wronged participant can supply the receipt Y (= Inv(K1)( C, K1( R1, Ka( R0, M ), A ))), the missing output X (= Ka( R0, M ), A ), and the retained string R1, such that K1( Y ) = C, K1( R1, X ). Becasue a mix will sign each output batch as a whole, the absence of an item X from a batch can be substantiated by a copy of the signed batch. The use of a "cascade", or series of mixes, offers the advantage that any single constituent mix is able to provide the secrecy of the correspondence between the inputs and the outputs of the entire cascade. Incrimination of a particular mix of a cascade that failed to properly process an item is accomplished as with a single mix, but only requires a receipt from the first mix of the cascade, since a mix can use the signed output of its predecessor to show the absence of an item from its own input. An item is prepared for a cascade of n mixes the same as for a single mix. It is then successively sealed for each succeeding mix: Kn( Rn, K<n-1>( R<n-1>, ... , K2( R2, K1( R1, Ka( R0, M ), A ))...)) -->. The fist mix yields a lexicographically ordered batch of items, each of the form K<n-1>( R<n-1>, ... , K2( R2, K1( R1, Ka( R0, M ), A ))...) -->. The items in the final output batch of a cascade are of the form Ka( R0, M ), A, the same as those of a single mix. Return Addresses The techniques just described allow participant x to send anonymous messages to participant y. What is needed now is a way for y to respond to x while still keeping the identity of x secret from y. A solution is for x to form an untraceable return address K1( R1, Ax), Kx, where Ax is its own real address, Kx is a public key chosen for the occasion, and R1 is a key that will also act as a random string for purposes of sealing. Then, x can send this return address to y as part of a message sent by the techniques already described. (In general, two participants can exchange return addresses through a chain of other participants, where at least one member of each adjacent pair knows the identity of the other member of the pair.) The following indicates how y uses this untraceable return address to form a response to x, via a new kind of mix: K1( R1, Ax ), Kx( R0, M ) --> Ax, R1( Kx( R0, M )). This mix uses the string of bits R1 that it finds after decrypting the address part K1( R1, Ax ) as a key to re-encrypt the message part Kx( R0, M ). Only the addressee x can decrypt the resulting output because x created both R1 and Kx. The mix must not allow address parts to be repeated - for the same reason that items of regular mail must not be repeated. This means that x must supply y with a return address for each item of mail x wishes to receive. Also notice that conventional as opposed to public key cryptography could be used for both encryptions of M. With a cascade of mixes, the message part is prepared the same as for a single mix, and the address part is as showin in the following input: K1( R1, K2( R2, ..., K<n-1>( R<n-1>, Kn( Rn, Ax ))...)), Kx( R0, M ) -->. The result of the first mix is K2( R2, ..., K<n-1>( R<n-1>, Kn( Rn, Ax ))...), R1( Kx( R0, M )) -->, and the final result of the remaining n-1 mixes is Ax, Rn( R<n-1> ... R2( R1( Kx( R0, M )))...). Untraceable return addresses allow the possiblity of "certified" mail: They can provide the sender of an anonymous letter with a receipt attesting to the fact that the letter appeared intact in the final output batch. The address A that is incorporated into a certified letter is expanded to include not only the usual address of the recipient, but also an untraceable return address for the sender. When this return address appears inthe output batch of the final mix, it is used to mail the sender a signed receipt which inlucdes the message as well as the address to which it was delivered. The receipt might be signed by each mix. Digital Pseudonyms A digital "pseudonym" is a public key used to verify signatures made by the anonymous holder of the corresponding private key. A "roster", or list of pseudonyms, is created by an authority that decides which applications for pseudonyms to accept, but is unable to trace the pseudonyms in the completed roster. The applications may be sent to the authority anonymously, by untraceable mail, for example, or they may be provided in some other way. Each application received by the authority contains all the information required for the acceptance decision and a special unaddressed digital letter (whose messages is the public key K, the applicant's proposed pseudonym.) In the case of a single mix, these letters are of the form K1( R1, K ). For a cascade of n mixes, they are of the form Kn( Rn, ..., K2( R2, K1( R1, K ))...). The authority will form an input batch containing only those unaddressed letters from the applications it accepts. This input batch will be supplied to a special cascade whose final output batch will be publically available. Since each entry in the final output batch of the cascade is a public key K from an accepted applicant, the signed output of the final mix is a roster of digital pseudonyms. Notification of applicants can be accomplished by also forming a roster for unaccepted applications and then using the technique of certified mail to return a single batch of receipts to both sets of applicants. Of course, repeats must not be allowed within or across batches. If only registered voters are accepted for a particular roster, then it can be used to carry out an election. For a single mix, each voter submits a ballot of the form K1( R1, K, Inv(K)( C, V )), where K is the voter's pseudonym and V is the actual vote. For a cascade of mixes, ballots are of the form Kn( Rn, ..., K2( R2, K1( R1, K, Inv(K)( C, V )))...). The ballots must be processed as a single batch, as were the letters used to form rosters. Items in the final lexicographically ordered output batch are of the form K, Inv(K)( C, V ). Since the roster of registered voters is also ordered on K, it is easy for anyone to count the votes by making a single pass through both batches at once. Each ballot is counted only after checking that the pseudonym K which forms its prefix, is also contained in the roster and that the pseudonym properly decrypts the signed vote V. An individual might be known to an organization only by a pseudonym that appears in a roster of acceptable clients. Clients can correspond with the organization via untraceable mail and the organization can correspond with the clients using untraceable return addresses. If applicants identify themselves in their applications, or if they sign applications with pseudonyms that appear in a roster issued by an authority that requires identification, then the organization is assured that the same client cannot come to it under different pseudonyms. Under special circumstances, such as default of payment, a particular pseudonym could be shown to correspond to a particular application (without revealing any other correspondences) if each mix in turn supplied the appropriate Rn. General Purpose Mail Systems One way to construct a general purpose, untraceable mail system is to require that every message pass through a cascade. Of course, mixes can operate continuously or periodically, and long messages will be encrypted first and then split into multiple items. In order to hide the number of messages sent, each participant supplies the same number of messages to each batch (some of which might be randomly addressed dummies). In order to hide the number of messages received, each participant privately searches the entire output for messages directed to it. Such a system may prove too costly for some participants. One way to reduce the cost is to allow mail to be addressed to subsets of participants, such as a local net. Participants that take advantage of such arrangements need search only the mail addressed to a particular subset. Another way to economize is for a participant to send for each batch only the number of dummy messages suggested by a random value (chosen from some suitable distribution), as opposed to always sending the maximal number of messages. This can substantially reduce message traffic and consequently, the size of output batches. While these techniques may open the door to some kinds of statistical attack, the system size that necessitated them may reduce the effectiveness of such attacks. In a large, general purpose mail system with many mixes, it may be impractical for every message to pass through every mix. In such a case, a sequence of mixes will be selected for each message, perhaps on the basis of network topology or trust. Notice that if a participant can choose mixes it trusts with its traffic volume data as early members of its sequences, then these mixes can discard dummies they receive from the participant and deliver small, fixed-sized batches (padded with dummies) directly to the participant. A new kind of mix will be presented here that allows a sequence of mixes to be selected for each message. It also (a) hides the number and identity of the mixes a message must pass through, (b) allows incrimination of a mix that does not properly forward items, and (c) makes no distinction between regular mail and mail sent by untraceable return address. It is based on the idea that every item of mail is composed of the same number of fixed-sized blocks. The operations performed by this new kind of mix are always the same. First it removes the first block and adds a random block J of junk to the end, to maintain the item's length of l blocks. Then, using its private key, the mix decrypts the block removed during the first step. This yields a key R, which the mix uses to encrypt each of the l blocks of the item (using either public-key or conventional cryptography). It also yields the address A (either of a recipient or of another mix) to which the item will be forwarded. The left-hand side of the following shows how an item is prepared to pass through a single mix: K2( R2, ..., K<n-1>( R<n-1>, Kn( Rn, Ax ))...), R1( Kx( R0, M )) -->, A1: [K<A1>( R<A1>, A )], [Inv(R<A1>)( M1 )], [Inv(R<A1>)( M2 )], ... [Inv(R<A1>)( M<l-1> )] --> A: [M1],...[M<l-1], [R<A1>( J<A1> )]. where square brackets show the extent of each block, and the sealed message Ka( R0, M ) is divided into pieces Mi, such that Ka( R0, M ) = M1, M2, ..., M<l-n>. The A1: indicates that the left-hand side is delivered to mix A1, while the A: means that the right-hand side is delivered to address A. Items with the same first block should be regarded as repeats. A message prepared to be passed through mixes A1 through An has the form A1: [K<A1>( R<A1>, A2 )], [Inv(R<A1>)( K<A2>( R<A2>, A3 ))], ..., [Inv(R<A1>)( Inv(R<A2>) ... Inv(R<A<n-1>>)( K<An>( R<An, A )) ... )], [Inv(R<A1>)( Inv(R<A2>) ... Inv(R<An>)( M1 )...)], ..., [Inv(R<A1>)( Inv(R<A2>) ... Inv(R<An>)( M<l-n> )...)] -->. The result leaving A1 is A2: [K<A2>( R<A2, A3 )], [Inv(R<A2>)( K<A3>( R<A3>, A4 ))], ..., [Inv(R<A2>)( Inv(R<A3>) ... Inv(R<A<n-1>>)( K<An>( R<An, A )) ... )], [Inv(R<A2>)( Inv(R<A3>) ... Inv(R<An>)( M1 )...)], ..., [Inv(R<A2>)( Inv(R<A3>) ... Inv(R<An>)( M<l-n> )...)], [R<A1>( J<A1> )] -->, and the final result leaving An is A: [M1], [M2], ..., [M<l-n>], [R<An>( R<A<n-1>> ... R<A1>( J<A1> )...)], ..., [R<An>( J<An> )]. An intermediate mix always knows which mix it received its input from - by assumption (2) - but if a mix broadcasts copies of its fixed-size output batches, then only individual recipient mixes need be able to recognize their own input in a broadcast batch. The untraceable return address x sends to y contains the key Kx that y uses to encrypt the message part. It also includes, in the case of a single mix, what y will use as the first block of the item it submits to the mix: A1: [K<A1>( R<A1>, Ax )], [M1], ..., [M<l-1>] --> Ax: [R<A1>( M1 )], ..., [R<A1>( M<l-1> )], [R<A1>( J<A1> )], where Kx( R0, M ) = M1, M2, ..., M<l-n>. Only x can decrypt the item it receives since it created R<A1> and Kx. When a message is to pass through n mixes, the untraceable return address contains the first n blocks: A1: [K<A1>( R<A1>, A2 )], [Inv(R<A1>)( K<A2>( R<A2>, A3 ))], ..., [Inv(R<A1>)( Inv(R<A2>) ... Inv(R<A<n-1>>)( K<An>( R<An>, Ax ))...)], [M1], [M2], ..., [M<l-n>] -->. After being operated on by mix A1 it will have the form A2: [K<A2>( R<A2>, A3 )], ..., [Inv(R<A2>)( Inv(R<A3>) ... Inv(R<A<n-1>>)( K<An>( R<An>, Ax ))...)], [R<A1>( M1 )], [R<A1>( M2 )], ..., [R<A1>( M<l-n> )], [R<A1>( J<A1> )] -->, and the final result leaving An is Ax: [R<An>( R<A<n-1>> ... R<A1>( M1 )...)], ..., [R<An>( R<A<n-1>> ... R<A1>( M<l-n> )...)], [R<An>( R<A<n-1>> ... R<A1>( J<A1> )...)], ..., [R<An>( J<An> )]. Summary and Conclusion A solution to the traffic analysis problem has been presented that allows any single intermediary to provide security for those messages passing through it. In addition, the solution allows messages to be sent or received pseudonymously. Through the notion of a roster of pseudonyms, it also provides some new and interesting kinds of limited anonymity. Acknowledgements. I owe a great deal to R. Fabry's outstanding and multifaceted support. Special thanks are due C. Sequin, who has read my work with great care and provided many stimulating discussions. I would also like to thank D. Gusfield, B. Mont-Reynaud, A. Moose, and S. Wecker for their comments and encouragement. The referees have been very helpful. Received 2/79; accepted 4/80; revised 10/80 References 1. Baran, P. On distributed communications: IX security secrecy and tamper-free considerations. Memo RM-3765-PR, Rand Corp., Santa Monica, CA, Aug. 1964. 2. Diffie, W. and Hellman, M.E. New directions in cryptography. IEEE Trans. Information Theory IT-22, 6 (Nov. 1976), 644-654. 3. Kahn, D. The Code Breakers, The Story of Secret Writing. Macmillan, New York, 1967. 4. Merkle, R.C. Secure communications over insecure channels. Comm. ACM 21, 4 (Apr. 1978), 294-299. 5. Rivest, R.L., Shamir, A., and Adleman, L. A method for obtaining digital signatures and public-key cryptosystems. Comm. ACM 21, 2 (Feb. 1977), 120-126.