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Received October 31, 1996
With the Compliments of Thomas P. Vartanian
Fried, Frank, Harris, Schriver & Jacobson
1001 Pennsylvania Avenue, N.W.
Washington, D.C. 20004-2505
Telephone: (202) 639-7200
HOW TO MAKE A MINT: THE CRYPTOGRAPHY OF ANONYMOUS ELECTRONIC CASH
Laurie Law, Susan Sabett, Jerry Solinas
National Security Agency Office of Information Security Research and Technology
Cryptology Division
18 June 1996
CONTENTS
INTRODUCTION
1. WHAT IS ELECTRONIC CASH?
1.1 Electronic Payment
1.2 Security of Electronic Payments
1.3 Electronic Cash
1.4 Multiple Spending
2. A CRYPTOGRAPHIC DESCRIPTION
2.1 Public-Key Cryptographic Tools
2.2 A Simplified Electronic Cash Protocol
2.3 Untraceable Electronic Payments
2.4 A Basic Electronic Cash Protocol
3. PROPOSED OFF-LINE IMPLEMENTATIONS
3.1 Including Identifying Information
3.2 Authentication and Signature Techniques
3.3 Summary of Proposed Implementations
4. OPTIONAL FEATURES OF OFF-LINE CASH
4. 1 Transferability
4.2 Divisibility
5. SECURITY ISSUES
5.1 Multiple Spending Prevention
5.2 Wallet Observers
5.3 Security Failures
5.4 Restoring Traceability
CONCLUSION
REFERENCES
INTRODUCTION
With the onset of the Information Age, our nation is becoming
increasingly dependent upon network communications. Computer-based
technology is significantly impacting our ability to access, store,
and distribute information. Among the most important uses of this
technology is electronic commerce: performing financial transactions
via electronic information exchanged over telecommunications lines. A
key requirement for electronic commerce is the development of secure
and efficient electronic payment systems. The need for security is
highlighted by the rise of the Internet, which promises to be a
leading medium for future electronic commerce.
Electronic payment systems come in many forms including digital
checks, debit cards, credit cards, and stored value cards. The usual
security features for such systems are privacy (protection from
eavesdropping), authenticity (provides user identification and message
integrity), and nonrepudiation (prevention of later denying having
performed a transaction) .
The type of electronic payment system focused on in this paper is
electronic cash. As the name implies, electronic cash is an attempt to
construct an electronic payment system modelled after our paper cash
system. Paper cash has such features as being: portable (easily
carried), recognizable (as legal tender) hence readily acceptable,
transferable (without involvement of the financial network),
untraceable (no record of where money is spent), anonymous (no record
of who spent the money) and has the ability to make "change." The
designers of electronic cash focused on preserving the features of
untraceability and anonymity. Thus, electronic cash is defined to be
an electronic payment system that provides, in addition to the above
security features, the properties of user anonymity and payment
untraceability..
In general, electronic cash schemes achieve these security goals via
digital signatures. They can be considered the digital analog to a
handwritten signature. Digital signatures are based on public key
cryptography. In such a cryptosystem, each user has a secret key and a
public key. The secret key is used to create a digital signature and
the public key is needed to verify the digital signature. To tell who
has signed the information (also called the message), one must be
certain one knows who owns a given public key. This is the problem of
key management, and its solution requires some kind of authentication
infrastructure. In addition, the system must have adequate network and
physical security to safeguard the secrecy of the secret keys.
This report has surveyed the academic literature for cryptographic
techniques for implementing secure electronic cash systems. Several
innovative payment schemes providing user anonymity and payment
untraceability have been found. Although no particular payment system
has been thoroughly analyzed, the cryptography itself appears to be
sound and to deliver the promised anonymity.
These schemes are far less satisfactory, however, from a law
enforcement point of view. In particular, the dangers of money
laundering and counterfeiting are potentially far more serious than
with paper cash. These problems exist in any electronic payment
system, but they are made much worse by the presence of anonymity.
Indeed, the widespread use of electronic cash would increase the
vulnerability of the national financial system to Information Warfare
attacks. We discuss measures to manage these risks; these steps,
however, would have the effect of limiting the users' anonymity.
This report is organized in the following manner. Chapter 1 defines
the basic concepts surrounding electronic payment systems and
electronic cash. Chapter 2 provides the reader with a high level
cryptographic description of electronic cash protocols in terms of
basic authentication mechanisms. Chapter 3 technically describes
specific implementations that have been proposed in the academic
literature. In Chapter 4, the optional features of transferability and
divisibility for off-line electronic cash are presented. Finally, in
Chapter 5 the security issues associated with electronic cash are
discussed.
The authors of this paper wish to acknowledge the following people for
their contribution to this research effort through numerous
discussions and review of this paper: Kevin Igoe, John Petro, Steve
Neal, and Mel Currie.
1. WHAT IS ELECTRONIC CASH?
We begin by carefully defining "electronic cash." This term is often
applied to any electronic payment scheme that superficially resembles
cash to the user. In fact, however, electronic cash is a specific kind
of electronic payment scheme, defined by certain cryptographic
properties. We now focus on these properties.
1.1 Electronic Payment
The term electronic commerce refers to any financial transaction
involving the electronic transmission of information. The packets of
information being transmitted are commonly called electronic tokens.
One should not confuse the token, which is a sequence of bits, with
the physical media used to store and transmit the information.
We will refer to the storage medium as a card since it commonly takes
the form of a wallet-sized card made of plastic or cardboard. (Two
obvious examples are credit cards and ATM cards.) However, the "card"
could also be, e.g., a computer memory.
A particular kind of electronic commerce is that of electronic
payment. An electronic payment protocol is a series of transactions,
at the end of which a payment has been made, using a token issued by a
third party. The most common example is that of credit cards when an
electronic approval process is used. Note that our definition implies
that neither payer nor payee issues the token.l
The electronic payment scenario assumes three kinds of players:2
a payer or consumer, whom we will name Alice.
a payee, such as a merchant. We will name the payee Bob.
a financial network with whom both Alice and Bob have accounts. We
will informally refer to the financial network as the Bank.
__________
1 In this sense, electronic payment differs from such systems as
prepaid phone cards and subway fare cards, where the token is issued
by the payee.
2 In 4.1, we will generalize this scenario when we discuss transfers.
1.2 Security of Electronic Payments
With the rise of telecommunications and the Internet, it is
increasingly the case that electronic commerce takes place using a
transmission medium not under the control of the financial system. It
is therefore necessary to take steps to insure the security of the
messages sent along such a medium.
The necessary security properties are:
Privacy, or protection against eavesdropping. This is obviously of
importance for transactions involving, e.g., credit card numbers sent
on the Internet.
User identification, or protection against impersonation. Clearly,
any scheme for electronic commerce must require that a user knows with
whom she is dealing (if only as an alias or credit card number).
Message integrity, or protection against tampering or
substitution. One must know that the recipient's copy of the message
is the same as what was sent.
Nonrepudiation, or protection against later denial of a
transaction. This is clearly necessary for electronic commerce, for
such things as digital receipts and payments.
The last three properties are collectively referred to as authenticity.
These security features can be achieved in several ways. The technique
that is gaining widespread use is to employ an authentication
infrastructure. In such a setup, privacy is attained by enciphering
each message, using a private key known only to the sender and
recipient. The authenticity features are attained via key management,
e.g., the system of generating, distributing and storing the users'
keys.
Key management is carried out using a certification authority, or a
trusted agent who is responsible for confirming a user's identity.
This is done for each user (including banks) who is issued a digital
identity certificate. The certificate can be used whenever the user
wishes to identify herself to another user. In addition, the
certificates make it possible to set up a private key between users in
a secure and authenticated way. This private key is then used to
encrypt subsequent messages. This technique can be implemented to
provide any or all of the above security features.
Although the authentication infrastructure may be separate from the
electronic-commerce setup, its security is an essential component of
the security of the electronic-commerce system. Without a trusted
certification authority and a secure infrastructure, the above four
security features cannot be achieved, and electronic commerce becomes
impossible over an untrusted transmission medium.
We will assume throughout the remainder of this paper that some
authentication infrastructure is in place, providing the four security
features.
1.3 Electronic Cash
We have defined privacy as protection against eavesdropping on one's
communications. Some privacy advocates such as David Chaum (see
[2],[3]), however, define the term far more expansively. To them,
genuine "privacy" implies that one's history of purchases not be
available for inspection by banks and credit card companies (and by
extension the government). To achieve this, one needs not just privacy
but anonymity. In particular, one needs
payer anonymity during payment,
payment untraceability so that the Bank cannot tell whose money is
used in a particular payment.
These features are not available with credit cards. Indeed, the only
conventional payment system offering it is cash. Thus Chaum and others
have introduced electronic cash (or digital cash), an electronic
payment system which offers both features. The sequence of events in
an electronic cash payment is as follows:
withdrawal, in which Alice transfers some of her wealth from her
Bank account to her card.
payment, in which Alice transfers money from her card to Bob's.
deposit, in which Bob transfers the money he has received to his
Bank account.
(See Figure 1.)
Diagram of electronic cash
Figure 1. The three types of transactions in a basic electronic cash model.
These procedures can be implemented in either of two ways:
On-line payment means that Bob calls the Bank and verifies the
validity of Alice's token3 before accepting her payment and delivering
his merchandise. (This resembles many of today's credit card
transactions.)
Off-line payment means that Bob submits Alice's electronic coin
for verification and deposit sometime after the payment transaction is
completed. (This method resembles how we make small purchases today by
personal check.)
Note that with an on-line system, the payment and deposit are not
separate steps. We will refer to on-line cash and off-line cash
schemes, omitting the word "electronic" since there is no danger of
confusion with paper cash.
__________
3 In the context of electronic cash, the token is usually called an
electronic coin.
1.4 Counterfeiting
As in any payment system, there is the potential here for criminal
abuse, with the intention either of cheating the financial system or
using the payment mechanism to facilitate some other crime. We will
discuss some of these problems in 5. However, the issue of
counterfeiting must be considered here, since the payment protocols
contain built-in protections against it.
There are two abuses of an electronic cash system analogous to
counterfeiting of physical cash:
Token forgery, or creating a valid-looking coin without making a
corresponding Bank withdrawal.
Multiple spending, or using the same token over again. Since an
electronic coin consists of digital information, it is as
valid-looking after it has been spent as it was before. (Multiple
spending is also commonly called re-spending, double spending, and
repeat spending.)
One can deal with counterfeiting by trying to prevent it from
happening, or by trying to detect it after the fact in a way that
identifies the culprit. Prevention clearly is preferable, all other
things being equal.
Although it is tempting to imagine electronic cash systems in which
the transmission and storage media are secure, there will certainly be
applications where this is not the case. (An obvious example is the
Internet, whose users are notoriously vulnerable to viruses and
eavesdropping.) Thus we need techniques of dealing with counterfeiting
other than physical security.
To protect against token forgery, one relies on the usual
authenticity functions of user identification and message integrity.
(Note that the "user" being identified from the coin is the issuing
Bank, not the anonymous spender.)
To protect against multiple spending, the Bank maintains a
database of spent electronic coins. Coins already in the database are
to be rejected for deposit. If the payments are on-line, this will
prevent multiple spending. If off-line, the best we can do is to
detect when multiple spending has occurred. To protect the payee, it
is then necessary to identify the payer. Thus it is necessary to
disable the anonymity mechanism in the case of multiple spending.
The features of authenticity, anonymity, and multiple-spender exposure
are achieved most conveniently using public-key cryptography. We will
discuss how this is done in the next two chapters.
2. A CRYPTOGRAPHIC DESCRIPTION
In this chapter, we give a high-level description of electronic cash
protocols in terms of basic authentication mechanisms. We begin by
describing these mechanisms, which are based on public-key
cryptography. We then build up the protocol gradually for ease of
exposition. We start with a simplified scheme which provides no
anonymity. We then incorporate the payment untraceability feature, and
finally the payment anonymity property. The result will be a complete
electronic cash protocol.
2.1 Public-Key Cryptographic Tools
We begin by discussing the basic public-key cryptographic techniques
upon which the electronic cash implementations are based.
One-Way Functions. A one-way function is a correspondence between two
sets which can be computed efficiently in one direction but not the
other. In other words, the function phi is one-way if, given s in the
domain of phi, it is easy to compute t = phi(s), but given only t, it
is hard to find s. (The elements are typically numbers, but could also
be, e.g., points on an elliptic curve; see [10].)
Key Pairs. If phi is a one-way function, then a key pair is a pair s,
t related in some way via phi. We call s the secret key and t the
public key. As the names imply, each user keeps his secret key to
himself and makes his public key available to all. The secret key
remains secret even when the public key is known, because the one-way
property of phi insures that t cannot be computed from s.
All public-key protocols use key pairs. For this reason, public-key
cryptography is often called asymmetric cryptography. Conventional
cryptography is often called symmetric cryptography, since one can
both encrypt and decrypt with the private key but do neither without
it.
Signature and Identification. In a public key system, a user
identifies herself by proving that she knows her secret key without
revealing it. This is done by performing some operation using the
secret key which anyone can check or undo using the public key. This
is called identification. If one uses a message as well as one's
secret key, one is performing a digital signature on the message. The
digital signature plays the same role as a handwritten signature:
identifying the author of the message in a way which cannot be
repudiated, and confirming the integrity of the message.
Secure Hashing. A hash function is a map from all possible strings of
bits of any length to a bit string of fixed length. Such functions are
often required to be collision-free: that is, it must be
computationally difficult to find two inputs that hash to the same
value. If a hash function is both one-way and collision-free, it is
said to be a secure hash.
The most common use of secure hash functions is in digital signatures.
Messages might come in any size, but a given public-key algorithm
requires working in a set of fixed size. Thus one hashes the message
and signs the secure hash rather than the message itself. The hash is
required to be one-way to prevent signature forgery, i.e.,
constructing a valid-looking signature of a message without using the
secret key.4 The hash must be collision-free to prevent repudiation,
i.e., denying having signed one message by producing another message
with the same hash.
__________
4 Note that token forgery is not the same thing as signature forgery.
Forging the Bank's digital signature without knowing its secret key is
one way of committing token forgery, but not the only way. A bank
employee or hacker, for instance, could "borrow" the Bank's secret key
and validly sign a token. This key compromise scenario is discussed in
5.3.
2.2 A Simplified Electronic Cash Protocol
We now present a simplified electronic cash system, without the
anonymity features.
PROTOCOL 1: On-line electronic payment.
Withdrawal:
Alice sends a withdrawal request to the Bank.
Bank prepares an electronic coin and digitally signs it.
Bank sends coin to Alice and debits her account.
Payment/Deposit:
Alice gives Bob the coin.
Bob contacts Bank5 and sends coin.
Bank verifies the Bank's digital signature.
Bank verifies that coin has not already been spent.
Bank consults its withdrawal records to confirm Alice's
withdrawal. (optional)
Bank enters coin in spent-coin database.
Bank credits Bob's account and informs Bob.
Bob gives Alice the merchandise.
__________
5 One should keep in mind that the term "Bank" refers to the financial
system that issues and clears the coins. For example, the Bank might
be a credit card company, or the overall banking system. In the latter
case, Alice and Bob might have separate banks. If that is so, then the
"deposit" procedure is a little more complicated: Bob's bank contacts
Alice's bank, "cashes in" the coin, and puts the money in Bob's
account.
PROTOCOL 2: Off-line electronic payment.
Withdrawal:
Alice sends a withdrawal request to the Bank.
Bank prepares an electronic coin and digitally signs it.
Bank sends coin to Alice and debits her account.
Payment:
Alice gives Bob the coin.
Bob verifies the Bank's digital signature. (optional)
Bob gives Alice the merchandise.
Deposit:
Bob sends coin to the Bank.
Bank verifies the Bank's digital signature.
Bank verifies that coin has not already been spent.
Bank consults its withdrawal records to confirm Alice's
withdrawal. (optional)
Bank enters coin in spent-coin database.
Bank credits Bob's account.
The above protocols use digital signatures to achieve authenticity.
The authenticity features could have been achieved in other ways, but
we need to use digital signatures to allow for the anonymity
mechanisms we are about to add.
2.3 Untraceable Electronic Payments
In this section, we modify the above protocols to include payment
untraceability. For this, it is necessary that the Bank not be able to
link a specific withdrawal with a specific deposit.6 This is
accomplished using a special kind of digital signature called a blind
signature.
We will give examples of blind signatures in 3.2, but for now we give
only a high-level description. In the withdrawal step, the user
changes the message to be signed using a random quantity. This step is
called "blinding" the coin, and the random quantity is called the
blinding factor. The Bank signs this random-looking text, and the user
removes the blinding factor. The user now has a legitimate electronic
coin signed by the Bank. The Bank will see this coin when it is
submitted for deposit, but will not know who withdrew it since the
random blinding factors are unknown to the Bank. (Obviously, it will
no longer be possible to do the checking of the withdrawal records
that was an optional step in the first two protocols.)
Note that the Bank does not know what it is signing in the withdrawal
step. This introduces the possibility that the Bank might be signing
something other than what it is intending to sign. To prevent this, we
specify that a Bank's digital signature by a given secret key is valid
only as authorizing a withdrawal of a fixed amount. For example, the
Bank could have one key for a $10 withdrawal, another for a $50
withdrawal, and so on.7
_________
6 In order to achieve either anonymity feature, it is of course
necessary that the pool of electronic coins be a large one.
7 0ne could also broaden the concept of "blind signature" to include
interactive protocols where both parties contribute random elements to
the message to be signed. An example of this is the "randomized blind
signature" occurring in the Ferguson scheme discussed in 3.3.
PROTOCOL 3: Untraceable On-line electronic payment.
Withdrawal:
Alice creates an electronic coin and blinds it.
Alice sends the blinded coin to the Bank with a withdrawal request.
Bank digitally signs the blinded coin.
Bank sends the signed blinded coin to Alice and debits her account.
Alice unblinds the signed coin.
Payment/Deposit:
Alice gives Bob the coin.
Bob contacts Bank and sends coin.
Bank verifies the Bank's digital signature.
Bank verifies that coin has not already been spent.
Bank enters coin in spent-coin database.
Bank credits Bob's account and informs Bob.
Bob gives Alice the merchandise.
PROTOCOL 4: Untraceable Off-line electronic payment.
Withdrawal:
Alice creates an electronic coin and blinds it.
Alice sends the blinded coin to the Bank with a withdrawal request.
Bank digitally signs the blinded coin.
Bank sends the signed blinded coin to Alice and debits her account.
Alice unblinds the signed coin.
Payment:
Alice gives Bob the coin.
Bob verifies the Bank's digital signature. (optional)
Bob gives Alice the merchandise.
Deposit:
Bob sends coin to the Bank.
Bank verifies the Bank's digital signature.
Bank verifies that coin has not already been spent.
Bank enters coin in spent-coin database.
Bank credits Bob's account.
2.4 A Basic Electronic Cash Protocol
We now take the final step and modify our protocols to achieve payment
anonymity. The ideal situation (from the point of view of privacy
advocates) is that neither payer nor payee should know the identity of
the other. This makes remote transactions using electronic cash
totally anonymous: no one knows where Alice spends her money and who
pays her.
It turns out that this is too much to ask: there is no way in such a
scenario for the consumer to obtain a signed receipt. Thus we are
forced to settle for payer anonymity.
If the payment is to be on-line, we can use Protocol 3 (implemented,
of course, to allow for payer anonymity). In the off-line case,
however, a new problem arises. If a merchant tries to deposit a
previously spent coin, he will be turned down by the Bank, but neither
will know who the multiple spender was since she was anonymous. Thus
it is necessary for the Bank to be able to identify a multiple
spender. This feature, however, should preserve anonymity for
law-abiding users.
The solution is for the payment step to require the payer to have, in
addition to her electronic coin, some sort of identifying information
which she is to share with the payee. This information is split in
such a way that any one piece reveals nothing about Alice's identity,
but any two pieces are sufficient to fully identify her.
This information is created during the withdrawal step. The withdrawal
protocol includes a step in which the Bank verifies that the
information is there and corresponds to Alice and to the particular
coin being created. (To preserve payer anonymity, the Bank will not
actually see the information, only verify that it is there.) Alice
carries the information along with the coin until she spends it.
At the payment step, Alice must reveal one piece of this information
to Bob. (Thus only Alice can spend the coin, since only she knows the
information.) This revealing is done using a challenge-response
protocol. In such a protocol, Bob sends Alice a random "challenge"
quantity and, in response, Alice returns a piece of identifying
information. (The challenge quantity determines which piece she
sends.) At the deposit step, the revealed piece is sent to the Bank
along with the coin. If all goes as it should, the identifying
information will never point to Alice. However, should she spend the
coin twice, the Bank will eventually obtain two copies of the same
coin, each with a piece of identifying information. Because of the
randomness in the challenge-response protocol, these two pieces will
be different. Thus the Bank will be able to identify her as the
multiple spender. Since only she can dispense identifying information,
we know that her coin was not copied and re-spent by someone else.
PROTOCOL 5: Off-line cash.
Withdrawal:
Alice creates an electronic coin, including identifying information.
Alice blinds the coin.
Alice sends the blinded coin to the Bank with a withdrawal request.
Bank verifies that the identifying information is present.
Bank digitally signs the blinded coin.
Bank sends the signed blinded coin to Alice and debits her account.
Alice unblinds the signed coin.
Payment:
Alice gives Bob the coin.
Bob verifies the Bank's digital signature.
Bob sends Alice a challenge.
Alice sends Bob a response (revealing one piece of identifying info).
Bob verifies the response.
Bob gives Alice the merchandise.
Deposit:
Bob sends coin, challenge, and response to the Bank.
Bank verifies the Bank's digital signature.
Bank verifies that coin has not already been spent.
Bank enters coin, challenge, and response in spent-coin database.
Bank credits Bob's account.
Note that, in this protocol, Bob must verify the Bank's signature
before giving Alice the merchandise. In this way, Bob can be sure that
either he will be paid or he will learn Alice's identity as a multiple
spender.
3. PROPOSED OFF-LINE IMPLEMENTATIONS
Having described electronic cash in a high-level way, we now wish to
describe the specific implementations that have been proposed in the
literature. Such implementations are for the off-line case; the
on-line protocols are just simplifications of them. The first step is
to discuss the various implementations of the public-key cryptographic
tools we have described earlier.
3.1 Including Identifying Information
We must first be more specific about how to include (and access when
necessary) the identifying information meant to catch multiple
spenders. There are two ways of doing it: the cut-and-choose method
and zero-knowledge proofs.
Cut and Choose. When Alice wishes to make a withdrawal, she first
constructs and blinds a message consisting of K pairs of numbers,
where K is large enough that an event with probability 2-K will never
happen in practice. These numbers have the property that one can
identify Alice given both pieces of a pair, but unmatched pieces are
useless. She then obtains signature of this blinded message from the
Bank. (This is done in such a way that the Bank can check that the K
pairs of numbers are present and have the required properties, despite
the blinding.)
When Alice spends her coins with Bob, his challenge to her is a string
of K random bits. For each bit, Alice sends the appropriate piece of
the corresponding pair. For example, if the bit string starts 0110. .
., then Alice sends the first piece of the first pair, the second
piece of the second pair, the second piece of the third pair, the
first piece of the fourth pair, etc. When Bob deposits the coin at the
Bank, he sends on these K pieces.
If Alice re-spends her coin, she is challenged a second time. Since
each challenge is a random bit string, the new challenge is bound to
disagree with the old one in at least one bit. Thus Alice will have to
reveal the other piece of the corresponding pair. When the Bank
receives the coin a second time, it takes the two pieces and combines
them to reveal Alice's identity.
Although conceptually simple, this scheme is not very efficient, since
each coin must be accompanied by 2K large numbers.
Zero-Knowledge Proofs. The term zero-knowledge proof refers to any
protocol in public-key cryptography that proves knowledge of some
quantity without revealing it (or making it any easier to find it). In
this case, Alice creates a key pair such that the secret key points to
her identity. (This is done in such a way the Bank can check via the
public key that the secret key in fact reveals her identity, despite
the blinding.) In the payment protocol, she gives Bob the public key
as part of the electronic coin. She then proves to Bob via a
zero-knowledge proof that she possesses the corresponding secret key.
If she responds to two distinct challenges, the identifying
information can be put together to reveal the secret key and so her
identity.
3.2 Authentication and Signature Techniques
Our next step is to describe the digital signatures that have been
used in the implementations of the above protocols, and the techniques
that have been used to include identifying information.
There are two kinds of digital signatures, and both kinds appear in
electronic cash protocols. Suppose the signer has a key pair and a
message M to be signed.
Digital Signature with Message Recovery. For this kind of
signature, we have a signing function SSK using the secret key SK, and
a verifying function VPK using the public key PK. These functions are
inverses, so that
(*) VPK (SSK (M)) = M
The function VPK is easy to implement, while SSK is easy if one
knows SK and difficult otherwise. Thus SSK is said to have a trapdoor,
or secret quantity that makes it possible to perform a cryptographic
computation which is otherwise infeasible. The function VPK is called
a trapdoor one-way function, since it is a one-way function to anyone
who does not know the trapdoor.
In this kind of scheme, the verifier receives the signed message
SSK (M) but not the original message text. The verifier then applies
the verification function VPK. This step both verifies the identity of
the signer and, by (*), recovers the message text.
Digital Signature with Appendix. In this kind of signature, the
signer performs an operation on the message using his own secret key.
The result is taken to be the signature of the message; it is sent
along as an appendix to the message text. The verifier checks an
equation involving the message, the appendix, and the signer's public
key. If the equation checks, the verifier knows that the signer's
secret key was used in generating the signature.
We now give specific algorithms.
RSA Signatures. The most well-known signature with message recovery is
the RSA signature. Let N be a hard-to-factor integer. The secret
signature key s and the public verification key v are exponents with
the property that
Msv = M (mod N)
for all messages M. Given v, it is easy to find s if one knows the
factors of N but difficult otherwise. Thus the "vth power (mod N)" map
is a trapdoor one-way function. The signature of M is
C := Ms (mod N);
to recover the message (and verify the signature), one computes
M := Cv (mod N).
Blind RSA Signatures. The above scheme is easily blinded. Suppose that
Alice wants the Bank to produce a blind signature of the message M.
She generates a random number r and sends
rv . M (mod N)
to the Bank to sign. The Bank does so, returning
r . Ms (mod N)
Alice then divides this result by r. The result is Ms (mod N), the
Bank's signature of M, even though the Bank has never seen M.
The Schnorr Algorithms. The Schnorr family of algorithms includes an
identification procedure and a signature with appendix. These
algorithms are based on a zero-knowledge proof of possession of a
secret key. Let p and q be large prime numbers with q dividing p - 1.
Let g be a generator; that is, an integer between 1 and p such that
gq = 1 (mod p).
If s is an integer (mod q), then the modular exponentiation operation on s is
phi : s -> gs (mod p).
The inverse operation is called the discrete logarithm function and is denoted
logg t <- t.
If p and q are properly chosen, then modular exponentiation is a
one-way function. That is, it is computationally infeasible to find a
discrete logarithm.
Now suppose we have a line
(**) y = mx + b
over the field of integers (mod q). A line can be described by giving
its slope m and intercept b, but we will "hide" it as follows. Let
c = gb (mod p),
n = gm (mod p).
Then c and n give us the "shadow" of the line under phi. Knowing c and
n doesn't give us the slope or intercept of the line, but it does
enable us to determine whether a given point (x, y) is on the line.
For if (x, y) satisfies (**), then it must also satisfy the relation
(***) gy = nx . c (mod p).
(Conversely, any point (x, y) satisfying (***) must be on the line.)
The relationship (***) can be checked by anyone, since it involves
only public quantities. Thus anyone can check whether a given point is
on the line, but points on the line can only be generated by someone
who knows the secret information.
The basic Schnorr protocol is a zero-knowledge proof that one
possesses a given secret quantity m. Let n be the corresponding public
quantity. Suppose one user (the "prover") wants to convince another
(the "verifier") that she knows m without revealing it. She does this
by constructing a line (**) and sending its shadow to the verifier.
The slope of the line is taken to be secret quantity m, and the prover
chooses the intercept at random, differently for each execution of the
protocol. The protocol then proceeds as follows.
Schnorr proof of possession:
1. Alice sends c (and n if necessary) to Bob.
2. Bob sends Alice a "challenge" value of x.
3. Alice responds with the value of y such that (x, y) is on the line.
4. Bob verifies via (**) that (x, y) is on the line.
Bob now knows that he is speaking with someone who can generate points
on the line. Thus this party must know the slope of the line, which is
the secret quantity m.
An important feature of this protocol is that it can be performed only
once per line. For if he knows any two points (xo, yo) and (x1, y1) on
the line, the verifier can compute the slope of the line using the
familiar "rise over the run" formula
m = yo - y1 / x1 - x1 (mod q),
and this slope is the secret quantity m. That is why a new intercept
must be generated each time. We call this the two-points-on-a-line
principle. This feature will be useful for electronic cash protocols,
since we want to define a spending procedure which reveals nothing of
a secret key if used once per coin, but reveals the key if a coin is
spent twice.
Schnorr identification. The above protocol can be used for
identification of users in a network. Each user is issued a key pair,
and each public key is advertised as belonging to a given user. To
identify herself, a user needs only prove that she knows her secret
key. This can be done using the above zero-knowledge proof, since her
public key is linked with her identity.
Schnorr Signature. It is easy to convert the Schnorr identification
protocol to produce a digital signature scheme. Rather than receiving
a challenge from an on-line verifier, the signer simply takes x to be
a secure hash of the message and of the shadow of the line. This
proves knowledge of his secret key in a way that links his key pair to
the message.
Blind Schnorr Signature. Suppose that Alice wants to obtain a blind
Schnorr signature for her coin, which she will spend with Bob. Alice
generates random quantities (mod q) which describe a change of
variables. This change of variables replaces the Bank's hidden line
with another line, and the point on the Bank's line with a point on
the new line. When Bob verifies the Bank's signature, he is checking
the new point on the new line. The two lines have the same slope, so
that the Bank's signature will remain valid. When the Bank receives
the coin for deposit, it will see the protocol implemented on the new
line, but it will not be able to link the coin with Alice's withdrawal
since only Alice knows the change of variables relating the two lines.
Chaum-Pederson Signature. A variant of Schnorr's signature scheme
given in [6] is used in electronic cash protocols. This modified
scheme is a kind of "double Schnorr" scheme. It involves a single line
and point but uses two shadows. This signature scheme can be blinded
in a way similar to the ordinary Schnorr signature.
Implementations of the Schnorr Protocols. We have described the
Schnorr algorithms in terms of integers modulo a prime p. The
protocols, however, work in any setting in which the analogue of the
discrete logarithm problem is difficult. An important example is that
of elliptic curves (see [10]). Elliptic curve based protocols are much
faster, and require the transmission of far less data, than
non-elliptic protocols giving the same level of security.
3.3 Summary of Proposed Implementations
We can now present summaries of the main off-line cash schemes from
the academic literature. There are three: those of Chaum-Fiat-Naor
[4], Brands [1], and Ferguson [9].
Chaum-Fiat-Naor. This was the first electronic cash scheme, and is the
simplest conceptually. The Bank creates an electronic coin by
performing a blind RSA signature to Alice's withdrawal request, after
having verified interactively that Alice has included her identifying
information on the coin. The prevention of multiple spending is
accomplished by the cut-and-choose method. For this reason, this
scheme is relatively inefficient.
Brands. Brands' scheme is Schnorr-based.8 Indeed, a Schnorr protocol
is used twice: at withdrawal, the Bank performs a blind Chaum-Pederson
signature, and then Alice performs a Schnorr possession proof as the
challenge-and-response part of the spending protocol.
The withdrawal step produces a coin which contains the Bank's
signature, authenticating both Alice's identifying information and the
shadow of the line to be used for the possession proof. This commits
Alice to using that particular line in the spending step. If she
re-spends the coin, she must use the same line twice, enabling the
Bank to identify her.
The Brands scheme is considered by many to be the best of the three,
for two reasons. First, it avoids the awkward cut-and-choose
technique. Second, it is based only on the Schnorr protocols, and so
it can be implemented in various settings such as elliptic curves.
Ferguson. Ferguson's scheme is RSA-based like Chaum-Fiat-Naor, but it
uses the "two-points-on-a-line" principle like Brands. The signature
it uses is not the blind RSA signature as described above, but a
variant called a randomized blind RSA signature. The ordinary blind
RSA scheme has the drawback that the Bank has absolutely no idea what
it is signing. As mentioned above, this is not a problem in the
cut-and-choose case, but in this case it can allow a payer to defeat
the mechanism for identifying multiple spenders. The randomized
version avoids this problem by having both Alice and the Bank
contribute random data to the message. The Bank still doesn't know
what it is signing, but it knows that the data was not chosen
maliciously.
The rest of the protocol is conceptually similar to Brands' scheme.
The message to be signed by the Bank contains, in addition to the
random data, the shadow of a line whose slope and intercept reveal
Alice's identity. During payment, Alice reveals a point on this line;
if she does so twice, the Bank can identify her.
Although Ferguson's scheme avoids the cut-and-choose technique, it is
the most complicated of the three (due largely to the randomized blind
RSA signature). Moreover, it cannot be implemented over elliptic
curves since it is RSA-based.
__________
8 For ease of exposition, we give a simplified account of Brands' protocol.
4. OPTIONAL FEATURES OF OFF-LINE CASH
Much of the recent literature on off-line cash has focused on adding
features to make it more convenient to use. In this chapter we will
discuss two of these features.
4.1 Transferability
Transferability is a feature of paper cash that allows a user to spend
a coin that he has just received in a payment without having to
contact the Bank in between. We refer to a payment as a transfer if
the payee can use the received coin in a subsequent payment. A payment
system is transferable if it allows at least one transfer per coin.
Figure 2 shows a maximum length path of a coin in a system which
allows two transfers. The final payment is not considered a transfer
because it must be deposited by the payee. Transferability would be a
convenient feature for an off-line cash system because it requires
less interaction with the Bank. (A transferable electronic cash system
is off-line by definition, since on-line systems require communication
with the Bank during each payment.)
Maximum length of path a coin in a system which allows 2 trasnfers per coin
Figure 2. A maximum length path of a coin in a system which allows 2
transfers per coin.
Transferable systems have received little attention in academic
literature. The schemes presented in 3.3 are not transferable because
the payee cannot use a received coin in another payment - his only
options are to deposit or to exchange it for new coins at the Bank.
Any transferable electronic cash system has the property that the coin
must "grow in size" (i.e., accumulate more bits) each time it is
spent. This is because the coin must contain information about every
person who has spent it so that the Bank maintains the ability to
identify multiple spenders. (See [5].) This growth makes it impossible
to allow an unlimited number of transfers. The maximum number of
transfers allowed in any given system will be limited by the allowable
size of the coin.
There are other concerns with any transferable electronic cash system,
even if the number of transfers per coin is limited, and we remove the
anonymity property. Until the coin is deposited, the only information
available to the Bank is the identity of the individual who originally
withdrew the coin. Any other transactions involving that withdrawal
can only be reconstructed with the cooperation of each consecutive
spender of that coin. This poses the same problems that paper cash
poses for detecting money laundering and tax evasion: no records of
the transactions are available.
In addition, each transfer delays detection of re-spent or forged
coins. Multiple spending will not be noticed until two copies of the
same coin are eventually deposited. By then it may be too late to
catch the culprit, and many users may have accepted counterfeit coins.
Therefore, detection of multiple spending after the fact may not
provide a satisfactory solution for a transferable electronic cash
system. A transferable system may need to rely on physical security to
prevent multiple spending. (See 5.1.)
4.2 Divisibility
Suppose that Alice is enrolled in a non-transferable, off-line cash
system, and she wants to purchase an item from Bob that costs, say,
$4.99. If she happens to have electronic coins whose values add up to
exactly $4.99 then she simply spends these coins. However, unless
Alice has stored a large reserve of coins of each possible
denomination, it is unlikely that she will have the exact change for
most purchases. She may not wish to keep such a large reserve of coins
on hand for the some of the same reasons that one doesn't carry around
a large amount of cash: loss of interest and fear of the cash being
stolen or lost. Another option is for Alice to withdraw a coin of the
exact amount for each payment, but that requires interaction with the
Bank, making the payment on-line from her point of view. A third
option is for Bob to pay Alice the difference between her payment and
the $4.99 purchase price. This puts the burden of having an exact
payment on Bob, and also requires Alice to contact the Bank to deposit
the "change."
A solution to Alice's dilemma is to use divisible coins: coins that
can be "divided" into pieces whose total value is equal to the value
of the original coin. This allows exact off-line payments to be made
without the need to store a supply of coins of different
denominations. Paper cash is obviously not divisible, but lack of
divisibility is not as much of an inconvenience with paper cash
because it is transferable. Coins that are received in one payment can
be used again in the next payment, so the supply of different
denominations is partially replenished with each transaction. (Imagine
how quickly a cashier would run out of change if paper cash were not
transferable and each payment was put in a separate bin set aside for
the next bank deposit!)
Three divisible off-line cash schemes have been proposed, but at a
cost of a longer transaction time and additional storage. Eng and
Okamoto's divisible scheme [7] is based on the "cut and choose"
method. Okamoto [11] is much more efficient and is based on Brands'
scheme but will also work on Ferguson's scheme. Okamoto and Ohta [12]
is the most efficient of the three, but also the most complicated. It
relies on the difficulty of factoring and on the difficulty of
computing discrete logarithms.
A binary tree for a divisible coin
Figure 3. A binary tree for a divisible coin worth $4.00, with a
minimum unit of $1.00. A $3.00 payment can be made by spending the
shaded nodes. Node 1I cannot be used in a subsequent payment because
it is an ancestor of nodes 2 and 6. Nodes 4 and 5 cannot be used
because they are descendants of node 2. Node 3 cannot be used because
it is an ancestor of node 6. Nodes 2 and 6 cannot be used more than
once, so node 7 is the only node which can be spent in a subsequent
payment.
All three of these schemes work by associating a binary tree with each
coin of value $w. (See Figure 3). Each node is assigned a monetary
value as follows: the unique root node (the node at level 0) has value
$w, the two nodes at level 1 each have value $w/2, the four nodes at
level 2 each have value $w/4, etc. Therefore, if w = 21, then the tree
has l+ 1 levels, and the nodes at level j each have value $w/2j. The
leaves of the tree are the nodes at level l, and have the minimum unit
of value.
To spend the entire amount of value $w, the root node is used. Amounts
less than $w can be spent by spending a set of nodes whose values add
up to the desired amount.
Initially, any whole dollar amount of up to $w can be spent.
Subsequent payments are made according to the following rules:
1. Once a node is used, all its descendant and ancestor9 nodes
cannot be used.
2. No node can be used more than once.
These two rules insure that no more than one node is used on any path
from the root to a leaf. If these two rules are observed, then it will
be impossible to spend more than the original value of the coin. If
either of these rules are broken, then two nodes on the same path are
used, and the information in the two corresponding payments can be
combined to reveal the identity of the individual that over-spent in
the same way that the identity of a multiple spender is revealed.
More specifically, in the Eng/Okamoto and Okamoto schemes, each user
has a secret value, s, which is linked to their identity (uncovering s
will uncover their identity, but not vice-versa.) Each node i is
assigned a secret value, ti. Hence, each node i corresponds to a line
y = sx + ti
When a payment is made using a particular node n, ti will be revealed
for all nodes i that are ancestors of node n. Then the payee sends a
challenge xi and the payer responds with
y1 = sx1 + tn .
This reveals a point (x1, y1) on the line y = sx + tn, but does not
reveal the line itself. If the same node is spent twice, then
responses to two independent challenges, x1 and x2, will reveal two
points on the same line: (x1, y1) and (x2, y2). Then the secret value
s can be recovered using the two-points-on-a-line principle described
in 3.2.
If someone tries to overspend a coin, then two nodes in the same path
will be used. Suppose that nodes n and m are in the same path, and
node n is farther from the root on this path. Spending node n will
reveal tm, since node m is an ancestor of node n. Now if node m is
also spent, then the response to a challenge x1 will be y1 = sx1 + tm.
But tm was revealed when tn was spent, so sx1 and hence s will be
revealed. Therefore, spending two nodes in the same path will reveal
the identity of the over-spender. The Okamoto/Ohta divisible scheme
also uses a binary tree with the same rules for using nodes to prevent
multiple and over-spending, but when nodes are used improperly, a
different technique is used to determine the identity of the spender.
Instead of hiding the user's identifying secret in a line for which a
point is revealed when a coin is spent, the user's identifying secret
is hidden in the factorization of an RSA modulus. Spending the same
node twice, or spending two nodes on the same path will provide enough
information for the Bank to factor the modulus (which is part of the
coin) and then compute the user's secret identifying information.
Although these three divisible schemes are untraceable, payments made
from the same initial coin may be "linked" to each other, meaning that
it is possible to tell if two payments came from the same coin and
hence the same person. This does not reveal the payer's identity if
both payments are valid (follow Rules 1 and 2, above), but revealing
the payer's identity for one purchase would reveal that payer's
identity for all other purchases made from the same initial coin.
These are three examples of off-line cash schemes that have divisible
coins. Although providing divisibility complicates the protocol, it
can be accomplished without forfeiting untraceability or the ability
to detect improper spenders. The most efficient divisible scheme has a
transaction time and required memory per coin proportional to the
logarithm of N, where N is the total coin value divided by the value
of the minimum divisible unit. More improvements in the efficiency of
divisible schemes are expected, since the most recent improvement was
just presented in 1995.
__________
9 A descendant of a node n is a node on a path from node n to a leaf.
An ancestor of node n is a node on the path from node n to the root
node.
5. SECURITY ISSUES
In this section we discuss some issues concerning the security of
electronic cash. First, we discuss ways to help prevent multiple
spending in off-line systems, and we describe the concept of wallet
observers. We also discuss the consequences of an unexpected failure
in the system?s security. Finally, we describe a solution to some of
the law enforcement problems that are created by anonymity.
5.1 Multiple Spending Prevention
In 1.3, we explained that multiple spending can be prevented in
on-line payments by maintaining a database of spent electronic coins,
but there is no cryptographic method for preventing an off-line coin
from being spent more than once. Instead, off-line multiple spending
is detected when the coin is deposited and compared to a database of
spent coins. Even in anonymous, untraceable payment schemes, the
identity of the multiple-spender can be revealed when the abuse is
detected. Detection after the fact may be enough to discourage
multiple spending in most cases, but it will not solve the problem. If
someone were able to obtain an account under a false identity, or were
willing to disappear after re-spending a large sum of money, they
could successfully cheat the system.
One way to minimize the problem of multiple spending in an off-line
system is to set an upper limit on the value of each payment. This
would limit the financial losses to a given merchant due to accepting
coins that have been previously deposited. However, this will not
prevent someone from spending the same small coin many times in
different places.
In order to prevent multiple spending in off-line payments, we need to
rely on physical security. A "tamper-proof" card could prevent
multiple spending by removing or disabling a coin once it is spent.
Unfortunately, there is no such thing as a truly "tamper-proof" card.
Instead, we will refer to a "tamper-resistant" card, which is
physically constructed so that it is very difficult to modify its
contents. This could be in the form of a smart card, a PC card10, or
any storage device containing a tamper-resistant computer chip. This
will prevent abuse in most cases, since the typical criminal will not
have the resources to modify the card. Even with a tamper-resistant
card, it is still essential to provide cryptographic security to
prevent counterfeiting and to detect and identify multiple spenders in
case the tamper-protection is somehow defeated. Also, setting limits
on the value of off-line payments would reduce the cost-effectiveness
of tampering with the card.
Tamper-resistant cards can also provide personal security and privacy
to the cardholder by making it difficult for adversaries to read or
modify the information stored on the card (such as secret keys,
algorithms, or records).
__________
10 Formerly PCMCIA, or Personal Computer Memory Card International Association.
5.2 Wallet Observers
All of the basic off-line cash schemes presented in 3.3 can
cryptographically detect the identity of multiple spenders, but the
only way to prevent off-line multiple spending is to use a
tamper-resistant device such as a smart card. One drawback of this
approach is that the user must put a great deal of trust in this
device, since the user loses the ability to monitor information
entering or leaving the card. It is conceivable that the
tamper-resistant device could leak private information about the user
without the user's knowledge.
Chaum and Pedersen [6] proposed the idea of embedding a
tamper-resistant device into a user-controlled outer module in order
to achieve the security benefits of a tamper-resistant device without
requiring the user to trust the device. They call this combination an
electronic wallet (see Figure 4). The outer module (such as a small
hand-held computer or the user's PC) is accessible to the user. The
inner module which cannot be read or modified is called the
"observer." All information which enters or leaves the observer must
pass through the outer module, allowing the user to monitor
information that enters or leaves the card. However, the outer module
cannot complete a transaction without the cooperation of the observer.
This gives the observer the power to prevent the user from making
transactions that it does not approve of, such as spending the same
coin more than once.
An electronic wallet
Figure 4. An electronic wallet.
Brands[1] and Ferguson[8] have both shown how to incorporate observers
into their respective electronic cash schemes to prevent multiple
spending. Brands' scheme incorporates observers in a much simpler and
more efficient manner. In Brands' basic scheme, the user's secret key
is incorporated into each of his coins. When a coin is spent, the
spender uses his secret to create a valid response to a challenge from
the payee. The payee will verify the response before accepting the
payment. In Brands' scheme with wallet observers, this user secret is
shared between the user and his observer. The combined secret is a
modular sum of the two shares, so one share of the secret reveals no
information about the combined secret. Cooperation of the user and the
observer is necessary in order to create a valid response to a
challenge during a payment transaction. This is accomplished without
either the user or the observer revealing any information about its
share of the secret to the other. It also prevents the observer from
controlling the response; hence the observer cannot leak any
information about the spender.
An observer could also be used to trace the user's transactions at a
later time, since it can keep a record of all transactions in which it
participates. However, this requires that the Bank (or whoever is
doing the tracing) must be able to obtain the observer and analyze it.
Also, not all types of observers can be used to trace transactions.
Brands and Ferguson both claim that they can incorporate observers
into their schemes and still retain untraceability of the users'
transactions, even if the observer used in the transactions has been
obtained and can be analyzed.
5.3 Security Failures
Types of failures.
In any cryptographic system, there is some risk of a security failure.
A security failure in an electronic cash system would result in the
ability to forge or duplicate money. There are a number of different
ways in which an electronic cash system could fail.
One of the most serious types of failure would be that the
cryptography (the protocol or the underlying mathematics) does not
provide the intended security.11 This could enable someone to create
valid looking coins without knowledge of an authorized bank's secret
key, or to obtain valid secret keys without physical access to them.
Anyone who is aware of the weakness could create coins that appear to
come from a legitimate bank in the system.
Another serious type of failure could occur in a specific
implementation of the system. For example, if the bank's random number
generator is not a-good one, one may be able to guess the secret
random number and use it to compute the secret keys that are used to
create electronic money.
Even if the cryptography and the implementation are secure, the
security could fail because of a physical compromise. If a computer
hacker, thief, dishonest bank employee, or a rogue state were to gain
access to the bank's secret key they could create counterfeit money.
If they gain access to a user's secret key they could spend that
user's money. If they could modify the user or bank's software they
could destroy the security of the system.
The above failure scenarios apply, not only to the electronic cash
system, but also to the underlying authentication infrastructure. Any
form of electronic commerce depends heavily on the ability of users to
trust the authentication mechanisms. So if, for example, an attacker
could demonstrate a forgery of the certification authority's digital
signature, it would undermine the users' trust in their ability to
identify each other. Thus the certification authorities need to be
secured as thoroughly as do the banks.
Consequences of a failure.
All three of the basic schemes described in this paper are anonymous,
which makes it impossible for anyone to connect a deposited coin to
the originating banks withdrawal record of that coin. This property
has serious consequences in the event of a security failure leading to
token forgery. When a coin is submitted for deposit, it is impossible
to determine if it is forged. Even the originating bank is unable to
recognize its own coins, preventing detection of the compromise. It is
conceivable that the compromise will not be detected until the bank
realizes that the total value of deposits of its electronic cash
exceeds the amount that it has created with a particular key. At this
point the losses could be devastating.
After the key compromise is discovered, the bank will still be unable
to distinguish valid coins from invalid ones since deposits and
withdrawals cannot be linked. The bank would have to change its secret
key and invalidate all coins which were signed with the compromised
key. The bank can replace coins that have not yet been spent, but the
validity of untraceable coins that have already been spent or
deposited cannot be determined without cooperation of the payer.
Payment untraceability prevents the Bank from determining the identity
of the payer, and payer anonymity prevents even the payee from
identifying the payer.
It is possible to minimize this damage by limiting the number of coins
affected by a single compromise. This could be done by changing the
Bank's public key at designated time intervals, or when the total
value of coins issued by a single key exceeds a designated limit.
However, this kind of compartmentation reduces the anonymity by
shrinking the pool of withdrawals that could correspond to a
particular deposit and vice versa.
__________
11 We are unaware of anything in the literature that would suggest
this type of failure with the protocols discussed in this paper.
5.4 Restoring Traceability
The anonymity properties of electronic cash pose several law
enforcement problems because they prevent withdrawals and deposits
from being linked to each other. We explained in the previous section
how this prevents detection of forged coins. Anonymity also makes it
difficult to detect money laundering and tax evasion because there is
no way to link the payer and payee. Finally, electronic cash paves the
way for new versions of old crimes such as kidnapping and blackmail
(see [13]) where money drops can now be carried out safely from the
criminal's home computer.12
One way to minimize these concerns is to require large transactions or
large numbers of transactions in a given time period to be traceable.
This would make it more difficult to commit crimes involving large
sums of cash. However, even a strict limit such as a maximum of $100 a
day on withdrawals and deposits can add up quickly, especially if one
can open several accounts, each with its own limit. Also, limiting the
amount spent in a given time period would have to rely on a
tamper-resistant device.
Another way to minimize these concerns is to provide a mechanism to
restore traceability under certain conditions, such as a court order.
Traceability can be separated into two types by its direction. For~ard
traceability is the ability to identify a deposit record (and hence
the payee), given a withdrawal record (and hence the identity of the
payer). In other words, if a search warrant is obtained for Alice,
forward tracing will reveal where Alice has spent her cash. Back~ard
traceability is the ability to identify a withdrawal record (and hence
the payer), given a deposit record (and hence the identity of the
payee). Backward tracing will reveal who Alice has been receiving
payments from.
A solution that conditionally restores both forward and backward
traceability into the cut-and-choose scheme is presented by Stadler,
Piveteau, and Camenisch in [14]. In the basic cut-and choose scheme,
an identifying number is associated with each withdrawal record and a
different identifying number is associated with each deposit record,
although there is no way to link these two records to each other. To
provide a mechanism for restoring backward traceability, the
withdrawal number (along with some other data which cannot be
associated with the withdrawal) is encrypted with a commonly trusted
entity's public key and incorporated into the coin itself. This
encrypted withdrawal number is passed to the payee as part of the
payment protocol, and then will be passed along to the bank when the
coin is deposited by the payee. The payer performs the encryption
during the withdrawal transaction, but the bank can insure that the
encryption was done properly. If the required conditions for tracing
are met, the payment or deposit can be turned over to the trusted
entity holding the secret key to decrypt the withdrawal number. This
withdrawal number will allow the bank to access its withdrawal
records, identifying the payer.
To provide a mechanism for restoring forward traceability, the payer
must commit to a deposit number at the time that the coin is
withdrawn. The payer encrypts this deposit number with a commonly
trusted entity's public key (along with some other data that cannot be
associated with the deposit) and must send this value to the bank as
part of the withdrawal protocol. The bank is able to determine that
the payer has not cheated, although it only sees the deposit number in
encrypted form. If the required conditions for tracing are met, the
withdrawal record can be turned over to the trusted entity holding the
secret key to decrypt the deposit number. The bank can use this
deposit number to identify the depositor (the payee).
Stadler et al. have shown that it is possible to provide a mechanism
for restoring traceability in either or both directions. This can be
used to provide users with anonymity, while solving many of the law
enforcement problems that exist in a totally untraceable system. The
ability to trace transactions in either direction can help law
enforcement officials catch tax evaders and money launderers by
revealing who has paid or has been paid by the suspected criminal.
Electronic blackmailers can be caught because the deposit numbers of
the victim's ill-gotten coins could be decrypted, identifying the
blackmailer when the money is deposited.
The ability to restore traceability does not solve one very important
law enforcement problem: detecting forged coins. Backwards tracing
will help identify a forged coin if a particular payment or deposit
(or depositor) is under suspicion. In that case, backwards tracing
will reveal the withdrawal number, allowing the originating bank to
locate its withdrawal record and verify the validity of the coin.
However, if a forged coin makes its way into the system it may not be
detected until the bank whose money is being counterfeited realizes
that the total value of its electronic cash deposits using a
particular key exceeds the values of its withdrawals. The only way to
determine which deposits are genuine and which are forged would
require obtaining permission to decrypt the withdrawal numbers for
each and every deposit of electronic cash using the compromised key.
This would violate the privacy that anonymous cash was designed to
protect.
Unfortunately, the scheme of [14] is not efficient because it is based
on the bulky cut-and-choose method. However, it may be possible to
apply similar ideas to restore traceability in a more efficient
electronic cash scheme.
__________
12 We will not focus on such crimes against individuals, concentrating
instead on crimes against the Government, the banking system, and the
national economy.
CONCLUSION
This report has described several innovative payment schemes which
provide user anonymity and payment untraceability. These electronic
cash schemes have cryptographic mechanisms in place to address the
problems of multiple spending and token forgery. However, some serious
concerns about the ability of an electronic cash system to recover
from a security failure have been identified. Concerns about the
impact of anonymity on money laundering and tax evasion have also been
discussed.
Because it is simple to make an exact copy of an electronic coin, a
secure electronic cash system must have a way to protect against
multiple spending. If the system is implemented on-line, then multiple
spending can be prevented by maintaining a database of spent coins and
checking this list with each payment. If the system is implemented
off-line, then there is no way to prevent multiple spending
cryptographically, but it can be detected when the coins are
deposited. Detection of multiple spending after-the-fact is only
useful if the identity of the offender is revealed. Cryptographic
solutions have been proposed that will reveal the identity of the
multiple spender while preserving user anonymity otherwise.
Token forgery can be prevented in an electronic cash system as long as
the cryptography is sound and securely implemented, the secret keys
used to sign coins are not compromised, and integrity is maintained on
the public keys. However, if there is a security flaw or a key
compromise, the anonymity of electronic cash will delay detection of
the problem. Even after the existence of a compromise is detected, the
Bank will not be able to distinguish its own valid coins from forged
ones. Since there is no way to guarantee that the Bank's secret keys
will never be compromised, it is important to limit the damage that a
compromise could inflict. This could be done by limiting the total
value of coins issued with a particular key, but lowering these limits
also reduces the anonymity of the system since there is a smaller pool
of coins associated with each key.
The untraceability property of electronic cash creates problems in
detecting money laundering and tax evasion because there is no way to
link the payer and payee. To counter this problem, it is possible to
design a system that has an option to restore traceability using an
escrow mechanism. If certain conditions are met (such as a court
order), a deposit or withdrawal record can be turned over to a
commonly trusted entity who holds a key that can decrypt information
connecting the deposit to a withdrawal or vice versa. This will
identify the payer or payee in a particular transaction. However, this
is not a solution to the token forgery problem because there may be no
way to know which deposits are suspect. In that case, identifying
forged coins would require turning over all of the Bank's deposit
records to the trusted entity to have the withdrawal numbers
decrypted.
We have also looked at two optional features of off-line electronic
cash: transferability and divisibility. Because the size of an
electronic coin must grow with each transfer, the number of transfers
allowed per coin must be limited. Also, allowing transfers magnifies
the problems of detecting counterfeit coins, money laundering, and tax
evasion. Coins can be made divisible without losing any security or
anonymity features, but at the expense of additional memory
requirements and transaction time.
In conclusion, the potential risks in electronic commerce are
magnified when anonymity is present. Anonymity creates the potential
for large sums of counterfeit money to go undetected by preventing
identification of forged coins. Anonymity also provides an avenue for
laundering money and evading taxes that is difficult to combat without
resorting to escrow mechanisms. Anonymity can be provided at varying
levels, but increasing the level of anonymity also increases the
potential damages. It is necessary to weigh the need for anonymity
with these concerns. It may well be concluded that these problems are
best avoided by using a secure electronic payment system that provides
privacy, but not anonymity.
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[End]
Thanks to the authors, Thomas Vartanian and anonymous others.
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