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Online Cash Checks
David Chaum
Centre for Mathematics and Computer Science
Kruislaan 413 1098SJ Amsterdam
INTRODUCTION
Savings of roughly an order of magnitude in space, storage, and bandwidth
over previously published online electronic cash protocols are achieved by
the techniques introduced here. In addition, these techniques can increase
convenience, make more efficient use of funds, and improve privacy.
"Offline" electronic money [CFN 88] is suitable for low value
transactions where "accountability after the fact" is sufficient to deter
abuse; online payment [C 89], however, remains necessary for transactions
that require "prior restraint" against persons spending beyond their
available funds.
Three online schemes are presented here. Each relies on the same
techniques for encoding denominations in signatures and for "devaluing"
signatures to the exact amount chosen at the time of payment. They differ
in how the unspent value is returned to the payer. In the first, all change is
accumulated by the payer in a single "cookie jar," which might be
deposited at the bank during the next withdrawal transaction. The second
and third schemes allow change to be distributed among unspent notes,
which can themselves later be spent. The second scheme reveals to the
shop and bank the maximum amount for which a note can be spent; the
third does not disclose this information.
DENOMINATIONS AND DEVALUING
For simplicity and concreteness, but without loss of generality, a
particular denomination scheme will be used here. It assigns the value of 1
cent to public exponent 3 in an RSA system, the value of 2 cents to
exponent 5, 4 cents to exponent 7, and so on; each successive power-of-two
value is represented by the corresponding odd prime public exponent, all
with the same modulus. Much as in [C 89], a third root of an image under
the one-way function f (together with the pre-image modulo the bank's
RSA composite) is worth 1 cent, a 7th root is worth 4 cents, and a 21st root
5 cents. In other words, a distinct public prime exponent is associated with
each digit of the binary integer representation of an amount of payment;
for a particular amount of payment, the product of all those prime
exponents corresponding to 1 's in the binary representation of the amount
is the public exponent of the signature.
A signature on an image under f is "devalued" by raising it to the
public powers corresponding to the coin values that should be removed.
For instance, a note having a 21st root could be devalued from its 5 cent
value, to 1 cent, simply by raising it to the 7th power.
In earlier online payment systems [C 89], the number of separate
signatures needed for a payment was in general the Hamming weight of the
binary representation of the amount. Since online systems would be used
for higher-value payments (as mentioned above), and extra resolution may
be desired to provide interest for unspent funds [C 89], an average of
roughly an order of magnitude is saved here.
COOKIE JAR
In this first scheme the payer periodically withdraws a supply of notes
from the bank, each with the system-wide maximum value. Consider an
example, shown in Figure 1.1, in which two notes are withdrawn. The n
and ri are random. The ri "blind" (from the bank) the images under the
public, one-way function f. The bank's signature corresponds to taking the
h-th root, where h = 3*5*7*11. As in all the figures, the payer sends
messages from the left and the bank sends from the right.
h h
f(n1) * r1 , f(n2) * r2
----------------------------------------->
PAYER BANK
<------------------------------------------
1/h 1/h
f(n1) * r1, f(n2) * r2
Fig. 1.1. Cookie-jar withdrawal
In preparing the first payment, the payer divides r1 out. The signature
is then raised to the 55th power to devalue it from 15 cents to 5 cents.
Figure 1.2 shows this first payment. Of course the shop is an intermediary
between the payer (left) and the bank (right) in every online payment, but
this is not indicated explicitly. Also not shown in the figures are messages
used to agree on the amounts of payment.
1/(3*7) 5*11
n1, f(n1) , f(j) * s1
----------------------------------------->
PAYER BANK
<------------------------------------------
1/(5*11)
f(j) * s1
Fig. 1.2. First cookie-jar payment
The first two residues sent in paying, n1 and its signed image under f,
are easily verified by the bank to be worth 5 cents. The third residue is a
blinded "cookie jar," a blinded image under f of a randomly chosen value
j. This cookie jar is modulo a second RSA composite that is only used for
cookie jars. Once the bank verifies the funds received, and that n1 has not
been spent previously, it signs and returns the blinded cookie jar (under
the cookie jar modulus) with public exponents corresponding to the
change due.
The second payment, shown in figure 1.3, is essentially the same as the
first, except that the amount is 3 cents and the cookie jar now has some
roots already on it. If more payments were to be made using the same
cookie jar, all resulting signatures for change would accumulate.
1/(3*5) 1/(5*11) 7*11
n2, f(n2) , f(j) * s2
----------------------------------------->
PAYER BANK
<------------------------------------------
1/(5*11*7*11)
f(j) * s2
Fig. 1.3. Second cookie-jar payment
The cookie jar might conveniently be deposited, as shown in figure
1.4, during the withdrawal of the next batch of notes. It is verified by the
bank much as a payment note would be: the roots must be present in the
claimed multiplicity and the pre-image under f must not have been
deposited before.
1/(5*7*11*11)
j, f(j)
----------------------------------------->
PAYER BANK
Fig. 1.4. Cookie-jar deposit
The cookie jar approach gives the effect of an online form of "offline
checks" [C 89], in that notes of a fixed value are withdrawn and the unspent
parts later credited to the payer during a refund transaction.
DECLARED NOTE VALUE
Figure 2 depicts a somewhat different scheme, which allows change to be
spent without an intervening withdrawal transaction. Withdrawals can be
just as in the cookie-jar scheme, but here a single modulus is used for
everything in the system. The products of public exponents representing
the various amounts are as follows: d is the amount paid, g is the note
value, the "change" c is g/d, and h is again the maximal amount, where d | g
| h. A payment (still to the bank through a shop) includes first and second
components that are the same as in the cookie-jar scheme. The third
component is the amount of change c the payer claims should be returned.
The fourth is a (blinded) number m, which could be an image under f used
in a later payment just as n is used in this one.
1/d c
n, f(n) , c, m*s
----------------------------------------->
PAYER BANK
<------------------------------------------
+------------+
1/c | 1/c | (Graphic padlock)
m * s * | f(f(n) ) |
+------------+
Fig. 2. Declared note value payment
The signature returned contains a "protection" factor (shown inside
the padlock). This factor ensures that the payer actually has the c-th root of
f(n), by requiring that the payer apply f to it before dividing the result out
of the signature. Without such protection, a payer could get the system-
wide maximum change, regardless of how much change is actually due;
with it, the change claimed can only be recovered if the corresponding
roots on n are in fact known to the payer.
DISTRIBUTING CHANGE
The change returned in a payment can be divided into parts that fill in
missing denominations in notes not yet spent. Suppose, for example, that
the last payment is spent with d = 5*11, c = 3*7, and that m is formed by
the payer as shown in the first line of Figure 3.1. Then unblinding after
the payment yields the a shown in the second line.
(Use === for "is equivalent to")
3 7
m === f(n1) * f(n2)
1/21
a === m
Fig. 3.1. Form of change returned
- From a, the two roots shown in the last two lines of Figure 3.2 are
readily computed. (This technique is easily extended to include any
number of separate roots.) Thus the values unused in the last payment fill
in roots missing in notes n1 and n2.
-1
u = 3 mod 7
v = 3u div 7
1/7 3 -1 u -v
f(n1) === (a * f(n2) ) * f(n1)
1/3 -1/7
f(n2) === a * f(n1)
Fig. 3.2. Distributing the change
Because overpayment allows change to be returned in any chosen
denominations (not shown), the payer has extra flexibility and is able to
use all funds held. This also increases convenience by reducing the need
for withdrawals.
HIDDEN NOTE VALUE
Although the combination of the previous two subsections is quite
workable, it may be desirable for the payer not to have to reveal c to the
shop or the bank. Figure 4 shows a system allowing this. The payment
message is just as in the declared note value protocol above, except that c is
not sent. The protection factor (shown again in a lock) is also placed under
the signature, but it is missing the extra f and is raised to a random power z
chosen by the bank
1/d c
n, f(n) , m*s
----------------------------------------->
PAYER BANK
<------------------------------------------
+----------+
d/h g/h | zd/h |
m * s * | f(n) |, z
+----------+
Fig. 4. Hidden note value payment
If z were known to the payer before payment, then the payer could
-z
cheat by including f(n) in the third component; this would yield the payer
the system-wide maximum change, even if none were due. Consider a
single change exponent q. If z mod q is guessed correctly by a cheating
payer, then the payer improperly gets the corresponding coin value. Thus
the chance of successful cheating is 1/q. If, however, the divisors of h are
chosen sufficiently large, quite practical security can be achieved. When
the possibilities of distributing change and refunding are included, this
scheme's privacy surpasses that of a coin system.
CONCLUSION
Combining online coins improves efficiency, use of funds, convenience,
and privacy.
REFERENCES
Chaum, D., "Privacy Protected Payments: Unconditional Payer and/or
Payee Anonymity," in Smart Card 2000, North-Holland, 1989, pp. 69-92.
Chaum, D., A. Fiat, & M. Naor, "Offline Electronic Cash," Proceedings
of Crypto '88.
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