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Digital Cash Mehdi Bazargan Fall 2004

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Introduction Definition Motivations Overview Properties Blind Signatures Brands Scheme Analysis

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Definition Since hard currency or paper cash carries total anonymity in transactions, the term digital cash is coined to refer to anonymous electronic token based payment systems. Digital Cash is meant to work as paper cash. There are different implementation of Digital Cash. Digital Cash is a technical product of anonymous digital commerce in strategic level. It is a highly political subject.

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Well… Anonymous? How can I prove I made my payments? Private? What keeps the bank from stealing from me? If a government doesn't know who pays whom, how can it collect an income tax? If the ownership of financial assets is indeterminate, what happens to taxes on financial assets?

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Motivations Comparing to paper cash, paper cash is: slow, vulnerable, costly, and difficult to transfer. Compared to credit cards, digital cash provides more anonymity and security.

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Overview Alice deposits cash into the bank 2. Alice receives some coins 3. Alice sends over the coins to Bob 4. Bob receives the coins 5. Bob cashes the coins and send Alice the product

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Overview There are several approaches in implementing digital cash: “Simple Anonymous Cash” by Fiat-Caum-Naor, “Traceable Anonymous Cash” by Ferguson, the Brands scheme, and “Auditable, Anonymous Electronic Cash” by Sander-Ta-Shama. The introduced methods have advantages and disadvantages. The Brands scheme provides reasonable security and anonymity; however, it is more complicated.

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Overview In Brands Scheme, we will mostly get benefit from a set of algorithms and mathematical toolkits: Prime Factorization: In short, it is hard to calculate prime factors of N=p.q where p and q are large primes. Discrete Log Problem: In short, if you have x= g a mod p, it is hard to find a where x and g are known.

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Overview Representation Issue in Groups with Prime Order Given a prime group G and a generator tuple of G (g1, g2,..gn), and constant h, it is hard to find a representation of h as Π k i=1 (g i a i ) where a i belongs to Z. However, it would be easy if you know the generator tuple and integers a i. Schnorr’s Digital Signature A method of signing messages and verifying validity of signatures.

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Properties Some important features of the system include these: The on-line system is a self-contained subset of the off-line system, and if the off-line features are not used, the remaining software-only system still could be efficiently implemented. Payments are private-- i.e. untraceable and unlinkable. The customer is protected from fraudulent bank claims that the customer is double-spending (i.e. protected from framing attempts by the bank), There is non-repudiation-- customers cannot deny having made a valid payment.

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Restrictive Blind Signature Let M denote a message. This message may be anything, including a piece of digital cash to be signed. To sign this message, the bank will raise it to the power x mod p, yielding [1] z = signed(M) = M^x. If we raise the message M to a random power w, we will call the result b a pseudo- signature. That is, [2] b = pseudo-signed(M) = M^w.

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Restrictive Blind Signature The public key of the signer is a generator g raised to the power x. So let's call the generator g raised to a random power w a pseudo-public key. Label this a. Thus we have: [3] public key h = g^x, [4] pseudo-public key a = g^w.

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Restrictive Blind Signature The steps in the restrictive blind signature protocol are as follows (all calculations in this protocol are done mod p, unless otherwise stated): Step 1: The customer, Alice, sends a message M to the bank. It is intended that the bank sign M with its secret key x: z = M^x The proof is to guarantee to the customer that the bank has signed M with a valid signature; namely with its secret key x.

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Restrictive Blind Signature Step 2: The bank, generates a random number w and sends to the receiver, Alice, the following elements: the signed message z = M^x the pseudo-public key a = g^w the pseudo-signed message b = M^w We shall see that b & a will be used in part to provide zero-knowledge proof for Alice that the bank’s signature is valid.

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Restrictive Blind Signature Step 3: The receiver generates a challenge c. To do this, the customer first generates four random numbers: s, t and u, v. Using s and t, the customer computes modifications of M and z, namely the blinded message M' and the signed blinded message z': [5] M' = M^s * g^t (blinded message) [6] z' = z^s*h^t = (M^x)^s*(g^x)^t = [M^s*g^t]^x = M'^x (signed blinded message)

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Restrictive Blind Signature Using u and v, the receiver (customer) computes modifications of a, and b, namely, a', and b': [7] a' = a^u*g^v = (g^w)^u*g^v = g^w', [8] b' = [a^(u*t)]*[b^(u*s)]*M'^v = [(g^w)^(u*t)]*[(M^w)^(u*s)]*M'^v = [(g^t)^(u*w)]*[(M^s)^(u*w)]*M'^v = [M'^(u*w)]*M'^v = M'^w'. where [9] w' = u*w + v mod q.

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Restrictive Blind Signature The customer then computes the hash value [10] c' = H(M', z', a', b'), and sends to the bank the challenge c: [11] c = c'/u mod q. Step 4: The signer (bank) responds with [12] r = w + c*x mod q. Notice this is a point on a line with slope x (the secret key) and intercept w.

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Restrictive Blind Signature Step 5: The receiver, Alice, uses the challenge c and the response r to check that [13] a*h^c = g^r and [14] b*z^c = M^r. If so, the receiver accepts the signature.

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Brand’s Scheme Uses the concepts in signature blinding as discussed. Brand’s implementation of Digital Cash considers: Opening an Account Withdrawal Deposit Payment

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Opening an Account The user has public/private key pairs. These are not used in the protocols that follow so will not be denoted by individual symbols. But we require that the user be able to send digitally signed messages to the bank. To open an account, the user U generates a random number u1 from Z(q)*, and computes an identifier or public key [15] hu = g1^u1 mod p.

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Opening an Account The user checks that hu*g2 is not equal to 1 mod p, and if so sends hu to the bank, keeping u1 secret. The bank stores hu along with any other information it requires on U. The bank computes and returns to the user U a signature with its secret key x as follows: [16] z = (hu*g2)^x mod p.

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Withdrawal Before the user U is allowed to withdraw a coin, U must first prove ownership of his account. Step 1: The bank generates a random number w from Z(q)*, and sends the pseudo-public key a and the pseudo-signed message b to the user U: [17] a= g^w mod p [18] b = (hu*g2)^w mod p

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Withdrawal Step 2: The user U generates three random numbers s, x1, and x2 from Z(q)*. These are used to calculate: [19] A = (hu*g2)^s mod p [20] B = g1^x1*g2^x2 mod p [21] z' = z^s mod p

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Withdrawal U also generates two random numbers u, v from Z(q)*. These are used to calculate [22] a' = a^u*g^v mod p [23] b' = b^(s*u)*A^v mod p The user U then computes the challenge c' as: [24] c' = H(A, B, z', a', b') then sends the blinded challenge c back to the bank: [25] c = c'/u mod q.

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Withdrawal The coin is the set of numbers {A, B, (z',a',b',r')}. (z',a',b',r') is Schnorr’s signature on A, B. Denominations… take different g for each different denomination.

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Withdrawal Step 3: The bank sends the response r : [26] r = w + c*x mod q and debits U's account in the amount equal to the value of one coin. Step 4: U accepts the debit only if [27] g^r = a*h^c mod p [28] (hu*g2)^r = b*z^c mod p. The user U also calculates r': [29] r' = v + r*u mod q.

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Payment When the user U is ready to spend the coin, the following protocol is enacted between the user and the shop S: Step 1: The user sends {A, B, (z',a',b',r')} to S. Step 2: The shop returns the challenge d: [30] d = Ho(A, B, SHOP-ID, DATE-TIME). Step 3: The user U calculates the responses r1, r2: [31] r1 = d*(u1*s) + x1 mod q [32] r2 = d*s + x2 mod q

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Payment Step 4: The shop S accepts the coin only if: [33] g^r' = a'*h^c' mod p [34] A^r' = b'*z'^c' mod p [35] A^d*B = g1^r1*g2^r2 mod p

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Deposit When the shop S is ready to deposit the coin at the bank, the shop sends the payment transcript consisting of the coin {A, B, (z',a',b',r')}, along with (r1, r2) and the DATE-TIME of the transaction. The bank already knows the SHOP-ID, which is used in the communication. Step 1: The bank verifies equations [33] to [35] to see that this is a valid coin.

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Deposit Step 2: If the coin is valid, the bank checks its database to see if the coin was spent previously. CASE A: If the coin is not in the database, then it was not previously spent. Hence the bank credits the account of S, and records the coin in the form {A, B, DATE-TIME, r1, r2}.

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Deposit CASE B: If the coin is already in the database, then a fraud has occurred. If S previously deposited the coin, and the DATE-TIME are the same, then S is trying to deposit the same coin or transcript twice. The deposit is rejected for that reason. The bank knows the identity of the shop S responsible.

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Deposit CASE C. Otherwise, the coin has been double-spent, and the bank takes steps to unmask the double-spender. The bank has two sets of information on the coin: {A, B, DATE-TIME, r1, r2}. {A, B, DATE-TIME', r'1, r'2}. Hence, the bank can calculate (r1 - r'1) / (r2 - r'2) = [d*(u1*s) - d'*(u1*s)] / [d*s - d's] = u1 mod q. Thus it can check its database for the user identity!

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Analysis Advantages: Security of this system rests on the difficulty in finding discrete logarithmic factors. Other systems rely on prime factorization used in RSA. So the ability in factoring for large primes would not break this system as it would be the case in other systems.

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Analysis Advantages: The major advantage of this mechanism is that the user does not need to keep track many copies of identity and many different bills as is the case in other systems.

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Analysis Disadvantages: This scheme is difficult to understand and is more complex compared to other mechanisms used such as Chaum’s system. Moreover, since we use discrete logarithmic signatures, we have to deal with larger signatures compared to other methods.

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References Jahanian Farsi, Mandana. Digital Cash. Retrieved: November Cormen, Leiserson, Rivest, and Stein. Introduction to Algorithms. Massachussetts: McGraw Hill, Sander, Ta-Shama. Auditable, Anonymous Electronic Cash. Retrieved: November. Bleumer, Gerrit. Electronic Cash. 25 April Orlin Grabbe,J. Stefan Brands' System of Digital Cash

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Questions, Comments… ?

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