**Information Security and Computer Fraud**

## Development of Self-Issuable (Divisible and Transferable) Offline Electronic Cash

**Shinsuke Tamura**^{1,}, **Hazim A. Haddad**^{1}

^{1}School of Engineering, University of Fukui, Fukui, Japan

Abstract | |

1. | Introduction |

2. | Environments and Requirements |

3. | Security Components |

4. | Behaviors of the e-Cash Scheme |

5. | Verification Phase |

6. | Conclusion |

References |

### Abstract

Based on anonymous tag based credentials and linear Mix-nets, this paper develops a scheme for e-cash systems that can be used in offline environments. The developed scheme makes e-cash holders anonymous while disabling them to use e-cash dishonestly. It also makes e-cash divisible and transferable. In detail, although no one except cash holders themselves can know correspondences between them and their e-cash, e-cash issuing authority can identify dishonest cash holders that had generated and/or spent e-cash illegitimately. In addition, cash holders can generate new e-cash of arbitrary values from their holding e-cash to make purchases of amounts less than original cash values. Also, cash holders can use e-cash that they had received from others as same as the one directly issued to them from the issuing authority.

**Keywords:** privacy, anonymous tags, anonymous credentials, linear mix-nets

Received July 23, 2015; Revised August 19, 2015; Accepted August 23, 2015

**Copyright**© 2015 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Shinsuke Tamura, Hazim A. Haddad. Development of Self-Issuable (Divisible and Transferable) Offline Electronic Cash.
*Information Security and Computer Fraud*. Vol. 3, No. 1, 2015, pp 15-24. https://pubs.sciepub.com/iscf/3/1/3

- Tamura, Shinsuke, and Hazim A. Haddad. "Development of Self-Issuable (Divisible and Transferable) Offline Electronic Cash."
*Information Security and Computer Fraud*3.1 (2015): 15-24.

- Tamura, S. , & Haddad, H. A. (2015). Development of Self-Issuable (Divisible and Transferable) Offline Electronic Cash.
*Information Security and Computer Fraud*,*3*(1), 15-24.

- Tamura, Shinsuke, and Hazim A. Haddad. "Development of Self-Issuable (Divisible and Transferable) Offline Electronic Cash."
*Information Security and Computer Fraud*3, no. 1 (2015): 15-24.

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### At a glance: Figures

### 1. Introduction

Together with credit card systems e-cash systems are one of the most convenient paying schemes in e-society and many schemes had been proposed already [1-7]^{[1]}. Among various features that existing e-cash schemes aim to achieve, anonymity, divisibility and transferability are most important, where anonymity ensures that correspondences between e-cash and their holders are not revealed, and divisibility and transferability enable each cash holder to divide its e-cash into ones with smaller cash values and to use e-cash that it had received from others, respectively.

But in offline environments where e-cash issuing authorities do not participate in individual purchases, to efficiently satisfy these requirements is not easy. Regarding the divisibility, existing schemes must assume the unit cash value for dividing original e-cash in advance and required computation and/or communication costs increase when the unit becomes small ^{[7]}. Also, several schemes cannot achieve complete anonymity, e.g. cash holders cannot conceal links among e-cash that they generated by dividing same e-cash ^{[1]}. In the same way, many existing transferable e-cash schemes cannot achieve complete anonymity ^{[4, 5]}. For example, authorities can identify cash holders that used illegitimately transferred e-cash even if they were honest ^{[5]}. In addition, they are not convenient enough, e.g. volume of information that constitutes each e-cash increases every time when the e-cash is transferred ^{[6]} or cash holders must maintain numbers of receipts obtained from payees even after they had spent their e-cash ^{[5]}.

While exploiting anonymous tag based credentials ^{[8, 9, 11]} and linear Mix-nets ^{[10, 11]}, this paper proposes a scheme that efficiently and effectively achieves the above 3 features, i.e. it achieves complete anonymity while enabling authorities to identify dishonest entities. Also, each cash holder can divide its e-cash into ones with arbitrary (even decimal) cash values to pay them to others provided that the total value of divided e-cash does not exceed the value of the original e-cash. In addition, volume of information that constitutes each e-cash or computation and communication cost for handling the e-cash does not increase even the e-cash is exchanged among many cash holders. Cash holders do not need to maintain numbers of receipts either.

### 2. Environments and Requirements

Figure 1 shows entities involved in the proposed offline e-cash scheme, they are e-ash issuing authority *A* and cash holders P_{1}, P_{2}, ---, P_{M}. Authority *A* issues e-cash to cash holders P_{1}, P_{2}, ---, P_{M}, and each P_{m} makes its purchase while paying its e-cash *C*(P_{m}, t, h) to other cash holder P_{k} (P_{m} may simply give *C*(P_{m}, t, h) to P_{k}). P_{m} also asks *A* to exchange its e-cash for real cash.

About e-cash, *C*(P_{m}, t, h) means that P_{m} pays it to P_{k} as the h-th division of its t-th e-cash, and P_{m} obtains its t-th e-cash as C(*A*, i) that was issued by authority *A* or *C*(P_{j}, s, v) that was paid by other cash holder P_{j}. Where, *C*(*A*, i) represents e-cash that authority *A* had issued directly to P_{m} as the i-th e-cash. Also an expiration time is defined for each e-cash, and cash holders must exchange their e-cash for real cash at *A* before they expire.

**Fig**

**ure**

**1.**Entities in the proposed e-cash scheme

An important thing is offline e-cash system schemes must satisfy the following requirements under conditions that cash holders P_{m}, P_{j} and P_{k} are anonymous and they pay or give their e-cash to other cash holders without the presence of authority *A*, and the proposed e-cash system scheme satisfies these requirements by exploiting anonymous tag based credentials and linear Mix-nets as explained in the following sections. The requirements are,

1. *Unforgeability** *Only cash issuing authority can generate valid e-cash,

2. *A**nonymity** *honest cash holders can conceal correspondences between them and their spending e-cash,

3. *Divisibility** *cash holders can divide their e-cash into e-cash with smaller cash values,

4. *Transferability** *cash holders can use e-cash that were paid to them by others,

5. *Unlinkability** *cash holders can conceal links among e-cash that they had spent, and

6. *S**ecurity** *cash holders cannot use e-cash illegitimately.

But about the security, illegitimate e-cash uses include double spending and using of e-cash owned by others, and in environments where individual cash holders are anonymous and authority *A* does not exist when cash holders pay their e-cash, payees cannot confirm that their receiving e-cash were certainly owned by payers or payers did not use same e-cash at other places. Therefore, to satisfy the security requirements, the proposed scheme detect illegitimately used e-cash after they were used and identify liable entities as same as other existing schemes.

### 3. Security Components

**3.1. Anonymous Tag based Credentials**

Provided that B, T_{P},* k* and *g* are integers defined by authority *A*, R and *w* are secret integers defined by entity P, d_{1} and d_{2} are 2 secret signing keys of *A* and S(d_{1}_{║}d_{2}, *x*) is a pair of RSA signatures {S(d_{1}, *x*) = *x*^{d1}_{mod B}, S(d_{2}, *x*) = *x*^{d2}_{mod B}}, signature pair T(*A*, T_{P}, R) = S(d_{1}_{║}d_{2}, T_{P}^{R+1}K_{w}G_{w}^{R}_{mod B}) is an anonymous tag based credential generated by *A* and given to P. Here, integers B, T_{P}, *k* and *g* are publicly known and they are defined so that to know integers q_{1}, q_{1*}, q_{2}, q_{2*}, q_{3}, q_{3*} that satisfy relations T_{P} = *k*^{q1}_{mod B}, T_{P}^{q1*}_{mod B} = *k*, *k* = *g*^{q2}_{mod B}, *k*^{q2*}_{mod B} = *g*, *g* = T_{P}^{q3}_{mod B},* **g*^{q3*}_{mod B} = T_{P} is computationally infeasible for entities other than *A*. Different from B, *k* and *g* that are common to all credentials, T_{P} and R are unique to T(*A*, T_{P}, R), and P calculates K_{w} and G_{w} as K_{w} = *k*^{w}_{mod B} and G_{w} = *g*^{w}_{mod B} based on publicly known *k*, *g* and secret integer *w*. About signing keys d_{1} and d_{2}, they can be maintained as secrets of *A* despite multiple verification keys are publicly disclosed because the signer is only *A*. Also it is easy to maintain uniqueness of P’s secret integer R as will be discussed in Sec. 4.1 ^{[8, 9, 11]}. In the remainder, notation _{mod B} is omitted when confusions can be avoided.

Important things are, firstly P can prove that credential T(*A*, T_{P}, R) is legitimate and it knows secret integer R in it by showing T(*A*, T_{P}, R)^{W}_{ }= S(d_{1}_{║}d_{2}, T_{P}^{R+1}K_{w}G_{w}^{R}_{mod B})^{W} without disclosing R itself, and secondly for given integer *U*, other entities can force P to calculate *U*^{R} honestly as a used seal of T(*A*, T_{P}, R) while using secret integer R. Here, W is P’s secret integer and entities that do not know W cannot know the correspondence between T(*A*, T_{P}, R) and T(*A*, T_{P}, R)^{W}, i.e. to calculate W from T(*A*, T_{P}, R) and T(*A*, T_{P}, R)^{W} is a discrete logarithm problem. Also, only P that knows R can calculate *U*^{R} from *U*. Then, P can convince others that it is a legitimate owner of the credential without revealing its identity by showing T(*A*, T_{P}, R)^{W}. On the other hand, other entities can use used seal *U*^{R} as an evidence that P had certainly shown credential T(*A*, T_{P}, R). But it must be noted that *U*^{R} and *U*^{R}^{*} may have a same value despite R ≠ R^{*}. Therefore to make used seals unique to credentials, actually T(*A*, T_{P}, R) is implemented as a set of values {S(d_{1}_{1}_{║}d_{2}_{1}, T_{P}^{R+1}K_{w}G_{w}^{R}_{mod B}_{1}), S(d_{1}_{2}_{║}d_{2}_{2}, T_{P}^{R+1}K_{w}G_{w}^{R}_{mod B}_{2})} by using different integers B_{1} and B_{2}. Namely, relation *U*^{R}_{mod B}_{1} = *U*^{R}^{*}_{mod B}_{1} does not holed even when relation *U*^{R}_{mod B}_{2} = *U*^{R}^{*}_{mod B}_{2} holds.

In conclusion, together with used seal *U*^{R} credential T(*A*, T_{P}, R) satisfies the following requirements. They are,

a) *U**nforgeability** *no one other than authority *A* can generate valid credentials,

b) *Soundness** **e*ntities that do not know integer R in credential T(*A*, T_{P}, R) cannot prove the ownership of T(*A*, T_{P}, R) to other entity Q. Also, when Q dishonestly accepts T(*A*, T_{P}, R) shown by other credential holder P_{*} possibly while conspiring with it, *A* can detect that and identify liable entities,

c) *A**nonymity** *anyone except P cannot identify P from credential form T(*A*, T_{P}, R)^{W},

d) *Unlinkability** *even if P shows credential T(*A*, T_{P}, R) n-times in forms T(*A*, T_{P}, R)^{W1}, T(*A*, T_{P}, R)^{W}^{2}, ---, T(*A*, T_{P}, R)^{W}^{n} while generating different secret integers W_{1}, W_{2}, ---, W_{n}, no one except P can know links between them,

e) *Revocability** **A* can invalidate credential T(*A*, T_{P}, R) without knowing secrets of honest entities, if its holder P behaved dishonestly while showing T(*A*, T_{P}, R)^{W} or if *A* reissued new credential to P as a replacement of T(*A*, T_{P}, R), and

f) *Verifiability* anyone can verify the validity of credential T(*A*, T_{P}, R), in other words, entities can verify the validity of T(*A*, T_{P}, R) without knowing any secret of *A*.

**3.2. Linear Mix-net**

Linear Mix-net *L* consists of a sequence of mutually independent mix-servers L_{1}, L_{2}, ---, L_{Z} and enables authority *A* to calculate the sum of attribute values D_{P}(1), D_{P}(2), ---, D_{P}(H_{P}) owned by same data holder P without knowing the correspondence between P and each D_{P}(h) or the calculated sum or links between individual attribute values D_{P}(1), D_{P}(2), ---, D_{P}(H_{P}) ^{[10, 11]}.

Conceptually, P generates triplet {h, D_{P}(h), *U*(h)^{R}} that includes its h-th attribute value D_{P}(h) for each h, puts triplets {1, D_{P}(1), *U*(1)^{R}}, {2, D_{P}(2), *U*(2)^{R}}, ---, {H_{P}, D_{P}(H_{P}), *U*(H_{P})^{R}} in *L* separately without revealing its identity, and mix-servers L_{1}, L_{2}, ---, L_{Z} repeatedly encrypt each {h, D_{P}(h), *U*(h)^{R}} to {h, E(D_{P}(h)), *U*(h)^{R}^{∙V}} by their secret encryption keys while shuffling their encryption results. After that, authority *A* gathers encrypted triplets {1, E(D_{P}(1)), *U*(1)^{R}^{∙V}}, {2, E(D_{P}(2)), *U*(2)^{R}^{∙V}}, ---, {H_{P}, E(D_{P}(H_{P})), *U*(H_{P})^{R}^{∙V}} that correspond to P and calculates sum E(D_{P}(*)) = E(D_{P}(1))+E(D_{P}(2))+ --- + E(D_{P}(H_{P})), constructs pair {E(D_{P}(*)), *U*(*)^{R}^{∙}^{V}}, and finally L_{Z}, L_{Z-1}, ---, L_{1} repeatedly decrypt each pair {E(D_{P}(*)), *U*(*)^{R}^{∙}^{V}} to {D_{P}(*) = D_{P}(1)+D_{P}(2)+ --- +D_{P}(H_{P}), *U*(*)^{R}^{∙}^{V}^{∙Y}} by their secret decryption keys while shuffling their decryption results.

In detail, P convinces *L* of its eligibility by showing credential T(*A*, T_{P}, R)^{W(P, }^{h}^{)} while generating secret integer W(P, h) to put its h-th attribute value D_{P}(h), and calculates used seal *U*(h)^{R} of T(*A*, T_{P}, R) from integer *U*(h) defined by L_{1}, L_{2}, ---, L_{Z}. About encryptions and decryptions of *U*(h)^{R} in {h, D_{P}(h), *U*(h)^{R}} and *U*(*)^{R}^{∙}^{V} in {E(D_{P}(*)), *U*(*)^{R}^{∙}^{V}}, L_{1}, L_{2}, ---, L_{Z} transform *U*(h)^{R} and *U*(*)^{R}^{∙}^{V} to *U*(h)^{R}^{∙V} = *U*(h)^{R}^{∙V(1)∙V(2)}^{ }^{---V(Z)} and* **U*(*)^{R}^{∙V∙Y} = *U*(*)^{R}^{∙V∙Y(Z)∙Y(Z-1)}^{ }^{---Y(1)} by using their secret integers V(1), Y(1), V(2), Y(2), ---, V(Z), Y(Z).

Therefore, no one except P can know correspondences between P and each D_{P}(h) or D_{P}(*), or links between D_{P}(1), D_{P}(2), ---, D_{P}(H_{P}) unless all L_{1}, L_{2}, ---, L_{Z} conspire (each L_{z} shuffles its encryption and decryption results). Nevertheless, *A* can identify triplets {1, E(D_{P}(1)), *U*(1)^{R}^{∙V}}, ---, {N_{P}, E(D_{P}(H_{P})), *U*(H_{P})^{R}^{∙V}} that correspond to same data holder P, and P can know the sum of its attribute values{D_{P}(*), *U*(*)^{R}^{∙}^{V}^{∙Y}}. To enable *A* to identify the triplets, provided that* U*(0) is a publicly known integer, L_{1}, L_{2}, ---, L_{Z} defines their secret integers X(1, h), X(2, h), ---, X(Z, h) for each h and calculate *U*(1), *U*(2), ---,* **U*(H_{P}), as* U*(1) = *U*(0)^{X(1, 1)∙X(}^{2}^{, }^{1}^{) --- X(}^{Z}^{, }^{1}^{)}, *U*(2) = *U*(1)^{X(}^{1}^{, }^{2}^{)∙X(2, }^{2}^{) --- X(}^{Z}^{, }^{2}^{)}, ---, *U*(H_{P}) = *U*(H_{P-1})^{X(}^{1, HP}^{)∙X(}^{2, HP}^{) --- X(}^{Z, HP}^{)}. Then, L_{1}, L_{2}, ---, L_{Z} calculate *U*(*)^{R}^{∙}^{V} = *U*(h)^{R}^{∙}^{V}^{∙X(}^{1, h+1}^{) --- X(}^{Z, h+1}^{)∙X(}^{1, h+2}^{) --- X(}^{Z, h+2}^{)}^{ --- }^{X(}^{1, HP}^{) --- X(}^{Z, HP}^{)} for each *U*(h)^{R}^{∙}^{V}, and *A* identifies triplets {1, E(D_{P}(1)), *U*(1)^{R}^{∙V}}, ---, {H_{P}, E(D_{P}(H_{P})), *U*(H_{P})^{R}^{∙V}} based on *U*(*)^{R}^{∙}^{V} calculated from *U*(1)^{R}^{∙V}, ---, *U*(H_{P})^{R}^{∙V}. On the other hand, P calculates *U*(*)^{R}^{∙}^{V}^{∙Y} from *U*(*)^{V}^{∙Y} by using its secret integer R to identify {D_{P}(*), *U*(*)^{R}^{∙}^{V}^{∙Y}}.

About calculation of D_{P}(*) = D_{P}(1)+D_{P}(2)+ --- +D_{P}(H_{P}), L_{1}, L_{2}, ---, L_{Z} encrypts each D_{P}(h) by additive encryption functions, as a result, E(D_{P}(1))+E(D_{P}(2))+ --- +E(D_{P}(H_{P})) is decrypted to D_{P}(*). *A* and L_{1}, L_{2}, ---, L_{Z} also can convince others of their honest handling of each triplet without disclosing their secret keys or integers provided that dishonesties bring losses to some entity.

### 4. Behaviors of the e-Cash Scheme

The proposed scheme of e-cash systems consists of 5 phases, i.e. registration, issuing, paying, reporting and verification phases. Here, authority *A* is accompanied by linear Mix-net *L* consists of mix-servers L_{1}, L_{2}, ---, L_{Z}. Also provided that H is the maximum number of payments that a single cash holder makes within a service period, each mix-server L_{z} maintains its secret integers X(z, 1), X(z, 2), ---, X(z, H), and for each h (0 < h ≤ H), L_{1}, L_{2}, ---, L_{Z} jointly calculates integer *U*_{h} as *U*_{h} = *U*_{h}_{-1}^{X(}^{h}^{)} = *U*_{h}_{-1}^{X(}^{1, h}^{)∙X(}^{2, h}^{) --- X(}^{Z, h)} to be disclosed publicly, as in the previous section. Therefore, no one can know integer X(h) unless all L_{1}, L_{2}, ---, L_{Z} conspire. Then, first 4 phases proceed as shown in Figure 2.

An important thing is, different from other e-cash schemes, validity of e-cash that each cash holder P_{m} pays is not ensured by authority *A*’s signature because in offline environments even *A*’s signature cannot disable cash holders to spend same e-cash multiple times. Instead, validity of e-cash P_{m} pays is ensured by the anonymous signature of P_{m}, in other words, the issuer of e-cash P_{m} pays is P_{m} itself. Here, anonymous signatures enable signers to conceal their identities.

**Fig**

**ure**

**2**

**.**First 4 phases of the proposed e-cash scheme

**4.1. Registration Phase**

Each cash holder P_{m} registers itself and obtains its credential and cash-IDs while revealing its identity as below. In the following phases, P_{m} uses its credential and cash-IDs to show its eligibility and to divide its e-cash respectively of course without revealing its identity. P_{m} uses credentials also to generate its anonymous signatures. On the other hand, authority *A* uses anonymous signatures and cash-IDs to identify dishonest cash holders and to calculate total cash values of e-cash generated by dividing same e-cash, respectively.

1. Pm shows A its identity.

2. If Pm is eligible, A issues credential T(A, TPm, Rm) and cash-IDs T*(A, TPm(1), Rm(1)), T*(A, TPm(2), Rm(2)), ---, T*(A, TPm(H), Rm(H)) to Pm.

3. Pm calculates initial used seal U_Rm from publicly disclosed constant integer U_ by using T(A, TPm, Rm).

4. A registers Pm as an authorized member with initial used seal U_Rm.

Here, *A* uses initial used seal* U*_{_}^{Rm} to disable P_{m} to use credentials of other cash holders. In detail, at a time when *A* requests P_{m} to calculate a used seal of T(*A*, T_{Pm}, R_{m}) from integer λ while showing its identity, *A* asks P_{m} to calculate a used seal also from integer *U*_{_}. Namely, if P_{m} calculated λ^{R*} by using a credential of other entity, P_{m} calculates *U*_{_}^{R}^{*} from* **U*_{_} that is different from initial used seal *U*_{_}^{Rm}. About cash-IDs, each T_{*}(*A*, T_{Pm}(h), R_{m}(h)) has the same structure as credential T(*A*, T_{Pm}, R_{m}). But to discriminate credentials from cash-IDs *A* defines different signing keys, i.e. *A* uses keys pairs {d_{1}, d_{2}} and {d_{1}_{*}, d_{2}_{*}} to generate credentials and cash-IDs, respectively.

Secret unique integers R_{m} = R_{m}(0), R_{m}(1), R_{m}(2), ---, R_{m}(H) in the credential and cash-IDs are generated through the cooperation among L_{1}, L_{2}, ---, L_{Z}. Namely provided that Ω is a publicly known integer, for each h (h = 0, 1, 2, ---, H), each mix-server L_{z} generates secret integer R_{m}(z, h) and calculates Ω^{R}^{m}^{(z*, h)} = Ω^{R}^{m}^{(}^{(z-1)*}^{, h)∙R}^{m}^{(}^{z}^{, h)} based on Ω^{R}^{m}^{(}^{(z-1)*}^{, h)} disclosed by L_{z-1}, and when finally disclosed Ω^{R}^{m}^{(}^{Z*}^{, h)} did not appear before each L_{z} informs P_{m} of R_{m}(z, h) so that P_{m} can calculate unique secret integer R_{m}(h) as product R_{m}(h) = R_{m}(1, h)∙R_{m}(2, h) --- R_{m}(Z, h). When Ω^{R}^{m}^{(}^{Z*}^{, h)} had appeared already, L_{1}, L_{2}, ---, L_{Z} generate different secret integers ^{[8, 11]}.

**4.2, Issuing Phase**

Authority *A* issues its i-th e-cash *C*(*A*, i) to cash holder P_{m} as t-th e-cash of P_{m} without knowing P_{m} as below.

1. P_{m} generates secret integers V_{m}(t) and V^{*}_{m}(t), and convinces *A* of its ownerships of credential T(*A*, T_{Pm}, R_{m}) and t-th cash-ID T_{*}(*A*, T_{Pm}(t), R_{m}(t)) by showing T(*A*, T_{Pm}, R_{m})^{Vm(t)} and T_{*}(*A*, T_{Pm}(t), R_{m}(t))^{V}^{*}^{m(t)}.

*2**. **A* defines cash value e(i) and expiration time d(i).

3. P_{m} calculates source-ID part value* **U*_{0}^{Rm(t)} and payee’s signature {*U*_{0}^{Rm(t)}+e(i)║d(i)}^{Rm} from integer *U*_{0}, e(i) and d(i) as used seals of T_{*}(*A*, T_{Pm}(t), R_{m}(t)) and T(*A*, T_{Pm}, R_{m}). Here, notation ║ represents concatenation, and cash value e(i) and expiration time d(i) are integers.

4. If *U*_{0}^{Rm(t)} did not appear before,* A* constructs cash issuing record *I*(i) = <e(i), d(i), 0, *U*_{0}^{Rm(t)}, {*U*_{0}^{Rm(t)}+e(i)║d(i)}^{Rm}> shown in Figure 3 (a) and gives *I*(i) to P_{m}. *A* also discloses *I*(i) publicly in its database.

5. P_{m}* *pays cash value e(i) through its anonymous credit card, or it simply pays real cash of value e(i) to *A*.

In the above, P_{m} chooses its t-th cash-ID for obtaining its t-th e-cash (actually P_{m} can choose an arbitrary cash-ID but step 4 disables P_{m} to use same cash-IDs repeatedly), as a result, P_{m} uses different cash IDs for different e-cash. Also, it changes values of secret integers V_{m}(t) and V^{*}_{m}(t) every time when it accesses *A*, therefore *A* cannot identify P_{m} or know links between individual e-cash P_{m} had obtained. But to maintain its anonymity P_{m} must buy *C*(*A*, i) by its anonymous credit card or by paying real cash.

About the issuing record in Figure 3 (a), division-No. part value 0 represents that *C*(*A*, i) is the original e-cash directly issued by *A* (i.e. it is the 0-th division). Together with division-ID part values in cash paying records shown in Figure 3 (b), source-ID part value *U*_{0}^{R}^{m}^{(t)} enables *A* to identify e-cash that are generated by dividing *C*(*A*, i) without knowing links between *C*(*A*, i) and individual divided e-cash. Payee’s signature {*U*_{0}^{R}^{m}^{(t)}+e(i)║d(i)}^{Rm} is used to identify dishonest cash holders without knowing secrets of honest entities, as will be discussed in Sec. 5.

**Fig**

**ure**

**3.**Configurations of cash records

Here, P_{m} calculates *U*_{0}^{Rm(t)} and {*U*_{0}^{Rm(t)}+e(i)║d(i)}^{Rm} as used seals of T_{*}(*A*, T_{Pm}(t), R_{m}(t)) and T(*A*, T_{Pm}, R_{m}), therefore although *A* does not know values of R_{m}(t) or R_{m}, it can convince itself that P_{m} calculates them honestly. Also {*U*_{0}^{R}^{m}^{(t)}+e(i)║d(i)}^{Rm} can be considered as an anonymous signature of P_{m} ^{[11]}, i.e. no one except P_{m} can know P_{m} from credential form T(*A*, T_{Pm}, R_{m})^{W}, {*U*_{0}^{R}^{m}^{(t)}+e(i)║d(i)}^{Rm} can be calculated only by P_{m} that knows integer R_{m}, correct calculation of {*U*_{0}^{R}^{m}^{(t)}+e(i)║d(i)}^{Rm} is ensured despite R_{m} is unknown, and by calculating a used seal of T(*A*, T_{Pm}, R_{m}) from integer *U*_{0}^{R}^{m}^{(t)}+e(i)║d(i), P_{m} can convince any entity that {*U*_{0}^{R}^{m}^{(t)}+e(i)║d(i)}^{Rm} was generated by it.

About the expiration time d(i) of *C*(*A*, i), because it may be used as a clue to identify P_{m}, all e-cash issued in a same service period have same expiration time. Also, *C*(*A*, i) and all e-cash generated by dividing *C*(*A*, i) have the same expiration time.

**4.3. Paying Phase**

Paying phase, in which P_{m} makes its purchase and pays e-cash *C*(P_{m}, t, h) to other cash holder P_{k} by dividing e-cash *C*(*A*, i) or *C*(P_{j}, s, v), proceeds as follow. Here, *C*(*A*, i) or *C*(P_{j}, s, v) is P_{m}’s t-th e-cash that it had received directly from issuing authority *A* or from other cash holder P_{j} respectively. Here as a special case, to exchange a part of *C*(*A*, i) or *C*(P_{j}, s, v) for real cash, P_{m} pays *C*(P_{m}, t, h) to itself. In the following, it is assumed that P_{m} generates* **C*(P_{m}, t, h) as the h-th division of *C*(*A*, i) or *C*(P_{j}, s, v), and P_{k} receives *C*(P_{m}, t, h) as its q-th e-cash. About cash-IDs, although P_{k} below chooses its q-th cash-ID to receive its q-th e-cash, it can choose an arbitrary unused cash-ID.

1. P_{m} generates secret integers W_{m}(t, h) and W^{*}_{m}(t, h), shows credential T(*A*, T_{P}_{m}, R_{m}) and t-th cash-ID T_{*}(*A*, T_{m}(t) R_{m}(t)) in forms T(*A*, T_{P}_{m}, R_{m})^{Wm}^{(}^{t, h}^{)} and T_{*}(*A*, T_{P}_{m}(t), R_{m}(t))^{W}^{*}^{m}^{(}^{t, h}^{)} and convinces P_{k} of its eligibility and the ownership of T_{*}(*A*, T_{P}_{m}(t), R_{m}(t)).

2. P_{k} generates secret integers V_{k}(q) and V^{*}_{k}(q), chooses q-th cash-ID T_{*}(*A*, T_{k}(q), R_{k}(q)), shows credential T(*A*, T_{P}_{k} R_{k}) and T_{*}(*A*, T_{k}(q), R_{k}(q)) in forms T(*A*, T_{P}_{k}, R_{k})^{Vk}^{(}^{q}^{)} and T_{*}(*A*, T_{P}_{k}(q), R_{k}(q))^{V}^{*}^{k}^{(}^{q}^{)}, and convinces P_{m} of its eligibility and the ownership of T_{*}(*A*, T_{P}_{k}(q), R_{k}(q)).

3. P_{m} defines cash value e(P_{m}, t, h) and expiration time d(P_{m}, t, h), calculates division-ID part value* **U*_{h}^{Rm(}^{t}^{)} from integer *U*_{h} by using cash-ID T_{*}(*A*, T_{P}_{m}(t), R_{m}(t)), and informs P_{k} of quadruplet <e(P_{m}, t, h), d(P_{m}, t, h), h, *U*_{h}^{Rm(}^{t}^{)}>.

4. P_{k} calculates source-ID part value *U*_{0}^{R}^{k}^{(}^{q}^{)} from *U*_{0} as a used seal of T_{*}(*A*, T_{P}_{k}(q), R_{k}(q)).

5. P_{m} calculates payer’s signature *S*(R_{m}, G(P_{m}, t, h)) = G(P_{m}, t, h)^{Rm} as a used seal of T(*A*, T_{P}_{m}, R_{m}), and constructs paying record *P*(P_{m}, t, h) = <e(P_{m}, t, h), d(P_{m}, t, h), h,* **U*_{h}^{Rm(}^{t}^{)}, *U*_{0}^{R}^{k}^{(}^{q}^{)}, *S*(R_{m}, G(P_{m}, t, h))> to give P_{k} as shown in Figure 3 (b). Where, G(P_{m}, t, h) = *U*_{h}^{R}^{m}^{(}^{t}^{)}+*U*_{0}^{R}^{k}^{(}^{q}^{)}+e(P_{m}, t, h)║d(P_{m}, t, h).

In the above, P_{m} and P_{k} show their credentials and cash-IDs while modifying them by their secret integers W_{m}(t, h), W^{*}_{m}(t, h), V_{k}(q) and V^{*}_{k}(q), therefore P_{m} and P_{k} can make them anonymous as same as in the issuing phase. Also P_{m} can conceal links among e-cash it generates from *C*(*A*, i) or *C*(P_{j}, s, v). Nevertheless, P_{k} can confirm correct calculations of division-ID part value* **U*_{h}^{R}^{m}^{(}^{t}^{)} and payer’s signature *S*(R_{m}, G(P_{m}, t, h)) because P_{m} calculates them as used seals of its cash-ID and credential. In the same way, P_{m} can confirm P_{k}’s correct calculation of source-ID part value *U*_{0}^{Rk(q)}.

Here as discussed in the previous subsection, together with the source-ID part value in issuing record *I*(i) and/or paying record *P*(P_{j}, s, v), division-ID part value* U*_{h}^{Rm(t)} in *P*(P_{m}, t, h) enables *A* to confirm that P_{m} had honestly divided its t-th e-cash *C*(*A*, i) or *C*(P_{j}, s, v) without knowing P_{m},* **C*(*A*, i) or *C*(P_{j}, s, v). *A* can also identify dishonest entities by exploiting payer’s signatures in paying records together with payee’s signatures that are calculated at step 2 in the reporting phase.

About dishonesties of cash holders, because P_{m} and P_{k} interact under offline environments, both of P_{m} and P_{k} can forge paying records without being noticed by P_{k} or P_{m}, e.g. P_{m} may calculate the division-ID part value while using a cash-ID different from T_{*}(*A*, T_{m}(t) R_{m}(t)), also P_{k} may modify the cash value, the expiration time or the division-ID part value in *P*(P_{m}, t, h) after it had received it from P_{m}. But these dishonesties are detected and liable entities are identified while exploiting division-ID and source-ID parts values and payer’s and payee’s signatures as will be discussed in Sec. 5.

**4.4. Reporting Phase**

In this phase, cash holder P_{k} that received its q-th e-cash *C*(P_{m}, t, h) from other cash holder P_{m} as paying record *P*(P_{m}, t, h) = <e(P_{m}, t, h), d(P_{m}, t, h), h, *U*_{h}^{Rm(}^{t}^{)}, *U*_{0}^{R}^{k}^{(}^{q}^{)}, *S*(R_{m}, G(P_{m}, t, h))>, reports *P*(P_{m}, t, h) to *A* by the expiration time of *C*(P_{m}, t, h). Also, *A* pays real cash of value e(P_{m}, t, h) to P_{k}, if P_{k} wants cashing of *C*(P_{m}, t, h). The reporting phase proceeds as below.

1. P_{k} generates secret integers W_{k}(q, 0) and W^{*}_{k}(q, 0), and shows its credential and q-th cash-ID in forms T(*A*, T_{P}_{k}, R_{k})^{Wk}^{(}^{q, }^{0}^{)} and T_{*}(*A*, T_{k}(q), R_{k}(q))^{W*k}^{(}^{q, }^{0}^{)} to convince *A* of its eligibility and the ownership of the cash-ID.

2. P_{k} shows paying record *P*(P_{m}, t, h) to *A*, and calculates source-ID part value* **U*_{0}^{R}^{k}^{(}^{q}^{)} from *U*_{0} and payee’s signature *S*(R_{k}, *S*(R_{m}, G(P_{m}, t, h))) =* **S*(R_{m}, G(P_{m}, t, h))^{Rk} = G(P_{m}, t, h)^{Rm∙Rk} from *S*(R_{m}, G(P_{m}, t, h)) by its q-th cash-ID and credential, respectively.

3. If P_{k} calculates source-ID part value *U*_{0}^{R}^{k}^{(}^{q}^{)} successfully and *U*_{0}^{R}^{k}^{(}^{q}^{)} did not appear before,* **A* accepts* **P*(P_{m}, t, h) and discloses it publicly with the payee’s signature, provided that *P*(P_{m}, t, h) does not expire.

4. When P_{k} wants cashing and no one exchanged *C*(P_{m}, t, h) for real cash yet, *A* pays real cash of value e(P_{m}, t, h) to P_{k}. *A* also modifies source-ID part value in* **P*(P_{m}, t, h) to 0. Where, source-ID part value 0 means P_{k} cannot generate new e-cash from *C*(P_{m}, t, h).

5. P_{k} calculates used seal of its q-th cash-ID from integer *U*_{_}, i.e. *U*_{_}^{R}^{k}^{(}^{q}^{)}, as a receipt, and *A* discloses *P*(P_{m}, t, h) with the receipt publicly.

In the above, because source-ID part value *U*_{0}^{R}^{k}^{(}^{q}^{)} can be calculated only by cash-ID T_{*}(*A*, T_{k}(q), R_{k}(q)) and P_{k} must calculate it honestly, anyone except P_{k} cannot report *P*(P_{m}, t, h) or exchange *C*(P_{m}, t, h) for real cash unless it is conspiring with P_{k}. In the same way, receipt* **U*_{_}^{R}^{k}^{(}^{q}^{)} disables P_{k} to exchange *C*(P_{m}, t, h) for real cash repeatedly. Also, P_{k} can convince *A* of correct calculation of payee’s signature without revealing its identity or secret integer R_{k}, then it can maintain its anonymity as same as in the paying phase.

About dishonest cash holders, because *A* discloses paying record *P*(P_{m}, t, h) at step 3, cash holders become able to use division-ID part value *U*_{h}^{Rm(}^{t}^{)} calculated by P_{m} to generate or modify their paying records. But it must be noted that step 3 disables cash holders to use same source-ID part values repeatedly. As a result, all paying records have different source-ID part values, and this means cash holders including P_{k} cannot forge paying records that include consistent payer’s signatures of P_{m} without the help of P_{m} (only P_{m} can generate consistent signatures of P_{m} on information that includes newly appearing source-ID part values). Therefore, when multiple paying records with consistent payer’s signatures and same division-ID part value *U*_{h}^{R}^{m}^{(}^{t}^{)} are detected, *A* can regard P_{m} as the liable entity. In other words, entities other than P_{m} cannot generate a paying record with division-ID part value *U*_{h}^{R}^{m}^{(}^{t}^{)} and consistent payer’s signature of P_{m} without the help of P_{m} despite *U*_{h}^{R}^{m}^{(}^{t}^{)} is disclosed.

### 5. Verification Phase

The verification phase that consists of dishonest record detection and dishonest entity identification stages detects dishonesties of entities and identifies entities that are liable for them.

In offline environments, payees cannot know whether payers are generating new e-cash so that their cash values do not exceed cash values of original e-cash or not. Authority *A* cannot sign on paying records that payers generate either. As a result, cash holders can behave dishonestly without being noticed by others. As shown below, there are 8 kinds of possibilities that entities involved behave dishonestly. Here in the following it is assumed that to pay P_{k} cash holder P_{m} generates e-cash* C*(P_{m}, t, h) and corresponding cash paying record* **P*(P_{m}, t, h) from e-cash *C*(P_{j}, s, v) that P_{m} had obtained from P_{j}. But* *illegitimate paying records are detected and liable entities are identified in the same way also in cases where* *P_{m} generates* C*(P_{m}, t, h) from *C*(*A*, i) directly issued from *A*.

a. P_{m} generates e-cash from *C*(P_{j}, s, v) more than the cash value of* **C*(P_{j}, s, v).

b. P_{m} generates paying record *P*(P_{m}, t, h) to give to P_{k} while forging the expiration time or calculating a division-ID part value by a cash-ID different from T_{*}(*A*, T_{P}_{m}(t), R_{m}(t)) (but P_{m} must use its cash-ID if P_{k} is honest, because P_{k} examines the division-ID part value at step 3 in the paying phase).

c. P_{m} does not report paying record *P*(P_{j}, s, v) to *A*.

d. P_{k} reports paying record *P*(P_{m}, t, h) to *A* while modifying it, e.g. changing its cash value, expiration time or division-ID part value. Or to use e-cash *C*(P_{m}, t, h) repeatedly, it reports *P*(P_{m}, t, h) multiple times while changing source-ID part values (source-ID part values must be unique to individual e-cash).

e. Other cash holder P_{j} reports *P*(P_{m}, t, h) to *A* while changing the source-ID part value in it. Where, P_{j} can obtain *P*(P_{m}, t, h) because *A* discloses it at step 3 in the reporting phase, or P_{j} may steal it by eavesdropping on interactions between P_{m} and P_{k}. But P_{j} must calculate the source-ID part value by its cash-ID because *A* examines the consistency between the source-ID part value and the cash-ID of P_{j}.

f. Authority *A* arbitrarily forges paying records.

Namely, because P_{m} or P_{k} cannot know e-cash generated and used in other places in offline environments and *A* in the reporting phase can examine only source-ID part values and payee’s signatures in individual paying records, P_{m} can generate e-cash from *C*(P_{j} s, v) more than its cash value, can generate *P*(P_{m}, t, h) while using a cash-ID different from T_{*}(*A*, T_{m}(t), R_{m}(t)) used to calculate the source-ID part value of *P*(P_{j}, s, v), or may not report *P*(P_{j}, s, v) to *A*. Also, P_{k} can report *P*(P_{m}, t, h) to *A* while modifying its cash value, expiration time and/or division-ID part value.

As the more vicious dishonesty, when other cash holder P_{j} steals *P*(P_{m}, t, h) given to P_{k} and reports it to *A* (of course while changing the source-ID part value), P_{j} becomes able to generate e-cash from P_{k}’s e-cash* **C*(P_{m}, t, h). Also, when authority *A* is dishonest, it can forge paying records arbitrarily to be registered in its database. Where about conspiracy between P_{m} and P_{k}, although it enables them to forge consistent paying records, they cannot reap any benefit as a total, i.e. one of P_{m} and P_{k} must compensate losses caused by the forged e-cash.

But it must be noted that cash holders must report paying records while generating source-ID part values and payee’s signatures by their legitimate cash-IDs and credentials, i.e. *A* examines them at steps 2 and 3 in the reporting phase. In addition, all paying records must have different source-ID values, and once authority *A* had accepted paying records, anyone including *A* and mix-servers in Mix-net *L* must honestly handle them, i.e. all paying records are publicly disclosed, and honest encryptions and decryptions of *L* are ensured ^{[10]}.

Then, without knowing secrets of honest entities, the dishonest record detection and the dishonest entity identification stages become able to detect above dishonesties and identify liable entities as in the following subsections.

**5.1. Dishonest Record Detection Stage**

The dishonest record detection stage detects dishonesties while exploiting division-ID and source-ID parts values in paying records and issuing records. Namely, although additional mechanisms are necessary so that P_{m} can conceal links between *C*(P_{m}, t, 1),* **C*(P_{m}, t, 2), ---, *C*(P_{m}, t, H(m, t)) it had generated from same e-cash* **C*(P_{j}, s, v) from others, while exploiting relation* **U*_{h}^{Rm(t)∙}^{X(h+1)}^{∙}^{X(h+2) --- X(H)} = *U*_{0}^{Rm(t)∙}^{X(1)}^{∙}^{X(2) --- X(H)} =* **U*_{0}^{Rm(t)∙}^{X*}, *A* can determine paying record *P*(P_{m}, t, h) accompanied by division-ID value *U*_{h}^{Rm(t)} was generated from *P*(P_{j}, s, v) when its source-ID part value *U*_{0}^{Rm(t)} is converted to *U*_{0}^{Rm(t)∙}^{X*}. Therefore, if *A* gathers *P*(P_{m}, t, 1),* **P*(P_{m}, t, 2), ---, *P*(P_{m}, t, H(m, t)), and compares cash values of *C*(P_{m}, t, 1),* **C*(P_{m}, t, 2), ---, *C*(P_{m}, t, H(m, t)) and *C*(P_{j}, s, v), *A* can examine whether all *C*(P_{m}, t, 1),* **C*(P_{m}, t, 2), ---, *C*(P_{m}, t, H(m, t)) were honestly generated and handled or not. Here, if P_{m} did not report *P*(P_{j}, s, v) to *A* or* **P*(P_{j}, s, v) had expired already, *A* cannot use relation* **U*_{h}^{Rm(t)∙}^{X(h+1) --- X(H)} = *U*_{0}^{Rm(t)∙}^{X*} to find* **C*(P_{j}, s, v) from which *C*(P_{m}, t, h) was generated. But in this case, *A* identifies each *C*(P_{m}, t, h) as an orphan paying record that cannot be corresponded to any paying or issuing record. As a result, every kind of dishonesties at the beginning of this section can be detected as inconsistent division-IDs and source-ID pairs or orphan paying records.

**Fig**

**ure 4**

**.**Expenditure, new-cash and total expenditure records

To implement the above strategies, authority *A* constructs expenditure record *D*(P_{m}, t, h) = <e(P_{m}, t, h), h, *U*_{h}^{Rm(}^{t}^{)}, G(P_{m}, t, h),* **S*(R_{m}, G(P_{m}, t, h)),* **S*(R_{k},* **S*(R_{m}, G(P_{m}, t, h)))> and new-cash record *N*(P_{m}, t, h) = <e(P_{m}, t, h), *U*_{0}^{R}^{k}^{(}^{q)}> from paying record *P*(P_{m}, t, h) = <e(P_{m}, t, h), d(P_{m}, t, h), h,* **U*_{h}^{Rm(}^{t}^{)}, *U*_{0}^{R}^{k}^{(}^{q}^{)}, *S*(R_{m}, G(P_{m}, t, h))> as shown in Figure 4 (a) and (b). Here, record characterizer part value G(P_{m}, t, h) and payer’s signature *S*(R_{m}, G(P_{m}, t, h)) are *U*_{h}^{R}^{m}^{(}^{t}^{)}+*U*_{0}^{R}^{k}^{(}^{q}^{)}+e(P_{m}, t, h)║d(P_{m}, t, h) and G(P_{m}, t, h)^{Rm} respectively, and payee’s signature* **S*(R_{k}, *S*(R_{m},G(P_{m}, t, h))) = G(P_{m}, t, h)^{Rm}^{∙}^{Rk} is calculated when P_{k} reports *P*(P_{m}, t, h) to *A*. In addition to the above records, total expenditure records are defined as shown in Figure 4 (c).

Authority *A* detects inconsistent division-IDs and source-ID pairs and orphan paying records without knowing secrets of honest entities while exploiting linear Mix-net *L* = {L_{1}, L_{2}, ---, L_{Z}} as follows.

1. For each paying record *P*(P_{m}, t, h) = <e(P_{m}, t, h), d(P_{m}, t, h), h, *U*_{h}^{Rm(}^{t}^{)},* **U*_{0}^{R}^{k}^{(}^{q}^{)}, *S*(R_{m}, G(P_{m}, t, h))>, which is valid in a considering service period, *A* constructs expenditure record *D*(P_{m}, t, h) = <e(P_{m}, t, h), h, *U*_{h}^{Rm(}^{t}^{)}, G(P_{m}, t, h),* **S*(R_{m}, G(P_{m}, t, h)),* **S*(R_{k},* **S*(R_{m}, G(P_{m}, t, h)))> and new-cash record *N*(P_{m}, t, h) = <e(P_{m}, t, h), *U*_{0}^{R}^{k}^{(}^{q)}>, and put *D*(P_{m}, t, h) in linear Mix-net *L*. Here, *S*(R_{k},* **S*(R_{m}, G(P_{m}, t, h))) is the payee’s signature accompanying *P*(P_{m}, t, h).

2. Mix-servers L_{1}, L_{2}, ---, L_{Z} repeatedly encrypt expenditure records put by *A* while shuffling their encryption results. Here, value part e(P_{m}, t, h) in each *D*(P_{m}, t, h) is encrypted to E(e(P_{m}, t, h)) by additive encryption functions of L_{1}, L_{2}, ---, L_{Z}, but *L* does not encrypt division-No. part value h as the attribute No. in Sec. 3.2. About division-ID part value* **U*_{h}^{Rm(}^{t}^{)}, provided that λ(1), λ(2), ---,λ(Z) are secret integers of L_{1}, L_{2}, ---, L_{Z}, λ_{*} =λ(1)∙λ(2) --- λ(Z), X_{*}(z, h) = X(z, h)∙X(z, h+1) --- X(z, H) and X_{*} = X(1)∙X(2) --- X(H), each L_{z} transforms (encrypts) *U*_{h}^{Rm(}^{t}^{)}^{∙{X*(1, h+1)∙}^{λ}^{(1)}∙{X*(2, h+1)∙}^{λ}^{(2)}---{X*(z-1, h+1)∙}^{λ}^{(z-1)}} calculated by L_{z-1} to *U*_{h}^{Rm(}^{t}^{)}^{∙{X*(1, h+1)∙}^{λ}^{(1)}∙{X*(2, h+1)∙}^{λ}^{(2)}---{X*(z-1, h+1)∙}^{λ}^{(z-1)}∙{X*(z, h+1)∙}^{λ}^{(z)}}, while referring to division No. part value h. As a result *U*_{h}^{Rm(}^{t}^{)} is transformed to *U*_{h}^{Rm(}^{t}^{)}^{∙X(h+1)∙X(h+2) ---X(H)∙}^{λ}^{(1)∙}^{λ}^{(2)---∙}^{λ}^{(z)} = *U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*} (as shown in Sec. 4, X(h) = X(1, h)∙X(2, h) --- X(Z, h)). In the same way, L_{1}, L_{2}, ---, L_{Z} transform record characterizer part value G(P_{m}, t, h), payer’s signature* **S*(R_{m}, G(P_{m}, t, h)) and payee’s signature* S*(R_{k},* S*(R_{m}, G(P_{m}, t, h))) to G(P_{m}, t, h)^{λ*}, *S*(R_{m}, G(P_{m}, t, h))^{λ}^{*} and *S*(R_{k},* **S*(R_{m}, G(P_{m}, t, h)))^{λ}^{*}. In conclusion, *D*(P_{m}, t, h) is encrypted to E(*D*(P_{m}, t, h)) = <E(e(P_{m}, t, h)), h, *U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*}, G(P_{m}, t, h)^{λ}^{*}, *S*(R_{m}, G(P_{m}, t, h))^{λ}^{*},* **S*(R_{k},* **S*(R_{m},G(P_{m}, t, h)))^{λ}^{*}>.

**3.*** **A* gathers encrypted expenditure records E(*D*(P_{m}, t, 1)), E(*D*(P_{m}, t, 2)), ---, E(*D*(P_{m}, t, H(m, t)) that include same division-ID part value *U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*}, calculates E(e(P_{m}, t, 1))+E(e(P_{m}, t, 2))+ --- +E(e(P_{m}, t, H(m, t))) = E(e(P_{m}, t, 1)+e(P_{m}, t, 2)+ --- +e(P_{m}, t, H(m, t))) =E(e_{*}(P_{m}, t)), and constructs encrypted total expenditure record E(Sum(P_{m}, t)) = <E(e_{*}(P_{m}, t)), *U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*}, G(P_{m}, t, 1)^{λ}^{*}, *S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*}> to put in *L*. Where, H(m, t) is the largest division No. part value generated from *C*(P_{j}, s, v).

4. L_{Z},* *L_{Z-1}, ---, L_{1} repeatedly decrypt each encrypted total expenditure record E(Sum(P_{m}, t)) put by *A* while shuffling their decryption results, and as a result E(Sum(P_{m}, t)) is decrypted to total expenditure record Sum(P_{m}, t) = <e_{*}(P_{m}, t) = e(P_{m}, t, 1)+e(P_{m}, t, 2)+ --- +e(P_{m}, t, H(m, t)), *U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*∙μ*}, G(P_{m}, t, 1)^{λ}^{*∙μ*}, *S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*∙μ*}>. Here mix-servers must shuffle also their decryption results to protect their additive encryption functions which are usually weak against plain text attacks. About the division-ID part value, provided that μ(1), μ(2), ---, μ(Z) are secret integers of L_{1}, L_{2}, ---, L_{Z} and μ_{*} = μ(1)∙μ(2) --- μ(Z), each L_{z} transforms (decrypts) *U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*∙μ(Z)∙μ(Z-1)---μ(z+1)} calculated by L_{z+1} to *U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*∙μ(Z)∙μ(Z-1)---μ(z+1)∙μ(z)}, and as a result *U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*} is transformed to *U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*∙μ*}. In the same way G(P_{m}, t, 1)^{λ}^{*} and *S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*} are transformed to G(P_{m}, t, 1)^{λ}^{*∙μ*} and *S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*∙μ*}.

5. For each issuing record *I*(i) = <e(i), d(i), 0, *U*_{0}^{Rm(t)}, {*U*_{0}^{Rm(t)}+e(i)║d(i)}^{Rm}> which does not expire, *A* constructs new-cash record *N*(*A*, i, 0) = <e(i), *U*_{0}^{Rm(t)}> and put it in *L* together with new cash records generated at step 1.

6. By using integers λ(1), λ(2), ---,λ(Z), X(1), X(2), ---, X(H), and μ(1), μ(2), ---, μ(Z), mix-servers L_{1}, L_{2}, ---, L_{Z} calculate* **U*_{0}^{R}^{k}^{(}^{q}^{)}^{∙X*∙}^{λ}^{*∙μ*} or* **U*_{0}^{Rm(t)∙X*∙}^{λ}^{*∙μ*} from *U*_{0}^{R}^{k}^{(}^{q)} or *U*_{0}^{Rm(t)} in each *N*(P_{j}, s, v) or *N*(*A*, i, 0) to transform it to *N*_{*}(P_{j}, s, v) = <e(P_{m}, t, h), *U*_{0}^{R}^{k}^{(}^{q}^{)}^{∙X*∙}^{λ}^{*∙μ*}> or *N*_{*}(*A*, i, 0) = <e(i), *U*_{0}^{Rm(t)∙X*∙}^{λ}^{*∙μ*}>.

7. For each total expenditure record Sum(P_{m}, t), *A* finds a transformed new-cash record that includes *U*_{0}^{Rm(t)∙X*∙λ*∙μ*} as its source-ID part value. Where, because all paying records have different source-ID part values, *A* can find at most one transformed new-cash record.

8. Provided that *N*_{*}(P_{j}, s, v) = <e(P_{j}, s, v), *U*_{0}^{Rm(t)∙X*∙λ*∙μ*}> is the found transformed new-cash record, *A* examines whether relation e_{*}(P_{m}, t) ≤ e(P_{j}, s, v) holds or not, and when the relation does not hold it determines that Sum(P_{m}, t) is an illegitimate total expenditure record. When no new-cash record that corresponds to Sum(P_{m}, t) is found, *A* determines Sum(P_{m}, t) is an orphan record.

In the above steps, because *U*_{1}, *U*_{2}, ---, *U*_{H} are calculated as *U*_{1} = *U*_{0}^{X(1)}, *U*_{2} = *U*_{0}^{X(1)}^{∙X(2)},---, *U*_{H} = *U*_{0}^{X(1)}^{∙X(2) --- X(H)}, division-ID part values of all expenditure records that correspond to e-cash generated from e-cash *C*(P_{j}, s, v) and source-ID part value* **U*_{0}^{Rm(}^{t}^{)} of *N*(P_{j}, s, v) are transformed to same value* **U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*∙μ*}. Therefore *A* can identify expenditure records generated from *C*(P_{j}, s, v) as the ones that are accompanied by division-ID part value* **U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*∙μ*}. Also, because linear Mix-net *L* encrypts e(P_{m}, t, 1), e(P_{m}, t, 2), ---, e(P_{m}, t, H(m, t)) in expenditure records by additive encryption functions, E(e(P_{m}, t, 1))+E(e(P_{m}, t, 2))+ --- +E(e(P_{m}, t, H(m, t))) is decrypted to e_{*}(P_{m}, t) = e(P_{m}, t, 1)+e(P_{m}, t, 2)+ --- +e(P_{m}, t, H(m, t)), i.e. the sum of cash values generated from *C*(P_{j}, s, v). As a result, *A* can determine that cash holder P_{m} had honestly divided its e-cash *C*(P_{j}, s, v) when e_{*}(P_{m}, t) does not exceed e(P_{j}, s, v).

Nevertheless, cash holder P_{m} can conceal its payments and links between its individual payments from others. Because integers R_{m}, R_{m(t)} are P_{m}’s secrets, other entities including *A* cannot know P_{m} from issuing record *I*(i) or paying record *P*(P_{m}, t, h). About links between *C*(P_{m}, t, 1), *C*(P_{m}, t, 2), ---,* **C*(P_{m}, t, H(m, t)), to calculate X(h) from *U*_{h-1}^{Rm(}^{t}^{)} and* **U*_{h-1}^{Rm(}^{t}^{)}^{∙X(h)} is a discrete logarithm problem, and entities other than P_{m} cannot identify sequence *U*_{0}^{Rm(}^{t}^{)}, *U*_{1}^{Rm(}^{t}^{)}, *U*_{2}^{Rm(}^{t}^{)}, ---, *U*_{H}^{Rm(}^{t}^{)}.

**5.2. Dishonest Entity Identification Stage**

As in the previous subsection dishonestly handled paying records are detected as inconsistent total expenditure and new-cash records pairs or orphan total expenditure records, and once dishonesties are detected, *A* identifies liable entities without knowing secrets of honest entities by the procedure below. In the following it is assumed that total expenditure record Sum(P_{m}, t) = <e_{*}(P_{m}, t), *U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*∙μ*}, G(P_{m}, t, 1)^{λ}^{*∙μ*}, *S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*∙μ*}> that corresponds to cash holder P_{m} is illegitimate, i.e. e_{*}(P_{m}, t) in Sum(P_{m}, t) is greater than e(P_{j}, s, v) in transformed new-cash record *N*_{*}(P_{j}, s, v) = <e(P_{j}, s, v), *U*_{0}^{R}^{m}^{(}^{t}^{)}^{∙X*∙}^{λ}^{*∙μ*}>, or Sum(P_{m}, t) is an orphan record.

Although additional steps are required to protect secret information of honest entities, conceptually, after detecting illegitimate total expenditure record Sum(P_{m}, t), *A* discloses Sum(P_{m}, t) and asks all cash holders to calculate used seals of their credentials from record characterizer part value G(P_{m}, t, 1)^{λ}^{*∙μ*} in Sum(P_{m}, t) while revealing their identities. Then, P_{m} that calculates *S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*∙μ*} = G(P_{m}, t, 1)^{Rm∙}^{λ}^{*∙μ*} is liable, i.e. only P_{m} that knows integer R_{m} can calculate *S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*∙μ*} from G(P_{m}, t, 1)^{λ}^{*∙μ*} and P_{m} must honestly calculate* **S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*∙μ*} from G(P_{m}, t, 1)^{λ}^{*∙μ*} as a used seal of its own credential.

However, if record characterizer and payers’ signature pair {G_{1}, G_{2}} in *P*(P_{m}, t, 1) is the one forged by P_{m} or someone else no one calculates G_{2}^{λ}^{*∙μ*} from G_{1}^{λ}^{*∙μ*}. Therefore in this case, *A* finds encrypted expenditure record E(*D*(P_{m}, t, 1)) that was used to calculate E(Sum(P_{m}, t)) at step 3 in the dishonest record detection stage, and asks all cash holders to calculate used seals of their credentials from payer’s signature G_{2}^{λ}^{*}, and determines P_{k} is liable when it calculates G_{2}^{λ}^{*∙Rk} that is equal to the payee’s signature G_{2}^{Rk∙}^{λ}^{*} in E(*D*(P_{m}, t, 1)).

Namely, authority *A* examines payee’s signature G_{2}^{Rk} when it accepts paying record* **P*(P_{m}, t, h), and P_{k} must honestly calculate it as a used seal of its credential. Also, P_{k} verifies the consistency of pair {G_{1}, G_{2}} at a time when P_{m} paid *C*(P_{m}, t, h) to it. In addition, anyone including *A* and mix-servers cannot handle paying records dishonestly after they are disclosed. Then, regardless that entities other than P_{k} itself had forged {G_{1}, G_{2}} or not, *A* can determine P_{k} is liable, i.e. P_{k} had accepted *P*(P_{m}, t, h) that includes inconsistent {G_{1}, G_{2}}. As an exception if it is conspiring with *A*, P_{k} does not need to calculate G_{2}^{Rk} honestly by using its credential when it reports *P*(P_{m}, t, h). But even in this case *A* cannot impute the liability to honest cash holders, i.e. *A* cannot forge payee’s signatures of honest cash holders and must compensate corresponding losses by itself.

In detail, the dishonest entity identification stage proceeds as follow.

1. *A* asks mix-servers L_{Z}, L_{Z-1}, ---, L_{1} to trace (partial) encryption forms of illegitimate total expenditure record Sum(P_{m}, t) = <e_{*}(P_{m}, t), *U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*∙μ*}, G(P_{m}, t, 1)^{λ}^{*∙μ*}, *S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*∙μ*}> calculated at step 4 in Sec. 5.1 back to E(Sum(P_{m}, t)) = <E(e_{*}(P_{m}, t)), *U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*}, G(P_{m}, t, 1)^{λ}^{*}, *S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*}>, and finds encrypted expenditure records E(*D*(P_{m}, t, 1)) = <E(e(P_{m}, t, 1)), 1, *U*_{0}^{R}^{m}^{(}^{t}^{)}^{∙X*∙}^{λ}^{*}, G(P_{m}, t, 1)^{λ}^{*}, *S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*}, *S*(R_{k},* **S*(R_{m}, G(P_{m}, t, 1)))^{λ}^{*}>, E(*D*(P_{m}, t, 2)), ---, E(*D*(P_{m}, t, H(m, t))) that constitute E(Sum(P_{m}, t)).

2.* **A* discloses illegitimate total expenditure record Sum(P_{m}, t) with individual encrypted expenditure records E(*D*(P_{m}, t, 1)), E(*D*(P_{m}, t, 2)), ---, E(*D*(P_{m}, t, H(m, t))) publicly.

3. For each illegitimate Sum(P_{m}, t) = <e_{*}(P_{m}, t), *U*_{0}^{Rm(}^{t}^{)}^{∙X*∙}^{λ}^{*∙μ*}, G(P_{m}, t, 1)^{λ}^{*∙μ*}, *S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*∙μ*}>, each P_{r} calculates G(P_{m}, t, 1)^{λ}^{*∙μ*∙Rr} from G(P_{m}, t, 1)^{λ}^{*∙μ*} by using its credential T(*A*, T_{Pr}, R_{r}).

4. When P_{m} calculates G(P_{m}, t, 1)^{λ}^{*∙μ*∙Rm} that coincides with *S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*∙μ*}, and P_{m} admits e_{*}(P_{m}, t) in Sum(P_{m}, t) is its correct total expenditure or expenditure records included in orphan record Sum(P_{m}, t) had already expired, P_{m} pays cash of value {e_{*}(P_{m}, t) - e(P_{j}, s, v)} to *A* possibly with arrears but without revealing its identity. In a case where Sum(P_{m}, t) is an orphan record P_{m} pays e_{*}(P_{m}, t).

5. On the other hand when P_{m} does not believe that e_{*}(P_{m}, t) is its correct total expenditure or expenditure records included in Sum(P_{m}, t) had expired, it examines payer’s signatures in disclosed individual encrypted expenditure records. In detail, for each encrypted record E(*D*(P_{m}, t, h)), P_{m} extracts pair {G(P_{m}, t, h)^{λ*}, *S*(R_{m}, G(P_{m}, t, h))^{λ*}} to examine whether relation G(P_{m}, t, h)^{λ*∙R}^{m} = *S*(R_{m}, G(P_{m}, t, h))^{λ*} holds or not. After that when relation G(P_{m}, t, h)^{λ*∙R}^{m} = *S*(R_{m}, G(P_{m}, t, h))^{λ*} does not hold, P_{m} without revealing its identity claims that E(*D*(P_{m}, t, h)) is not its record while convincing *A* that *S*(R_{m}, G(P_{m}, t, h))^{λ*} is not calculated by its credential (*A* receives this claim at step 9).

6. In a case where relation G(P_{m}, t, h)^{λ*∙R}^{m} = *S*(R_{m}, G(P_{m}, t, h))^{λ*} holds but P_{m} does not believe that E(*D*(P_{m}, t, h)) had expired, P_{m} picks *P*(P_{j}, s, v) from which it had generated* **P*(P_{m}, t, h), and claims that the expiration time in *P*(P_{j}, s, v) is incorrect without revealing its identity.

7. When P_{m} claims that the expiration time in *P*(P_{j}, s, v) = <e(P_{j}, s, v), d(P_{j}, s, v), v, *U*_{v}^{R}^{j}^{(}^{s}^{)},* **U*_{0}^{R}^{m}^{(}^{t}^{)}, *S*(R_{j}, G(P_{j}, s, v))> is incorrect, *A* requests all cash holders to calculate used seal of G(P_{j}, s, v) while revealing their identities.

8. *A* determines cash holder P_{j} that calculates payer’s signature *S*(R_{j}, G(P_{j}, s, v)) from G(P_{j}, s, v) had included a wrong value as the expiration time in *P*(P_{j}, s, v) to give to P_{m}, and P_{j} pays e(P_{j}, s, v) to *A* with penalty fine. But *A* determines P_{m} had accepted incorrect *P*(P_{j}, s, v) intentionally when no one calculates *S*(R_{j}, G(P_{j}, s, v)), and to identify P_{m} proceeds to step 11.

9. When anonymous P_{m} claims that E(*D*(P_{m}, t, h)) is not its expenditure record, *A* requests all cash holders to calculate used seals of their credentials from payer’s signature *S*(R_{m}, G(P_{m}, t, h))^{λ*} while revealing their identities.

10. *A* determines P_{k} that calculates payee’s signature *S*(R_{k},* **S*(R_{m}, G(P_{m}, t, h)))^{λ}^{*} = *S*(R_{m}, G(P_{m}, t, h))^{λ*}^{∙Rk} from* **S*(R_{m}, G(P_{m}, t, h))^{λ*} is liable and forces P_{k} to pay cash of value e(P_{m}, t, h)} with penalty fine.

11. If no one paid for illegitimate expenditure records corresponding to Sum(P_{m}, t) or claimed they were incorrect, or no one calculates payer’s signature *S*(R_{j}, G(P_{j}, s, v)) at step 8, firstly, *A* requests all cash holders to calculate used seals of their credentials from record characterizer value G(P_{m}, t, 1)^{λ*} while revealing their identities. After that, *A* determines P_{m} that calculated *S*(R_{m}, G(P_{m}, t, 1))^{λ*} is liable, and forces P_{m} to pay cash of value {e_{*}(P_{m}, t) - e(P_{j}, s, v)} with penalty fine. When Sum(P_{m}, t) is an orphan record P_{m} pays e_{*}(P_{m}, t).

12. If no one calculates* **S*(R_{m}, G(P_{m}, t, 1))^{λ*} in the previous step, *A* picks encrypted expenditure record E(*D*(P_{m}. t, 1)) = <E(e(P_{m}, t, 1)), 1, *U*_{0}^{Rm(t)∙X*∙λ*}, G(P_{m}, t, 1)^{λ*}, *S*(R_{m}, G(P_{m}, t, 1))^{λ*}, *S*(R_{k},* **S*(R_{m}, G(P_{m}, t, 1)))^{λ*}> that constitutes Sum(P_{m}, t), and requests all cash holders to calculate used seals of their credentials from payer’s signature *S*(R_{m}, G(P_{m}, t, 1))^{λ*} while revealing their identities.

13. *A* forces P_{k} that calculates payee’s signature* **S*(R_{k},* **S*(R_{m}, G(P_{m}, t, 1)))^{λ}^{*} = *S*(R_{m}, G(P_{m}, t, 1))^{Rk}^{∙λ*} from* **S*(R_{m}, G(P_{m}, t, 1))^{λ*} to pay {e_{*}(P_{m}, t) - e(P_{j}, s, v)} with penalty fine, but in a case where Sum(P_{m}, t) is an orphan record, P_{k} pays e_{*}(P_{m}, t).

If cash holders honestly use their e-cash and handle their paying records, in other words, if cash values, expiration times, division-ID and source-ID parts values and payer’s signatures in individual paying records are honestly generated, reported and disclosed, apparently the above procedure successfully identifies dishonest entities while maintaining sensitive data as their secrets provided that authority *A* is honest (about payee’s signatures, cash holders must calculate them honestly as discussed before). When cash holder P_{m} generated new e-cash from its e-cash *C*(P_{j}, s, v) excessively by mistake, firstly at step 3, P_{m} knows that disclosed illegitimate total expenditure record Sum(P_{m}, t) corresponds to it, i.e. payer’s signature *S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*∙μ*} in Sum(P_{m}, t) is calculated from record characterizer G(P_{m}, t, 1)^{λ}^{*∙μ*} only by using its credential, and at step 4, P_{m} pays the difference between its actual expenditure and the original cash value e_{*}(P_{m}, t) - e(P_{j}, s, v) to *A*. Here, P_{m} is not requested to reveal its identity when it calculates* **S*(R_{m}, G(P_{m}, t, 1))^{λ}^{*∙μ*}, and still can conceal the correspondence between it and its e-cash regardless that its mistake is intentional or not.

The procedure also identifies dishonest entities even when paying records are dishonestly generated and/or handled. Namely, if payee P_{k} is honest, because P_{k} examines the division-ID part value and the payer’s signature in paying record* **P*(P_{m}, t, h) when payer P_{m} pays *C*(P_{m}, t, h), P_{m} must generate the payer’s signature honestly while using its credential. Therefore, *A* can determine that P_{m} is dishonest by examining payer’s signatures at step 11 even when *P*(P_{m}, t, h) was forged by P_{m}. On the other hand, in a case where P_{k} modified *P*(P_{m}, t, h) after having received it from P_{m}, P_{k} cannot include consistent payer’s signature of P_{m} in* **P*(P_{m}, t, h) because all paying records must have different source-ID part values despite P_{k} does not know integer R_{m}. Then P_{m} claims that* **P*(P_{m}, t, h) is incorrect at step 5, and because P_{k}* *must calculate its payee’s signature honestly when it reports *P*(P_{m}, t, h), *A* can determine that P_{k} is dishonest at steps 9 and 10, i.e. P_{k} had received an inconsistent paying record regardless that other entities are conspiring with it or not. In the same way, *A* can identify also other entities when they forge *P*(P_{m}, t, h), i.e. anyone other than P_{m} cannot forge paying record* **P*(P_{m}, t, h) so that it becomes consistent with P_{m}’s credential.

About a case where P_{m} does not report paying record *P*(P_{j}, s, v) to *A*, each *P*(P_{m}, t, h) becomes an orphan record, and if P_{k} is honest, P_{m} that calculates payer’s signature* **S*(R_{m}, G(P_{m}, t, 1))^{λ*} is determined as liable, i.e. only P_{m} can calculate *S*(R_{m}, G(P_{m}, t, 1))^{λ*} at step 11 as same as in the above. On the other hand when P_{k} is dishonest, P_{m} does not calculate *S*(R_{m}, G(P_{m}, t, 1))^{λ*} because P_{k} cannot generate P_{m}’s payer’s signature consistently, and P_{k} is identified as an entity that had accepted inconsistent payer’s signature at step 12. In the same way, P_{j} that had paid expired e-cash *C*(P_{j}, s, v) to P_{m} while modifying the expiration time is identified at step 8. Lastly in a case where authority *A* is dishonest, even *A* cannot forge payer’s or payee’s signatures of honest cash holders consistently. Then, it cannot identify any liable entity and must compensate the corresponding losses by itself.

About the anonymity of cash holders, cash holders in Step 5 do not reveal their identities. Although each cash holder calculates used seals of its credential from G(P_{m}, t, h)^{λ*} or *S*(R_{m}, G(P_{m}, t, h))^{λ*} while revealing its identity at Steps 7, 9, 11 and 12, it did not calculate G(P_{m}, t, h)^{λ*}^{∙Rj}, G(P_{m}, t, h)^{λ*}^{∙Rm}, G(P_{m}, t, h)^{λ*}^{∙Rk} or *S*(R_{m}, G(P_{m}, t, h))^{λ*}^{∙Rk} before if it is honest. Therefore no one including *A* can identify payments of honest cash holders.

As another type of dishonesty although protection of this dishonesty is out of the scope of the proposed scheme, cash holders may disappear during interactions between them, e.g. a payee can disappear after it receives e-cash despite a payer does not complete its purchase yet. But these threats also can be removed easily by the secure object exchange scheme ^{[11]}.

### 6. Conclusion

As discussed in previous sections the proposed scheme successfully satisfy anonymity, divisibility and transferability of e-cash in offline environments. Namely, honest cash holders can conceal correspondences between them and their e-cash and links between e-cash they had spent. Nevertheless, cash issuing authority *A* can identify liable entities when e-cash were illegitimately generated or used.

In addition, P_{m} can generate new e-cash *C*(P_{m}, t, 1), *C*(P_{m}, t, 2), ---,* C*(P_{m}, t, H(m, t)) of any values from its e-cash *C*(P_{j}, s, v) that it had received from other cash holder P_{j} provided that the total cash value of them does not exceed the cash value of *C*(P_{j}, s, v) as above, i.e. *C*(P_{j}, s, v) is divisible and transferable. But different from in other schemes, volume of information included in each e-cash does not increase even when it is transferred multiple times. Cash holders do not need to maintain receipts for transferring their e-cash either. In addition, an honest e-cash holder can conceal the correspondence between it and its e-cash even when the e-cash is a dishonestly transferred one. About divisibility, authority *A* does not need to define the minimum cash value unit in advance, and as a result, costs for handling e-cash can be decreased (the costs do not increase with the minimum cash value unit).

Drawbacks of the scheme are, firstly when cash holder P_{k} receives e-cash* **C*(P_{m}, t, h) from P_{m}, it must report paying record *P*(P_{m}, t, h) to authority *A* through online communication channels, and secondly to generate new e-cash from existing e-cash each P_{m} must obtain numbers of cash-IDs in advance. But P_{k} is not required to report *P*(P_{m}, t, h) immediately, i.e. it can report it together with other paying records at its convenient time. Therefore, inconvenience caused by the reporting can be mitigated. Also, although cash holder P_{m} obtains all cash-IDs in the registration phase in Sec.4, it can obtain convenient number of cash-IDs anytime, e.g. at a time when it reports paying records.

As other drawbacks, each cash holder is required to examine individual illegitimate records even if it is honest, and different from real cash, cash holders must use their e-cash before they expire. About these drawbacks, numbers of illegitimate records can be decreased by high penalty fines. Also, although the proposed scheme defines expiration time of e-cash *C*(P_{m}, t, h) as that of *C*(P_{j}, s, v) from which *C*(P_{m}, t, h) was generated, it is possible to make *C*(P_{m}, t, h) valid during a fixed duration after *C*(P_{m}, t, h) was generated.

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