taler-docs

024-age-restriction.rst (31252B)

---

 DD 24: Anonymous Age Restriction Extension
 ##########################################
 
 Summary
 =======
 
 This document presents and discusses an extension to GNU Taler that provides
 anonymous age-restriction.
 
 Motivation
 ==========
 
 Merchants are legally obliged to perform age verification of customers when
 they buy certain goods and services.  Current mechanisms for age verification
 are either ID-based or require the usage of credit/debit cards.  In all cases
 sensitive private information is disclosed.
 
 We want to offer a better mechanism for age-restriction with GNU Taler that
 
 * ensures anonymity and unlinkability of purchases
 * can be set to particular age groups by parents/wardens at withdrawal
 * is bound to particular coins/tokens
 * can be verified by the merchant at purchase time
 * persists even after refresh
 
 The mechanism is presented as an 'extension' to GNU Taler, that is, as an
 optional feature that can be switched on by the exchange operator.
 
 Requirements
 ============
 
 * legal requirements for merchants must allow for this kind of mechanism
 
 
 Proposed Solution
 =================
 
 We propose an extension to GNU Taler for age-restriction that can be enabled by
 an Exchange¹).
 
 Once enabled, coins with age restrictions can be withdrawn by parents/warden
 who can choose to **commit** the coins to a certain maximum age out of a
 predefined list of age groups.
 
 The minors/wards receive those coins and can now **attest** a required minimum
 age (provided that age is less or equal to the committed age of the coins) to
 merchants, who can **verify** the minimum age.
 
 For the rest values (change) after an transaction, the minor/ward can
 **derive** new age-restricted coins.  The exchange can **compare** the equality
 of the age-restriction of the old coin with the new coin (in a zero-knowledge
 protocol, that gives the minor/ward a 1/κ chance to raise the minimum age for
 the new coin).
 
 The proposed solution maintains the guarantees of GNU Taler with respect to
 anonymity and unlinkability.  We have published a paper
 `Zero Knowledge Age Restriction for GNU Taler <https://link.springer.com/chapter/10.1007/978-3-031-17140-6_6>`_
 with the details.
 
 ¹) Once the feature is enabled and the age groups are defined, the exchange has
 to stick to that decision until the support for age restriction is disabled.
 We might reconsider this design decision at some point.
 
 
 Main ideas and building blocks
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 
 The main ideas are as follows:
 
 #. The exchange defines and publishes M+1 different *age groups* of increasing
    order: :math:`0 < a_1 < \ldots < a_M` with :math:`a_i \in \mathbb{N}`.  The
    zeroth age group is :math:`\{0,\ldots,a_1-1\}`.
 
 #. An **unrestricted age commitment** is defined as a vector of length M of
    pairs of Edx25519_ public and private keys on Curve25519. In other words: one
    key pair for each age group after the zeroth: :math:`\bigl\langle (q_1,
    p_1), \ldots, (q_M, p_M) \bigr\rangle`.  Here, :math:`q_i` are the public keys
    (mnemonic: **q-mitments**), :math:`p_i` are the private keys.
 
 #. A **restricted age commitment** *to age group m* is derived from an
    unrestricted age commitment by removing all private keys for
    indices larger than m: :math:`\bigl\langle (q_1, p_1), \ldots, (q_m, p_m),
    \, (q_{m+1}, \perp), \ldots, (q_M, \perp )\bigr\rangle`.  F.e. if *none* of
    the private keys is provided, the age commitment would be restricted to the
    zeroth age group.
 
 #. The act of restricting an unrestricted age commitment is performed by the
    parent/ward.
 
 #. An *age commitment* (without prefix) is just the vector of public keys:
    :math:`\vec{Q} := \langle q_1, \ldots, q_M \rangle`.  Note that from
    just the age commitment one can not deduce if it originated from an
    unrestricted or restricted one (and what age).
 
 #. An *attestation of age group k* is essentially the signature to any message
    with the private key for slot k, if the corresponding private key is
    available in a restricted age commitment.  (Unrestricted age commitments can
    attest for any age group).
 
 #. An age commitment is *bound to a particular coin* by incorporating the
    SHA256 hash value of the age commitment (i.e. the M public keys) into the
    signature of the coin.  So instead of signing :math:`\text{FDH}_N(C_p)` with
    the RSA private key of a denomination with support for age restriction, we
    sign :math:`\text{FDH}_N(C_p, h_Q)`.  Here, :math:`C_p` is the EdDSA public
    key of a coin and :math:`h_Q` is the hash of the age commitment :math:`\vec{Q}`.
    **Note:** A coin with age restriction can only be validated when both, the
    public key of the coin itself **and** the hash of the age commitment, are
    present.  This needs to be supported in each subsystem: Exchange, Wallet and
    Merchant.
 
 
 TODO: Summarize the design based on the five functions ``Commit()``,
 ``Attest()``, ``Verify()``, ``Derive()``, ``Compare()``, once the paper from
 Özgür and Christian is published.
 
 Changes in the Exchange API
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^
 
 The necessary changes in the exchange involve
 
 * indication of support for age restriction as an extension
 * modification of the refresh protocol (both, commit and reveal phase)
 * modification of the deposit protocol
 
 
 Extension for age restriction
 -----------------------------
 
 .. note::
 
   Registering an extension is defined in
   :doc:`design document 006 ― Extensions <006-extensions>`.
 
 
 The exchange indicates support for age-restriction in response to ``/keys`` by
 registering the extension ``age_restriction`` with a value type
 ``ExtensionAgeRestriction``:
 
 .. ts:def:: ExtensionAgeRestriction
 
    interface ExtensionAgeRestriction {
       // The field ``critical`` is mandatory for an extension.
       // Age restriction is not required to be understood by an client, so
       // ``critical`` will be set to ``false``.
       critical: false;
 
       // The field ``version`` is mandatory for an extension.  It is of type
       // `LibtoolVersion`.
       version: "1";
 
       // Age restriction specific configuration
       config: ConfigAgeRestriction;
    }
 
 .. ts:def:: ConfigAgeRestriction
 
    interface ConfigAgeRestriction {
       // The age groups.  This field is mandatory and binding in the sense
       // that its value is taken into consideration when signing the
       // age restricted denominations in the `ExchangeKeysResponse`
       age_groups: AgeGroups;
    }
 
 Age Groups
 ~~~~~~~~~~
 
 Age groups are represented as a finite list of positive, increasing integers
 that mark the beginning of the *next* age group.  The value 0 is omitted but
 implicitly marks the beginning of the *zeroth* age group and the first number
 in the list marks the beginning of the *first* age group.  Age groups are
 encoded as a colon separated string of integer values.  They are referred to by
 their *slot*, i.e.  "age group 3" is the age group that starts with the 3.
 integer in the list.
 
 For example: the string "8:10:12:14:16:18:21" represents the age groups
 
 0. {0,1,2,3,4,5,6,7}
 #. {8,9}
 #. {10,11}
 #. {12,13}
 #. {14,15}
 #. {16,17}
 #. {18,19,20}
 #. {21, ⋯ }
 
 The field ``age_groups`` of type `AgeGroups` is mandatory and binding in the
 sense that its value is taken into consideration when signing the denominations
 in ``ExchangeKeysResponse.age_restricted_denoms``.
 
 .. ts:def:: AgeGroups
 
    // Representation of the age groups as colon separated edges: Increasing
    // from left to right, the values mark the beginning of an age group up
    // to, but not including the next value.  The initial age group starts at
    // 0 and is not listed.  Example: "8:10:12:14:16:18:21".
    type AgeGroups = string;
 
 
 Age restricted denominations
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 If age-restriction is registered as extension ``age_restriction``, as described
 above, the root-object ``ExchangeKeysResponse`` in response to ``/keys`` MUST
 be extended by an additional field ``age_restricted_denoms``.  This is an
 *additional* list of denominations that must be used during the modified
 ``refresh`` and ``deposit`` operations (see below).
 
 The data structure for those denominations is the same as for the regular ones
 in ``ExchangeKeysResponse.denoms``.  **However**, the following differences
 apply for each denomination in the list:
 
 1. The value of ``TALER_DenominationKeyValidityPS.denom_hash``
    is taken over the public key of the denomination **and** the string in
    ``ExtensionAgeRestriction.age_groups`` from the corresponding extension
    object (see above).
 
 2. The value of ``TALER_DenominationKeyValidityPS.purpose`` is set to
    ``TALER_SIGNATURE_MASTER_AGE_RESTRICTED_DENOMINATION_KEY_VALIDITY``.
 
 And similar to ``.denoms``, if the query parameter ``last_issue_date`` was
 provided by the client, the exchange will only return the keys that have
 changed since the given timestamp.
 
 
 .. ts:def:: ExchangeKeysResponse
 
    interface ExchangeKeysResponse {
    //...
 
    // List of denominations that support age-restriction with the age groups
    // given in age_groups.  This is only set **iff** the extension
    // ``age_restriction`` is registered under ``entensions`` with type
    // ``ExtensionAgeRestriction``.
    //
    // The data structure for each denomination is the same as for the
    // denominations in ExchangeKeysResponse.denoms.  **However**, the
    // following differences apply for each denomination in the list:
    //
    //  1. The value of ``TALER_DenominationKeyValidityPS.denom_hash``
    //     is taken over the public key of the denomination __and__ the
    //     string in ``ExtensionAgeRestriction.age_groups`` from the
    //     corresponding extension object.
    //
    //  2. The value of ``TALER_DenominationKeyValidityPS.purpose`` is set to
    //     ``TALER_SIGNATURE_MASTER_AGE_RESTRICTED_DENOMINATION_KEY_VALIDITY``
    //
    // Similar as for ``.denoms``, if the query parameter ``last_issue_date``
    // was provided by the client, the exchange will only return the keys that
    // have changed since the given timestamp.
    age_restricted_denoms: DenomCommon[];
 
    //...
    }
 
 
 SQL schema
 -----------
 
 The exchange has to mark denominations with support for age restriction as such
 in the database.  Also, during the melting phase of the refresh operation, the
 exchange will have to persist the SHA256 hash of the age commitment of the
 original coin.
 
 The schema for the exchange is changed as follows:
 
 .. sourcecode:: sql
 
    -- Everything in one big transaction
    BEGIN;
    -- Check patch versioning is in place.
    SELECT _v.register_patch('exchange-TBD', NULL, NULL);
 
    -- Support for age restriction is marked per denomination.
    ALTER TABLE denominations
      ADD COLUMN age_restricted BOOLEAN NOT NULL DEFAULT (false);
    COMMENT ON COLUMN denominations.age_restriced
      IS 'true if this denomination can be used for age restriction';
 
    -- During the melting phase of the refresh, the wallet has to present the
    -- hash value of the age commitment (only for denominations with support
    -- for age restriction).
    ALTER TABLE refresh_commitments
      ADD COLUMN age_commitment_h BYTEA CHECK (LENGTH(age_commitment_h)=64);
    COMMENT ON COLUMN refresh_commitments.age_commitment_h
      IS 'SHA256 hash of the age commitment of the old coin, iff the corresponding
          denomimination has support for age restriction, NULL otherwise.';
    COMMIT;
 
 Note the constraint on ``refresh_commitments.age_commitment_h``:  It can be
 NULL, but only iff the corresponding denomination (indirectly referenced via
 table ``known_coins``) has ``.age_restricted`` set to true.  This constraint
 can not be expressed reliably with SQL.
 
 
 Protocol changes
 ----------------
 
 Withdraw
 ~~~~~~~~
 
 The withdraw protocol is affected in the following situations:
 
 - A wire transfer to the exchange (to fill a reserve) was marked by the
   originating bank as coming from a bank account of a minor, belonging to a of
   a specific age group, or by other means.
 - A KYC-process has been performed with the owner of a reserve and the user has
   been identified as being a minor.
 - A Peer-to-Peer transaction was performed between customers.  The receiving
   customer's KYC result tells the exchange that the customer belongs to a
   specific age group.
 
 In these cases, the wallet will have to perform a zero-knowledge protocol with
 exchange as part of the the withdraw protocol, which we sketch here.  Let
 
 - :math:`\kappa` be the same cut-and-choose parameter for the refresh-protocol.
 - :math:`\Omega \in E` be a published, nothing-up-my-sleeve, constant
   group-element on the elliptic curve.
 - :math:`a \in \{1,\ldots,M\}` be the maximum age (group) for which the wallet
   has to prove its commitment.
 
 The values :math:`\kappa`, :math:`\Omega` and :math:`a` are known to the
 Exchange and the Wallet.  Then, Wallet and Exchange run the following protocol
 for the withdrawal of one coin:
 
 - *Wallet*
    1. creates planchets :math:`C_i` for :math:`i \in \{1,\ldots,\kappa\}` as candidates for *one* coin.
    #. creates age-commitments :math:`\vec{Q}^i` for :math:`i \in \{1,\ldots,\kappa\}` as follows:
 
       a) creates :math:`a`-many Edx25519-keypairs :math:`(p^i_j, q^i_j)`
          randomly for :math:`j \in \{1,\ldots,a\}` (with public keys :math:`q^i_j`),
       #) chooses randomly :math:`(M - a)`-many scalars :math:`s^i_j` for :math:`j \in \{a+1,\ldots,M\}`,
       #) calculates :math:`\omega^i_j = s^i_j*\Omega` for :math:`j \in \{a+1,\ldots,M \}`,
       #) sets :math:`\vec{Q}^i := (q^i_1,\ldots,q^i_a,\omega^i_{a+1},\ldots,\omega^i_M)`
 
    #. calculates :math:`f_i := \text{FDH}(C_i, H(\vec{Q}^i))` for :math:`i \in \{ 1,\ldots,\kappa \}`.
    #. chooses random blindings :math:`\beta_i(.)` for :math:`i \in \{1,\ldots,\kappa\}`.  The blinding functions depend on the cipher (RSA, CS).
    #. sends :math:`(\beta_1(f_1),\ldots,\beta_\kappa(f_\kappa))` to the Exchange
 
 - *Exchange*
    7. receives :math:`(b_1,\ldots,b_\kappa)`
    #. calculates :math:`F := \text{H}(b_1||\ldots||b_\kappa)`
    #. chooses randomly :math:`\gamma \in \{1,\ldots,\kappa\}` and
    #. signs :math:`r := b_\gamma` resulting in signature :math:`\sigma_r`
    #. stores :math:`F \mapsto (r, \sigma_r)`
    #. sends :math:`\gamma` to the Wallet.
 
 - *Wallet*
    10. receives :math:`\gamma`
    #. sends to the Exchange the tuple :math:`\left(F, \vec{\beta}, \vec{\vec{Q}}, \vec{\vec{S}}\right)` with
 
       - :math:`F := \text{H}(\beta_1(f_1)||\ldots||\beta_\kappa(f_\kappa))`
       - :math:`\vec{\beta} := (\beta_1,\ldots,\beta_{\gamma-1},\bot,\beta_{\gamma+1},\ldots,\beta_\kappa)`
       - :math:`\vec{\vec{Q}} := (\vec{Q}^1,\ldots,\vec{Q}^{\gamma-1},\bot,\vec{Q}^{\gamma+1},\ldots,\vec{Q}^\kappa)`
       - :math:`\vec{\vec{S}} := (\vec{S}^1,\ldots,\vec{S}^{\gamma-1},\bot,\vec{S}^{\gamma+1},\ldots,\vec{S}^\kappa)`
         with :math:`\vec{S}^i := (s^i_j)`
 
 - *Exchange*
    12. receives :math:`\left(F, (\beta_i), (\vec{Q}^i), (\vec{B}^i) \right)`
    #. retrieves :math:`(r, \sigma_r)` from :math:`F` or bails out if not present
    #. calculates :math:`b_i := \beta_i\left(\text{FDH}(\vec{Q}^i)\right)` for :math:`i \neq \gamma`
    #. compares :math:`F \overset{?}{=} \text{H}(b_1||\ldots||b_{\gamma - 1}||r||b_{\gamma+1}||\ldots||b_\kappa)` and bails out on inequality
    #. for each :math:`\vec{B}^i, i \neq \gamma`
 
       i. calculates :math:`\tilde{\omega}^i_j := b^i_j * \Omega` for :math:`j \in \{a+1,\ldots,M\}`
       #. compares each :math:`\tilde{\omega}^i_j` to :math:`q^i_j` from :math:`\vec{Q}^i = (q^i_1, \ldots, q^i_M)` and bails out on inequality
    #. sends (blinded) signature :math:`\sigma_r` to Wallet
 
 - *Wallet*
    18. receives :math:`\sigma_r`
    #. calculates (unblinded) signature :math:`\sigma_\gamma := \beta^{-1}_\gamma(\sigma_r)` for coin :math:`C_\gamma`.
 
 
 Note that the batch version of withdraw allows the withdrawal of *multiple*
 coins at once.  For that scenario the protocol sketched above is adapted to
 accomodate for handling multiple coins at once -- thus multiplying the amount
 of data by the amount of coins in question--, but all with the same value of
 :math:`\gamma`.
 
 The *actual* implementation of the protocol above will have major optimizations
 to keep the bandwidth usage to a minimum and also ensure that a denomination in
 the commitment doesn't expire before the reveal.
 
 Instead of generating and sending the age commitment (array of public keys) and
 blindings for each coin, the wallet *MUST* derive the corresponding blindings
 and the age commitments from the coin's private key itself as follows:
 
 Let
 
 - :math:`s` be the master secret of the coin, from which the private key :math:`c_s`, blinding :math:`\beta` and nonce :math:`n` are derived as usual in the wallet core
 - :math:`m \in \{1,\ldots,M\}` be the maximum age (according to the reserve)
   that a wallet can commit to during the withdrawal.
 - :math:`P` be a published constant Edx25519-public-key to which the private
   key is not known to any client.
 
 For the age commitment, calculate:
 
 1. For age group :math:`a \in \{1,\ldots,m\}`, set
 
 .. math::
            s_a &:= \text{HDKF}(s, \text{"age-commitment"}, a) \\
            p_a &:= \text{Edx25519\_generate\_private}(s_a) \\
            q_a &:= \text{Edx25519\_public\_from\_private}(p_a)
 
 2. For age group :math:`a \in \{m,\ldots,M\}`, set
 
 .. math::
            f_a &:= \text{HDKF}(s, \text{"age-factor"}, a) \\
            q_a &:= \text{Edx25519\_derive\_public}(P, f_a).
 
 Then the vector :math:`\vec{q} = \{q_1,\ldots,q_M\}` is then the age commitment
 associated to the coin's private key :math:`c_s`.  For the non-disclosed coins,
 the wallet can use the vector :math:`(p_1,\ldots,p_m,\bot,\ldots,\bot)` of
 private keys for the attestation.
 
 Provided with the secret :math:`s`, the exchange can therefore calculate the
 private key :math:`c_s`, the blinding :math:`\beta`, the nonce :math:`n` (if
 needed) and the age commitment :math:`\vec{q}`, along with the coin's public
 key :math:`C_p` and use the value of
 
 .. math::
 
   \text{TALER\_CoinPubHashP}(C_p, \text{age\_commitment\_hash}(\vec{q}))
 
 during the verification of the original age-withdraw-commitment.
 
 For the withdrawal with age restriction, a sketch of the corresponding database
 schema in the exchange is given here:
 
 .. graphviz::
 
 	digraph deposit_policies {
 		rankdir = LR;
 		splines = true;
 		fontname="monospace"
 		node [
 			fontname="monospace"
 			shape=record
 		]
 
 		subgraph cluster_commitments {
 			label=<<B>age_withdraw</B>>
 			margin=20
 			commitments [
 			  label="age_withdraw_id\l|<hc>h_commitment\l|amount_with_fee_val\l|amount_with_fee_frac\l|noreveal_index\l|max_age\l|<res>reserve_pub\l|reserve_sig\l|<denom>[n] denominations_serials\l|[n] h_blind_evs\l|[n] denom_sigs\l"
 			]
 		}
 
 		commitments:res->reserves:id [ label="n:1"; fontname="monospace"];
 		commitments:denom -> denominations:id [ label="n:1"; fontname="monospace"] ;
 	}
 
 
 Refresh - melting phase
 ~~~~~~~~~~~~~~~~~~~~~~~
 
 During the melting phase of the refresh, the wallet has to present the hash
 value of the age commitment (for denominations with support for age
 restriction).  Therefore, in the ``/coins/$COIN_PUB/melt`` POST request, the
 ``MeltRequest`` object is extended with an optional field
 ``age_commitment_hash``:
 
 .. ts:def:: AgeMeltRequest
 
    interface AgeMeltRequest {
       ...
 
       // SHA256 hash of the age commitment of the coin, IFF the denomination
       // has age restriction support.  MUST be omitted otherwise.
       age_commitment_hash?: AgeCommitmentHash;
 
       ...
    }
 
 .. ts:def:: AgeCommitmentHash
 
    type AgeCommitmentHash = SHA256HashCode;
 
 The responses to the POST request remain the same.
 
 For normal denominations *without* support for age restriction, the calculation
 for the signature check is as before (borrowing notation from
 `Florian's thesis <https://taler.net/papers/thesis-dold-phd-2019.pdf>`_):
 
 .. math::
    \text{FDH}_N(C_p)\; \stackrel{?}{=}\; \left(\sigma_C\right)^{e} \;\;\text{mod}\,N
 
 Here, :math:`C_p` is the EdDSA public key of a coin, :math:`\sigma_C` is its
 signature and :math:`\langle e, N \rangle` is the RSA public key of the
 denomination.
 
 For denominations *with* support for age restriction, the exchange takes the
 hash value ``age_commitment_hash`` (abbreviated as :math:`h_a`) into account
 when verifying the coin's signature:
 
 .. math::
    \text{FDH}_N(C_p, h_a)\; \stackrel{?}{=}\; \left(\sigma_C\right)^{e} \;\;\text{mod}N
 
 
 
 
 Refresh - reveal phase
 ~~~~~~~~~~~~~~~~~~~~~~
 
 During the reveal phase -- that is upon POST to ``/refreshes/$RCH/reveal`` --
 the client has to provide the original age commitment of the old coin (i.e. the
 vector of public keys), iff the corresponding denomination had support for age
 restriction.  The size of the vector is defined by the Exchange implicetly as
 the amount of age groups defined in the field ``.age_groups`` of the
 ``ExtensionAgeRestriction``.
 
 .. ts:def:: RevealRequest
 
    interface RevealRequest {
       ...
 
       // Iff the corresponding denomination has support for age restriction,
       // the client MUST provide the original age commitment, i.e. the vector
       // of public keys.
       // The size of the vector is defined by the Exchange implicetly as the
       // amount of age groups defined in the field ``.age_groups`` of the
       // ``ExtensionAgeRestriction``.
       old_age_commitment?: Edx25519PublicKey[];
 
 
       ...
    }
 
 
 The exchange can now check if the provided public keys ``.old_age_commitment``
 have the same SHA256 hash value when hashed in sequence as the
 ``age_commitment_hash`` of the original coin from the call to melt.
 
 The existing `cut&choose protocol during the reveal phase
 </core/api-exchange.html#post--refreshes-$RCH-reveal>`__ is extended to perform
 the following additional computation and checks:
 
 Using the κ-1 transfer secrets :math:`\tau_i` from the reveal request, the
 exchange derives κ-1 age commitments from the ``old_age_commitment`` by calling
 ``Edx25519_derive_public()`` on each `Edx25519PublicKey`, with :math:`\tau_i`
 as the seed, and then calculates the corresponding κ-1 hash values :math:`h_i`
 of those age commitments.
 
 It then calculates the κ-1 blinded hashes
 :math:`m_i = r^{e_i}\text{FDH}_N(C^{(i)}_p, h_i)` (using the notation from Florian's
 thesis) of the disclosed coins and together with the :math:`m_\gamma` of the
 undisclosed coin, calculates the hash
 :math:`h'_m = H(m_1,\cdots,m_\gamma,\cdots,m_\kappa)` which is then used in the
 final verification step of the cut&choose protocol.
 
 
 Deposit
 ~~~~~~~
 
 As always, the merchant has to provide the public key of a coin during a POST
 to ``/coins/$COIN_PUB/deposit``.  However, for coins with age restriction, the
 signature check requires the hash of the age commitment.  Therefore the request
 object ``DepositRequest`` is extended by an optional field
 ``age_commitment_hash`` which MUST be set (with the SHA256 hash of the age
 commitment), iff the corresponding denomination had support for age restriction
 enabled.  The merchant has received this value prior from the customer during
 purchase.
 
 .. ts:def:: DepositRequest
 
    interface DepositRequest {
    ...
 
    // Iff the corresponding denomination had support for age restriction
    // enabled, this field MUST contain the SHA256 value of the age commitment that
    // was provided during the purchase.
    age_commitment_hash?: AgeCommitmentHash;
 
    ...
    }
 
 Again, the exchange can now check the validity of the coin with age restriction
 by evaluating
 
 .. math::
    \text{FDH}_N(C_p, h_a)\; \stackrel{?}{=}\; \left(\sigma_C\right)^{e} \;\;\text{mod}N
 
 Also again, :math:`C_p` is the EdDSA public key of a coin, :math:`\sigma_C` is
 its signature, :math:`\langle e, N \rangle` is the RSA public key of the
 denomination and :math:`h_a` is the value from ``age_commitment_hash``.
 
 
 
 Changes in the Merchant API
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^
 
 
 Claiming the order
 ------------------
 
 If an order requires a minimum age, the merchant MUST express that required
 minimum age in response to order claim by the wallet, that is, a POST to
 ``[/instances/$INSTANCE]/orders/$ORDER_ID/claim``.
 
 The object ``ContractTerms`` is extended by an optional field
 ``minimum_age`` that can be any integer greater than 0.  In reality
 this value will not be smaller than, say, 8, and not larger than, say, 21.
 
 .. ts:def:: DD24ContractTerms
 
    interface DD24ContractTerms {
    ...
 
    // If the order requires a minimum age greater than 0, this field is set
    // to the integer value of that age.  In reality this value will not be
    // smaller than, say, 8, and not larger than, say, 21.
    minimum_age?: Integer;
 
    ...
    }
 
 By sending the contract term with the field ``minimum_age`` set to an
 non-zero integer value, the merchant implicetly signals that it understands the
 extension ``age_restriction`` for age restriction from the exchange.
 
 
 Making the payment
 ------------------
 
 If the ``ContractTerms`` had a non-zero value in field
 ``minimum_age``, the wallet has to provide evidence of that minimum
 age by
 
 #. *either* using coins which are of denominations that had *no* age support
    enabled,
 
 #. *or* using coins which are of denominations that have support for age
    restriction enabled
 
    * and then  ―for each such coin― it has the right private key of the
      restricted age commitment to the age group into which the required minimum
      age falls (i.e.  a non-empty entry at the right index in vector of Edx25519
      keys, see above).
 
    * and signs the required minimum age with each coin's private key
      corresponding to the age group,
 
    * and sends ―for each coin― the complete age commitment and the signature to
      the merchant.
 
 The object ``CoinPaySig`` used within a ``PayRequest`` during a POST to
 ``[/instances/$INSTANCE]/orders/$ORDER_ID/pay`` is extended as follows:
 
 .. ts:def:: AgeCoinPaySig
 
    export interface AgeCoinPaySig extends CoinPaySig {
    ...
 
    // If a minimum age was required by the order and the wallet had coins that
    // are at least committed to the corresponding age group, this is the
    // signature of the minimum age as a string, using the private key to the
    // corresponding age group.
    minimum_age_sig?: Edx25519Signature;
 
    // If a minimum age was required by the order, this is age commitment bound
    // to the coin, i.e. the complete vector of Edx25519_ public keys, one for each
    // age group (as defined by the exchange).
    age_commitment?: Edx25519PublicKey[];
 
    }
 
 
 The merchant can now verify
 
 #. the validity of each (age restricted) coin by evaluating
 
    .. math:: \text{FDH}_N(C_p, h_a)\; \stackrel{?}{=}\; \left(\sigma_C\right)^{e} \;\;\text{mod}N
 
    Again, :math:`C_p` is the EdDSA public key of a coin, :math:`\sigma_C` is
    its signature, :math:`\langle e, N \rangle` is the RSA public key of the
    denomination and :math:`h_a` is the SHA256 hash value of the vector in
    ``age_commitment``.
 
 #. the minimum age requirement by checking the signature in ``minimum_age_sig``
    against the public key ``age_commitment[k]`` of the corresponding age group,
    say, ``k``.  (The minimum age must fall into the age group at index ``k`` as
    defined by the exchange).
 
 **Note**: This applies only to coins for denominations that have support for
 age restriction.  Denominations *without* support for age restriction *always*
 satisfy any minimum age requirement.
 
 
 
 Changes in the Wallet
 ^^^^^^^^^^^^^^^^^^^^^
 
 A wallet implementation SHOULD support denominations with age restriction.  In
 that case it SHOULD allow to select an age group as upper bound during
 withdraw.
 
 
 Alternatives
 ============
 
 * ID-based systems
 * credit/debit card based systems
 
 
 Drawbacks
 =========
 
 * age groups, once defined, are set permanently
 
 Also discuss:
 
 * storage overhead
 * computational overhead
 * bandwidth overhead
 * legal issues?
 
 Discussion / Q&A
 ================
 
 We had some very engaged discussions on the GNU Taler `mailing list <taler@gnu.org>`__:
 
 * `Money with capabilities <https://lists.gnu.org/archive/html/taler/2021-08/msg00005.html>`_
 
 * `On age-restriction (was: online games in China) <https://lists.gnu.org/archive/html/taler/2021-09/msg00006.html>`__
 
 * `Age-restriction is about coins, not currencies <https://lists.gnu.org/archive/html/taler/2021-09/msg00021.html>`__
 
 * The published paper: `Zero Knowledge Age Restriction for GNU Taler <https://link.springer.com/chapter/10.1007/978-3-031-17140-6_6>`_
 
 
 .. _Edx25519:
 
 Edx25519
 ========
 
 Edx25519 is a variant of EdDSA on curve25519 which allows for repeated
 derivation of private and public keys, independently.  It is implemented in
 `GNUNET with commit ce38d1f6c9bd7857a1c3bc2094a0ee9752b86c32.
 <https://git.gnunet.org/gnunet.git/commit/?id=ce38d1f6c9bd7857a1c3bc2094a0ee9752b86c32>`__
 
 The private keys in Edx25519 initially correspond to the data after expansion
 and clamping in EdDSA.  However, this correspondence is lost after deriving
 further keys from existing ones.  The public keys and signature verification
 are compatible with EdDSA.
 
 The scheme is as follows:
 
 ::
 
     /* Private keys in Edx25519 are pairs (a, b) of 32 byte each.
      * Initially they correspond to the result of the expansion
      * and clamping in EdDSA.
      */
 
     Edx25519_generate_private(seed) {
       /* EdDSA expand and clamp */
       dh := SHA-512(seed)
       a := dh[0..31]
       b := dh[32..64]
       a[0]  &= 0b11111000
       a[31] &= 0b01111111
       a[31] |= 0b01000000
 
       return (a, b)
     }
 
     Edx25519_public_from_private(private) {
       /* Public keys are the same as in EdDSA */
       (a, _) := private
       return [a] * G
     }
 
     Edx25519_blinding_factor(P, seed) {
       /* This is a helper function used in the derivation of
        * private/public keys from existing ones.  */
       h1 := HKDF_32(P, seed)
 
       /* Ensure that h == h % L */
       h := h1 % L
 
       /* Optionally: Make sure that we don't create weak keys. */
       P' := [h] * P
       if !( (h!=1) && (h!=0) && (P'!=E) ) {
         return Edx25519_blinding_factor(P, seed+1)
       }
 
       return h
     }
 
     Edx25519_derive_private(private, seed) {
       /* This is based on the definition in
        * GNUNET_CRYPTO_eddsa_private_key_derive.  But it accepts
        * and returns a private pair (a, b) and allows for iteration.
        */
       (a, b) := private
       P := Edx25519_public_key_from_private(private)
       h := Edx25519_blinding_factor(P, seed)
 
       /* Carefully calculate the new value for a */
       a1 := a / 8;
       a2 := (h  * a1) % L
       a' := (a2 *  8) % L
 
       /* Update b as well, binding it to h.
          This is an additional step compared to GNS. */
       b' := SHA256(b ∥ h)
 
       return (a', b')
     }
 
     Edx25519_derive_public(P, seed) {
       h := Edx25519_blinding_factor(P, seed)
       return [h]*P
     }
 
     Edx25519_sign(private, message) {
       /* As in Ed25519, except for the origin of b */
       (d, b) := private
       P := Edx25519_public_from_private(private)
       r := SHA-512(b ∥ message)
       R := [r] * G
       s := r + SHA-512(R ∥ P ∥ message) * d % L
 
       return (R,s)
     }
 
     Edx25519_verify(P, message, signature) {
       /* Identical to Ed25519 */
       (R, s) := signature
       return [s] * G == R + [SHA-512(R ∥ P ∥ message)] * P
     }