12.24. DD24: Anonymous Age Restriction Extension for GNU Taler¶

12.24.1. Summary¶

This document presents and discusses an extension to GNU Taler that provides anonymous age-restriction.

12.24.2. 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.

12.24.3. Requirements¶

legal requirements for merchants must allow for this kind of mechanism

12.24.4. 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Once enabled, coins with age restrictions can be withdrawn by parents/warden who can choose to commit the coins to a certain maximum age out of a predefined list of age groups.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 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: $0 < a_1 < \ldots < a_M$ with $a_i \in \mathbb{N}$ . The zeroth age group is $\{0,\ldots,a_1-1\}$ .
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: $\bigl\langle (q_1, p_1), \ldots, (q_M, p_M) \bigr\rangle$ . Here, $q_i$ are the public keys (mnemonic: q-mitments), $p_i$ 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: $\bigl\langle (q_1, p_1), \ldots, (q_m, p_m), \, (q_{m+1}, \perp), \ldots, (q_M, \perp )\bigr\rangle$ . 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: $\vec{Q} := \langle q_1, \ldots, q_M \rangle$ . 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 $\text{FDH}_N(C_p)$ with the RSA private key of a denomination with support for age restriction, we sign $\text{FDH}_N(C_p, h_Q)$ . Here, $C_p$ is the EdDSA public key of a coin and $h_Q$ is the hash of the age commitment $\vec{Q}$ . 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 design document 006 ― Extensions.

The exchange indicates support for age-restriction in response to /keys by registering the extension age_restriction with a value type 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;
}

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

{0,1,2,3,4,5,6,7}
{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.

// 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:

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).
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.

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: Denom[];

//...
}

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:

-- 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 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

$\kappa$ be the same cut-and-choose parameter for the refresh-protocol.
$\Omega \in E$ be a published, nothing-up-my-sleeve, constant group-element on the elliptic curve.
$a \in \{1,\ldots,M\}$ be the maximum age (group) for which the wallet has to prove its commitment.

The values $\kappa$ , $\Omega$ and $a$ 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 $C_i$ for $i \in \{1,\ldots,\kappa\}$ as candidates for one coin.
2. creates age-commitments for as follows:
  1. creates $a$ -many Edx25519-keypairs $(p^i_j, q^i_j)$ randomly for $j \in \{1,\ldots,a\}$ (with public keys $q^i_j$ ),
  2. chooses randomly $(M - a)$ -many scalars $s^i_j$ for $j \in \{a+1,\ldots,M\}$ ,
  3. calculates $\omega^i_j = s^i_j*\Omega$ for $j \in \{a+1,\ldots,M \}$ ,
  4. sets $\vec{Q}^i := (q^i_1,\ldots,q^i_a,\omega^i_{a+1},\ldots,\omega^i_M)$
3. calculates $f_i := \text{FDH}(C_i, H(\vec{Q}^i))$ for $i \in \{ 1,\ldots,\kappa \}$ .
4. chooses random blindings $\beta_i(.)$ for $i \in \{1,\ldots,\kappa\}$ . The blinding functions depend on the cipher (RSA, CS).
5. sends $(\beta_1(f_1),\ldots,\beta_\kappa(f_\kappa))$ to the Exchange
Exchange
1. receives $(b_1,\ldots,b_\kappa)$
2. calculates $F := \text{H}(b_1||\ldots||b_\kappa)$
3. chooses randomly $\gamma \in \{1,\ldots,\kappa\}$ and
4. signs $r := b_\gamma$ resulting in signature $\sigma_r$
5. stores $F \mapsto (r, \sigma_r)$
6. sends $\gamma$ to the Wallet.
Wallet
1. receives $\gamma$
2. sends to the Exchange the tuple with
  - $F := \text{H}(\beta_1(f_1)||\ldots||\beta_\kappa(f_\kappa))$
  - $\vec{\beta} := (\beta_1,\ldots,\beta_{\gamma-1},\bot,\beta_{\gamma+1},\ldots,\beta_\kappa)$
  - $\vec{\vec{Q}} := (\vec{Q}^1,\ldots,\vec{Q}^{\gamma-1},\bot,\vec{Q}^{\gamma+1},\ldots,\vec{Q}^\kappa)$
  - $\vec{\vec{S}} := (\vec{S}^1,\ldots,\vec{S}^{\gamma-1},\bot,\vec{S}^{\gamma+1},\ldots,\vec{S}^\kappa)$ with $\vec{S}^i := (s^i_j)$
Exchange
1. receives $\left(F, (\beta_i), (\vec{Q}^i), (\vec{B}^i) \right)$
2. retrieves $(r, \sigma_r)$ from $F$ or bails out if not present
3. calculates $b_i := \beta_i\left(\text{FDH}(\vec{Q}^i)\right)$ for $i \neq \gamma$
4. compares $F \overset{?}{=} \text{H}(b_1||\ldots||b_{\gamma - 1}||r||b_{\gamma+1}||\ldots||b_\kappa)$ and bails out on inequality
5. for each
  1. calculates $\tilde{\omega}^i_j := b^i_j * \Omega$ for $j \in \{a+1,\ldots,M\}$
  2. compares each $\tilde{\omega}^i_j$ to $q^i_j$ from $\vec{Q}^i = (q^i_1, \ldots, q^i_M)$ and bails out on inequality
6. sends (blinded) signature $\sigma_r$ to Wallet
Wallet
1. receives $\sigma_r$
2. calculates (unblinded) signature $\sigma_\gamma := \beta^{-1}_\gamma(\sigma_r)$ for coin $C_\gamma$ .

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 $\gamma$ .

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

$s$ be the master secret of the coin, from which the private key $c_s$ , blinding $\beta$ and nonce $n$ are derived as usual in the wallet core
$m \in \{1,\ldots,M\}$ be the maximum age (according to the reserve) that a wallet can commit to during the withdrawal.
$P$ be a published constant Edx25519-public-key to which the private key is not known to any client.

For the age commitment, calculate:

For age group $a \in \{1,\ldots,m\}$ , set

$s_a &:= \text{HDKF}(s, \text{"age-commitment"}, a) \\ p_a &:= \text{Edx25519\_generate\_private}(s_a) \\ q_a &:= \text{Edx25519\_public\_from\_private}(p_a)$

For age group $a \in \{m,\ldots,M\}$ , set

$f_a &:= \text{HDKF}(s, \text{"age-factor"}, a) \\ q_a &:= \text{Edx25519\_derive\_public}(P, f_a).$

Then the vector $\vec{q} = \{q_1,\ldots,q_M\}$ is then the age commitment associated to the coin’s private key $c_s$ . For the non-disclosed coins, the wallet can use the vector $(p_1,\ldots,p_m,\bot,\ldots,\bot)$ of private keys for the attestation.

Provided with the secret $s$ , the exchange can therefore calculate the private key $c_s$ , the blinding $\beta$ , the nonce $n$ (if needed) and the age commitment $\vec{q}$ , along with the coin’s public key $C_p$ and use the value of

$\text{TALER\_CoinPubHashP}(C_p, \text{age\_commitment\_hash}(\vec{q}))$

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:

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:

interface MeltRequest {
   ...

   // SHA256 hash of the age commitment of the coin, IFF the denomination
   // has age restriction support.  MUST be omitted otherwise.
   age_commitment_hash?: 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):

$\text{FDH}_N(C_p)\; \stackrel{?}{=}\; \left(\sigma_C\right)^{e} \;\;\text{mod}\,N$

Here, $C_p$ is the EdDSA public key of a coin, $\sigma_C$ is its signature and $\langle e, N \rangle$ 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 $h_a$ ) into account when verifying the coin’s signature:

$\text{FDH}_N(C_p, h_a)\; \stackrel{?}{=}\; \left(\sigma_C\right)^{e} \;\;\text{mod}N$

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.

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 is extended to perform the following additional computation and checks:

Using the κ-1 transfer secrets $\tau_i$ 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 $\tau_i$ as the seed, and then calculates the corresponding κ-1 hash values $h_i$ of those age commitments.

It then calculates the κ-1 blinded hashes $m_i = r^{e_i}\text{FDH}_N(C^{(i)}_p, h_i)$ (using the notation from Florian’s thesis) of the disclosed coins and together with the $m_\gamma$ of the undisclosed coin, calculates the hash $h'_m = H(m_1,\cdots,m_\gamma,\cdots,m_\kappa)$ 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.

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

$\text{FDH}_N(C_p, h_a)\; \stackrel{?}{=}\; \left(\sigma_C\right)^{e} \;\;\text{mod}N$

Also again, $C_p$ is the EdDSA public key of a coin, $\sigma_C$ is its signature, $\langle e, N \rangle$ is the RSA public key of the denomination and $h_a$ 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.

interface ContractTerms {
...

// 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:

export interface 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

$\text{FDH}_N(C_p, h_a)\; \stackrel{?}{=}\; \left(\sigma_C\right)^{e} \;\;\text{mod}N$

Again, $C_p$ is the EdDSA public key of a coin, $\sigma_C$ is its signature, $\langle e, N \rangle$ is the RSA public key of the denomination and $h_a$ 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.

12.24.5. Alternatives¶

ID-based systems
credit/debit card based systems

12.24.6. Drawbacks¶

age groups, once defined, are set permanently

Also discuss:

storage overhead
computational overhead
bandwidth overhead
legal issues?

12.24.7. Discussion / Q&A¶

We had some very engaged discussions on the GNU Taler mailing list:

12.24.8. 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.

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
}

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