RFC Refactor: Sphinx Packet Format (#173)

vac/raw/mix.md

@@ -120,6 +120,18 @@ contents as packets are forwarded hop-by-hop.
 Sphinx packets are fixed-size and indistinguishable from one another, providing
 unlinkability and metadata protection.
 
+- **Initialization Vector (IV)**
+  A fixed-length input used to initialize block ciphers to add randomness to the
+  encryption process. It ensures that encrypting the same plaintext with the same
+  key produces different ciphertexts. The IV is not secret but must be unique for
+  each encryption.
+
+- **Single-Use Reply Block (SURB)**
+A pre-computed Sphinx header that encodes a return path back to the sender.
+SURBs are generated by the sender and included in the Sphinx packet sent to the recipient.
+It enables the recipient to send anonymous replies,
+without learning the sender’s identity, the return path, or the forwarding delays.
+
 ## 3. Motivation and Background
 
 libp2p enables modular peer-to-peer applications, but it lacks built-in support
@@ -194,7 +206,7 @@ describe the core components of a mix network.
 
 The Mix Protocol relies on two core design elements to achieve sender
 unlinkability and metadata
-protection: a mixing strategy and a a cryptographically secure mix packet
+protection: a mixing strategy and a cryptographically secure mix packet
 format.
 
 ### 4.1 Mixing Strategy
@@ -297,7 +309,7 @@ layered encryption and per-hop delay provides resistance to traffic analysis and
 enables message-level
 unlinkability.
 
-Unlike typical custom libp2p protocols, the Mix protocol is stateless—it
+Unlike typical custom libp2p protocols, the Mix Protocol is stateless—it
 does not establish
 persistent streams, negotiate protocols, or maintain sessions. Each message is
 self-contained
@@ -458,8 +470,8 @@ multistream handshake for protocol negotiation.
 
 While libp2p supports multiplexing multiple streams over a single transport
 connection using
-stream muxers such as mplex and yamux, it does not natively support reusing a
-stream over multiple
+stream muxers such as mplex and yamux, it does not natively support reusing the
+same stream over multiple
 message transmissions. However, stream reuse may be desirable in the mixnet
 setting to reduce overhead
 and avoid hitting per protocol stream limits between peers.
@@ -512,9 +524,9 @@ adversarial routing bias.
 
 While no existing mechanism provides unbiased sampling by default,
 [Waku’s ambient discovery](https://rfc.vac.dev/waku/standards/core/33/discv5/)
-— an extension
+—an extension
 over [Discv5](https://github.com/ethereum/devp2p/blob/master/discv5/discv5.md)
-— demonstrates
+—demonstrates
 an approximate solution. It combines topic-based capability advertisement with
 periodic
 peer sampling. A similar strategy could potentially be adapted for the Mix
@@ -618,7 +630,7 @@ _unobservability_ where
 a passive adversary cannot determine whether a node is sending real messages or
 not.
 
-In the Mix Protocol, cover traffic is limited to _loop messages_ — dummy
+In the Mix Protocol, cover traffic is limited to _loop messages_—dummy
 Sphinx packets
 that follow a valid mix path and return to the originating node. These messages
 carry no application
@@ -627,7 +639,7 @@ routing behavior.
 
 Cover traffic MAY be generated by either mix nodes or senders. The strategy for
 generating
-such traffic — such as timing and frequency — is pluggable and not
+such traffic—such as timing and frequency—is pluggable and not
 specified
 in this document.
 
@@ -667,9 +679,9 @@ deter participation by compromised or transient peers.
 
 The Mix Protocol does not mandate any form of payment, token exchange, or
 accounting. More
-sophisticated economic models — such as stake-based participation,
+sophisticated economic models—such as stake-based participation,
 credentialed relay networks,
-or zero-knowledge proof-of-contribution systems — MAY be layered on top of
+or zero-knowledge proof-of-contribution systems—MAY be layered on top of
 the protocol or
 enforced via external coordination.
 
@@ -779,6 +791,11 @@ destination origin protocol instance.
 
 The node MUST NOT retain decrypted content after forwarding.
 
+The routing behavior described in this section relies on the use of
+Sphinx packets to preserve unlinkability and confidentiality across
+hops. The next section specifies their structure, cryptographic
+components, and construction.
+
 ## 8. Sphinx Packet Format
 
 The Mix Protocol uses the Sphinx packet format to enable unlinkable, multi-hop
@@ -789,8 +806,10 @@ encapsulated in a Sphinx packet constructed by the initiating mix node. The
 packet is encrypted in
 layers such that each hop in the mix path can decrypt exactly one layer and
 obtain the next-hop
-routing information and delay value, without learning the complete path or the
+routing information and forwarding delay, without learning the complete path or the
 message origin.
+Only the final hop learns the destination, which is encoded in the innermost
+routing layer.
 
 Sphinx packets are self-contained and indistinguishable on the wire, providing
 strong metadata
@@ -820,12 +839,10 @@ subsections.
 
 ### 8.1 Packet Structure Overview
 
-Each Sphinx packet consists of three fixed-length header fields — $α$,
-$β$, and $γ$ —
-followed by a fixed-length encrypted payload $δ$. Together, these components
-enable per-hop message
-processing with strong confidentiality and integrity guarantees in a stateless
-and unlinkable manner.
+Each Sphinx packet consists of three fixed-length header fields— $α$,
+$β$, and $γ$ —followed by a fixed-length encrypted payload $δ$.
+Together, these components enable per-hop message processing with strong
+confidentiality and integrity guarantees in a stateless and unlinkable manner.
 
 - **$α$ (Alpha)**: An ephemeral public value. Each mix node uses its private key
 and $α$ to
@@ -833,8 +850,9 @@ derive a shared session key for that hop. This session key is used to decrypt
 and process
 one layer of the packet.
 - **$β$ (Beta)**: The nested encrypted routing information. It encodes the next
-hop address, the forwarding delay,
-integrity check $γ$ for the next hop, and the $β$ for subsequent hops.
+hop address, the forwarding delay, integrity check $γ$ for the next hop, and
+the $β$ for subsequent hops. At the final hop, $β$ encodes the destination
+address and fixed-length zero padding to preserve uniform size.
 - **$γ$ (Gamma)**: A message authentication code computed over $β$ using the
 session key derived
 from $α$. It ensures header integrity at each hop.
@@ -843,12 +861,12 @@ fixed maximum length and
 encrypted in layers corresponding to each hop in the mix path.
 
 At each hop, the mix node derives the session key from $α$, verifies the header
-integrity
-using $γ$, decrypts one layer of $β$ to extract the next hop and delay, and
-decrypts one layer
-of $δ$. It then constructs a new packet with updated values of $α$, $β$, $γ$,
-and $δ$, and
-forwards it to the next hop.
+integrity using $γ$, decrypts one layer of $β$ to extract the next hop and
+delay, and decrypts one layer of $δ$. It then constructs a new packet with
+updated values of $α$, $β$, $γ$, and $δ$, and forwards it to the next hop. At
+the final hop, the mix node decrypts the innermost layer of $β$ and $δ$, which
+yields the destination address and the original application message
+respectively.
 
 All Sphinx packets are fixed in size and indistinguishable on the wire. This
 uniform format,
@@ -866,20 +884,29 @@ This section defines the cryptographic primitives used in Sphinx packet
 construction and processing.
 
 - **Security Parameter**: All cryptographic operations target a minimum of
-$\kappa = 128$ bits of
+$κ = 128$ bits of
 security, balancing performance with resistance to modern attacks.
 
 - **Elliptic Curve Group $\mathbb{G}$**:
   - **Curve**: Curve25519
+  - **Notation**: Let $g$ denote the canonical base point (generator) of $\mathbb{G}$.
   - **Purpose**: Used for deriving Diffie–Hellman-style shared key at each hop
 using $α$.
   - **Representation**: Small 32-byte group elements, efficient for both
 encryption and key exchange.
+  - **Scalar Field**: The curve is defined over the finite field
+$\mathbb{Z}_q$, where $q = 2^{252} + 27742317777372353535851937790883648493$.
+Ephemeral exponents used in Sphinx packet construction are selected uniformly
+at random from $\mathbb{Z}_q^*$, the multiplicative subgroup of $\mathbb{Z}_q$.
+
+- **Hash Function**:
+  - **Construction**: SHA-256
+  - **Notation**: The hash function is denoted by $H(\cdot)$ in subsequent sections.
 
 - **Key Derivation Function (KDF)**:
   - **Purpose**: To derive encryption keys, IVs, and MAC key from the shared
 session key at each hop.
-  - **Construction**: SHA-256 hash with output truncated to 128 bits.
+  - **Construction**: SHA-256 hash with output truncated to $128$ bits.
   - **Key Derivation**: The KDF key separation labels (_e.g.,_ `"aes_key"`,
 `"mac_key"`)
 are fixed strings and MUST be agreed upon across implementations.
@@ -889,9 +916,457 @@ are fixed strings and MUST be agreed upon across implementations.
   - **Keys and IVs**: Each derived from the session key for the hop using the KDF.
 
 - **Message Authentication Code (MAC)**:
-  - **Construction**: HMAC-SHA-256 with output truncated to 128 bits.
+  - **Construction**: HMAC-SHA-256 with output truncated to $128$ bits.
   - **Purpose**: To compute $γ$ for each hop.
   - **Key**: Derived using KDF from the session key for the hop.
 
 These primitives are used consistently throughout packet construction and
 decryption, as described in the following sections.
+
+### 8.3 Packet Component Sizes
+
+This section defines the size of each component in a Sphinx packet, deriving them
+from the security parameter and protocol parameters introduced earlier. All Sphinx
+packets MUST be fixed in length to ensure uniformity and indistinguishability on
+the wire. The serialized packet is structured as follows:
+
+```text
++--------+----------+--------+----------+
+|   α    |     β    |   γ    |    δ     |
+| 32 B   | variable | 16 B   | variable |
++--------+----------+--------+----------+
+```
+
+#### 8.3.1 Header Field Sizes
+
+The header consists of the fields $α$, $β$, and $γ$, totaling a fixed size per
+maximum path length:
+
+- **$α$ (Alpha)**: 32 bytes  
+  The size of $α$ is determined by the elliptic curve group representation used
+  (Curve25519), which encodes group elements as 32-byte values.
+
+- **$β$ (Beta)**: $((t + 1)r + 1)κ$ bytes  
+  The size of $β$ depends on:
+  - **Maximum path length ($r$)**: The recommended value of $r=5$ balances
+  bandwidth versus anonymity tradeoffs.
+  - **Combined address and delay width ($tκ$)**: The recommended $t=6$
+  accommodates standard libp2p relay multiaddress representations plus a
+  2-byte delay field. While the actual multiaddress and delay fields may be
+  shorter, they are padded to $tκ$ bytes to maintain fixed field size. The
+  structure and rationale for the $tκ$ block and its encoding are specified in
+  [Section 8.4](#84-address-and-delay-encoding).
+
+    Note: This expands on the original
+    [Sphinx packet format]((https://cypherpunks.ca/~iang/pubs/Sphinx_Oakland09.pdf)),
+    which embeds a fixed $κ$-byte mix node identifier per hop in $β$.
+    The Mix Protocol generalizes this to $tκ$ bytes to accommodate libp2p
+    multiaddresses and forwarding delays while preserving the cryptographic
+    properties of the original design.
+
+  - **Per-hop $γ$ size ($κ$)** (defined below): Accounts for the integrity tag
+  included with each hop’s routing information.
+
+  Using the recommended value of $r=5$ and $t=6$, the resulting $β$ size is
+  $576$ bytes. At the final hop, $β$ encodes the destination address in the
+  first $tκ-2$ bytes and the remaining bytes are zero-padded.
+
+- **$γ$ (Gamma)**: $16$ bytes  
+  The size of $γ$ equals the security parameter $κ$, providing a $κ$-bit integrity
+  tag at each hop.
+
+Thus, the total header length is:
+
+$`
+\begin{aligned}
+|Header| &= α + β + γ \\
+  &= 32 + ((t + 1)r + 1)κ + 16
+\end{aligned}
+`$
+
+Notation: $|x|$ denotes the size (in bytes) of field $x$.
+
+Using the recommended value of $r = 5$ and $t = 6$, the header size is:
+
+$`
+\begin{aligned}
+|Header| &= 32 + 576 + 16 \\
+  &= 624 \ bytes
+\end{aligned}
+`$
+
+#### 8.3.2 Payload Size
+
+This subsection defines the size of the encrypted payload $δ$ in a Sphinx packet.
+
+$δ$ contains the application message, padded to a fixed maximum length to ensure
+all packets are indistinguishable on the wire. The size of $δ$ is calculated as:
+
+$`
+\begin{aligned}
+|δ| &= TotalPacketSize - HeaderSize
+\end{aligned}
+`$
+
+The recommended total packet size is $4608$ bytes, chosen to:
+
+- Accommodate larger libp2p application messages, such as those commonly
+observed in Status chat using Waku (typically ~4KB payloads),
+- Allow inclusion of additional data such as SURBs without requiring fragmentation,
+- Maintain reasonable per-hop processing and bandwidth overhead.
+
+This recommended total packet size of \$4608\$ bytes yields:
+
+$`
+\begin{aligned}
+Payload &= 4608 - 624 \\
+  &= 3984\ bytes
+\end{aligned}
+`$
+
+Implementations MUST account for payload extensions, such as SURBs,
+when determining the maximum message size that can be encapsulated in a
+single Sphinx packet. Details on SURBs are defined in
+[Section X.X].
+
+The following subsection defines the padding and fragmentation requirements for
+ensuring this fixed-size constraint.
+
+#### 8.3.3 Padding and Fragmentation
+
+Implementations MUST ensure that all messages shorter than the maximum payload size
+are padded before Sphinx encapsulation to ensure that all packets are
+indistinguishable on the wire. Messages larger than the maximum payload size MUST
+be fragmented by the origin protocol or top-level application before being passed
+to the Mix Protocol. Reassembly is the responsibility of the consuming application,
+not the Mix Protocol.
+
+#### 8.3.4 Anonymity Set Considerations
+
+The fixed maximum packet size is a configurable parameter. Protocols or
+applications that choose to configure a different packet size (either larger or
+smaller than the default) MUST be aware that using unique or uncommon packet sizes
+can reduce their effective anonymity set to only other users of the same size.
+Implementers SHOULD align with widely used defaults to maximize anonymity set size.
+
+Similarly, parameters such as $r$ and $t$ are configurable. Changes to these
+parameters affect header size and therefore impact payload size if the total packet
+size remains fixed. However, if such changes alter the total packet size on the
+wire, the same anonymity set considerations apply.
+
+The following subsection defines how the next-hop or destination address and
+forwarding delay are encoded within $β$ to enable correct routing and mixing
+behavior.
+
+### 8.4 Address and Delay Encoding
+
+Each hop’s $β$ includes a fixed-size block containing the next-hop address and
+the forwarding delay, except for the final hop, which encodes the destination
+address and a delay-sized zero padding. This section defines the structure and
+encoding of that block.
+
+The combined address and delay block MUST be exactly $tκ$ bytes in length,
+as defined in [Section 8.3.1](#831-header-field-sizes), regardless of the
+actual address or delay values. The first $(tκ - 2)$ bytes MUST encode the
+address, and the final $2$ bytes MUST encode the forwarding delay.
+This fixed-length encoding ensures that packets remain indistinguishable on
+the wire and prevents correlation attacks based on routing metadata structure.
+
+Implementations MAY use any address and delay encoding format agreed upon
+by all participating mix nodes, as long as the combined length is exactly
+$tκ$ bytes. The encoding format MUST be interpreted consistently by all
+nodes within a deployment.
+
+For interoperability, a recommended default encoding format involves:
+
+- Encoding the next-hop or destination address as a libp2p multi-address:
+  - To keep the address block compact while allowing relay connectivity, each mix
+  node is limited to one IPv4 circuit relay multiaddress. This ensures that most
+  nodes can act as mix nodes, including those behind NATs or firewalls.
+  - In libp2p terms, this combines transport addresses with multiple peer
+  identities to form an address that describes a relay circuit:  
+  `
+  /ip4/<ipv4>/tcp/<port>/p2p/<relayPeerID>/p2p-circuit/p2p/<relayedPeerID>
+  `  
+  Variants may include directly reachable peers and transports such as
+  `/quic-v1`, depending on the mix node's supported stack.
+  - IPv6 support is deferred, as it adds $16$ bytes just for the IP field.
+  - Future revisions may extend this format to support IPv6 or DNS-based
+  multiaddresses.
+  
+  With these constraints, the recommended encoding layout is:
+  - IPv4 address (4 bytes)
+  - Protocol identifier _e.g.,_ TCP or QUIC (1 byte)
+  - Port number (2 bytes)
+  - Peer IDs (39 bytes, post-Base58 decoding)
+
+- Encoding the forwarding delay as an unsigned 16-bit integer (2 bytes) in
+  milliseconds, using big endian network byte order.
+
+If the encoded address or delay is shorter than its respective allocated
+field, it MUST be padded with zeros. If it exceeds the allocated size, it
+MUST be rejected or truncated according to the implementation policy.
+
+Note: Future versions of the Mix Protocol may support address compression by
+encoding only the peer identifier and relying on external peer discovery
+mechanisms to retrieve full multiaddresses at runtime. This would allow for
+more compact headers and greater address flexibility, but requires fast and
+reliable lookup support across deployments. This design is out of scope for
+the current version.
+
+With the field sizes and encoding conventions established, the next section describes
+how a mix node constructs a complete Sphinx packet when initiating the Mix Protocol.
+
+### 8.5 Packet Construction
+
+This section defines how a mix node constructs a Sphinx packet when initiating
+the Mix Protocol on behalf of a local origin protocol instance.
+The construction process wraps the message in a sequence of encryption
+layers&mdash;one for each hop&mdash;such that only the corresponding mix node
+can decrypt its layer and retrieve the routing instructions for that hop.
+
+#### 8.5.1 Inputs
+
+To initiate the Mix Protocol, the origin protocol instance submits a message
+to the Mix Entry Layer on the same node. This layer forwards it to the local
+Mix Protocol instance, which constructs a Sphinx packet
+using the following REQUIRED inputs:
+
+- **Application message**: The serialized message provided by the origin
+  protocol instance. The Mix Protocol instance applies any configured spam
+  protection mechanism and attaches one or two SURBs prior to encapsulating
+  the message in the Sphinx packet. The initiating node MUST ensure that
+  the resulting payload size does not exceed the maximum supported size
+  defined in [Section 8.3.2](#832-payload-size).
+- **Origin protocol codec**: The libp2p protocol string corresponding to the
+  origin protocol instance. This is included in the payload so that
+  the exit node can route the message to the intended destination protocol
+  after decryption.
+- **Mix Path length $L$**: The number of mix nodes to include in the path.
+  The mix path MUST consist of at least three hops, each representing a
+  distinct mix node.
+- **Destination address $Δ$**: The routing address of the intended recipient
+  of the message. This address is encoded in $(tκ - 2)$ bytes as defined in
+  [Section 8.4](#84-address-and-delay-encoding) and revealed only at the last hop.
+
+#### 8.5.2 Construction Steps
+
+This subsection defines how the initiating mix node constructs a complete
+Sphinx packet using the inputs defined in
+[Section 8.5.1](#851-inputs). The construction MUST
+follow the cryptographic structure defined in
+[Section 8.1](#81-packet-structure-overview), use the primitives specified in
+[Section 8.2](#82-cryptographic-primitives), and adhere to the component sizes
+and encoding formats from [Section 8.3](#83-packet-component-sizes) and
+[Section 8.4](#84-address-and-delay-encoding).
+
+The construction MUST proceed as follows:
+
+1. **Prepare Application Message**
+
+   - Apply any configured spam protection mechanism (_e.g.,_ PoW, VDF, RLN)
+  to the serialized message. Spam protection mechanisms are pluggable as defined
+  in [Section 6.3](#63-spam-protection).
+   - Attach one or more SURBs, if required. Their format and processing are
+  specified in [Section X.X].
+   - Append the origin protocol codec.
+   - Pad the result to the maximum application message length of $3968$ bytes
+  using a deterministic padding scheme. This value is derived from the fixed
+  payload size in [Section 8.3.2](#832-payload-size) ($3984$ bytes) minus the
+  security parameter $κ = 16$ bytes defined in
+  [Section 8.2](#82-cryptographic-primitives). The chosen scheme MUST yield a
+  fixed-size padded output and MUST be consistent across all mix nodes to
+  ensure correct interpretation during unpadding. For example, schemes that
+  explicitly encode the padding length and prepend zero-valued padding bytes
+  MAY be used.
+   - Let the resulting message be $m$.  
+
+2. **Select A Mix Path**
+
+   - First obtain an unbiased random sample of live, routable mix nodes using
+  some discovery mechanism. The choice of discovery mechanism is
+  deployment-specific as defined in [Section 6.1](#61-discovery). The
+  discovery mechanism MUST be unbiased and provide, at a minimum, the
+  multiaddress and X25519 public key of each mix node.
+   - From this sample, choose a random mix path of length $L \geq 3$. As defined
+  in [Section 2](#2-terminology), a mix path is a non-repeating sequence of
+  mix nodes.
+   - For each hop $i \in \{0 \ldots L-1\}$:
+     - Retrieve the multiaddress and corresponding X25519 public key $y_i$ of
+  the $i$-th mix node.
+     - Encode the multiaddress in $(tκ - 2)$ bytes as defined in
+  [Section 8.4](#84-address-and-delay-encoding). Let the resulting encoded
+  multiaddress be $\mathrm{addr\_i}$.
+
+3. **Wrap Plaintext Payload In Sphinx Packet**
+
+   a. **Compute Ephemeral Secrets**
+
+   - Choose a random private exponent $x \in_R \mathbb{Z}_q^*$.
+   - Initialize:  
+     $`
+     \begin{aligned}
+     α_0 &= g^x \\
+     s_0 &= y_0^x \\
+     b_0 &= H(α_0\ |\ s_0)
+     \end{aligned}
+     `$
+   - For each hop $i$ (from $1$ to $L-1$), compute:  
+     $`
+     \begin{aligned}
+     α_i &= α_{i-1}^{b_{i-1}} \\
+     s_i &= y_{i}^{x\prod_{\text{j=0}}^{\text{i-1}} b_{j}} \\
+     b_i &= H(α_i\ |\ s_i)
+     \end{aligned}
+     `$
+
+   Note that the length of $α_i$ is $32$ bytes as defined in
+   [Section 8.3](#83-packet-component-sizes).
+
+   b. **Compute Per-Hop Filler Strings**  
+   Filler strings are encrypted strings that are appended to the header during
+   encryption. They ensure that the header length remains constant across hops,
+   regardless of the position of a node in the mix path.
+
+   To compute the sequence of filler strings, perform the following steps:
+
+   - Initialize $Φ_0 = \epsilon$ (empty string).
+   - For each $i$ (from $1$ to $L-1$):
+
+     - Derive per-hop AES key and IV:
+
+       $`
+       \begin{array}{l}
+       Φ_{\mathrm{aes\_key}_{i-1}} =
+       \mathrm{KDF}(\text{"aes\_key"} \mid s_{i-1})\\
+       Φ_{\mathrm{iv}_{i-1}} =
+       \mathrm{KDF}(\text{"iv"} \mid s_{i-1})
+       \end{array}
+       `$
+
+     - Compute the filler string $Φ_i$ using $\text{AES-CTR}^\prime_i$,
+       which is AES-CTR encryption with the keystream starting from
+       index $((t+1)(r-i)+t+2)\kappa$ :
+
+       $`
+       \begin{array}{l}
+       Φ_i = \mathrm{AES\text{-}CTR}'_i\bigl(Φ_{\mathrm{aes\_key}_{i-1}},
+       Φ_{\mathrm{iv}_{i-1}}, Φ_{i-1} \mid 0_{(t+1)\kappa} \bigr),
+       \text{where notation $0_x$ defines the string of $0$ bits of length $x$.}
+       \end{array}
+       `$
+
+   Note that the length of $Φ_i$ is $(t+1)i\kappa$.
+
+   c. **Construct Routing Header**
+   The routing header as defined in
+   [Section 8.1](#81-packet-structure-overview) is the encrypted structure
+   that carries the forwarding instructions for each hop. It ensures that a
+   mix node can learn only its immediate next hop and forwarding delay without
+   inferring the full path.
+
+   Filler strings computed in the previous step are appended during encryption
+   to ensure that the header length remains constant across hops. This prevents
+   a node from distinguishing its position in the path based on header size.
+
+   To construct the routing header, perform the following steps for each hop
+   $i = L-1$ down to $0$, recursively:
+
+   - Derive per-hop AES key, MAC key, and IV:
+
+     $`
+     \begin{array}{l}
+     β_{\mathrm{aes\_key}_i} =
+     \mathrm{KDF}(\text{"aes\_key"} \mid s_i)\\
+     \mathrm{mac\_key}_i =
+     \mathrm{KDF}(\text{"mac\_key"} \mid s_{i})\\
+     β_{\mathrm{iv}_i} =
+     \mathrm{KDF}(\text{"iv"} \mid s_i)
+     \end{array}
+     `$
+  
+   - Set the per hop two-byte encoded delay $\mathrm{delay}_i$ as defined in
+  [Section 8.4](#84-address-and-delay-encoding):
+     - If final hop (_i.e.,_ $i = L - 1$), encode two byte zero padding.
+     - For all other hop $i,\ i < L - 1$, sample a forwarding delay
+  using the delay strategy configured by the application and encode it in two bytes.
+  The delay strategy is pluggable as defined in [Section 6.2](#62-delay-strategy).
+  
+   - Using the derived keys and encoded forwarding delay, compute the nested
+  encrypted routing information $β_i$:
+
+     - If $i = L-1$ (_i.e.,_ exit node):
+
+       $`
+       \begin{array}{l}
+       β_i = \mathrm{AES\text{-}CTR}\bigl(β_{\mathrm{aes\_key}_i},
+       β_{\mathrm{iv}_i}, Δ \mid \mathrm{delay}_i \mid 0_{((t+1)(r-L)+2)\kappa}
+       \bigr) \bigm| Φ_{L-1}
+       \end{array}
+       `$
+
+     - Otherwise (_i.e.,_ intermediary node):
+
+       $`
+       \begin{array}{l}
+       β_i = \mathrm{AES\text{-}CTR}\bigl(β_{\mathrm{aes\_key}_i},
+       β_{\mathrm{iv}_i}, \mathrm{addr}_{i+1} \mid $\mathrm{delay}_i$
+       \mid γ_{i+1} \mid β_{i+1 \, [0 \ldots (r(t+1) - t)\kappa - 1]} \bigr)
+       \end{array}
+       `$
+
+     Note that the length of $\beta_i$ is $(r(t+1)+1)\kappa$, $0 \leq i \leq L-1$
+     as defined in [Section 8.3](#83-packet-component-sizes).
+
+     - Compute the message authentication code $γ_i$:
+
+       $`
+       \begin{array}{l}
+       γ_i = \mathrm{HMAC\text{-}SHA\text{-}256}\bigl(\mathrm{mac\_key}_i,
+       β_i \bigr)
+       \end{array}
+       `$
+
+     Note that the length of $\gamma_i$ is $\kappa$ as defined in
+     [Section 8.3](#83-packet-component-sizes).
+
+   d. **Encrypt Payload**
+   The encrypted payload $δ$ contains the message $m$ defined in Step 1,
+   prepended with a $κ$-byte string of zeros. It is encrypted in layers such that
+   each hop in the mix path removes exactly one layer using the per-hop session
+   key. This ensures that only the final hop (_i.e.,_ the exit node) can fully
+   recover $m$, validate its integrity, and forward it to the destination.
+   To compute the encrypted payload, perform the following steps for each hop
+   $i = L-1$ down to $0$, recursively:
+
+   - Derive per-hop AES key and IV:
+
+     $`
+     \begin{array}{l}
+     δ_{\mathrm{aes\_key}_i} =
+     \mathrm{KDF}(\text{"δ\_aes\_key"} \mid s_i)\\
+     δ_{\mathrm{iv}_i} =
+     \mathrm{KDF}(\text{"δ\_iv"} \mid s_i)
+     \end{array}
+     `$
+  
+   - Using the derived keys, compute the encrypted payload $δ_i$:
+
+     - If $i = L-1$ (_i.e.,_ exit node):
+
+       $`
+       \begin{array}{l}
+       δ_i = \mathrm{AES\text{-}CTR}\bigl(δ_{\mathrm{aes\_key}_i},
+       δ_{\mathrm{iv}_i}, 0_{\kappa} \mid m
+       \bigr)
+       \end{array}
+       `$
+
+     - Otherwise (_i.e.,_ intermediary node):
+
+       $`
+       \begin{array}{l}
+       δ_i = \mathrm{AES\text{-}CTR}\bigl(δ_{\mathrm{aes\_key}_i},
+       δ_{\mathrm{iv}_i}, δ_{i+1} \bigr)
+       \end{array}
+       `$