issuer keys implementation

.circleci/config.yml

@@ -0,0 +1,29 @@
+# Use the latest 2.1 version of CircleCI pipeline process engine.
+# See: https://circleci.com/docs/2.0/configuration-reference
+version: 2.1
+
+# Define a job to be invoked later in a workflow.
+# See: https://circleci.com/docs/2.0/configuration-reference/#jobs
+jobs:
+  cargo-test:
+    # Specify the execution environment. You can specify an image from Dockerhub or use one of our Convenience Images from CircleCI's Developer Hub.
+    # See: https://circleci.com/docs/2.0/configuration-reference/#docker-machine-macos-windows-executor
+    docker:
+      - image: cimg/rust:1.59.0
+    # Add steps to the job
+    # See: https://circleci.com/docs/2.0/configuration-reference/#steps
+    steps:
+      - checkout
+      - run:
+          name: "cargo test"
+          command: |
+            cargo version;
+            cargo test;
+
+
+# Invoke jobs via workflows
+# See: https://circleci.com/docs/2.0/configuration-reference/#workflows
+workflows:
+  build-and-test:
+    jobs:
+      - cargo-test

src/keys.rs

@@ -18,7 +18,7 @@ use zcash_note_encryption::EphemeralKeyBytes;
 
 use crate::{
     address::Address,
-    primitives::redpallas::{self, SpendAuth},
+    primitives::redpallas::{self, SpendAuth, VerificationKey},
     spec::{
         commit_ivk, diversify_hash, extract_p, ka_orchard, prf_nf, to_base, to_scalar,
         NonIdentityPallasPoint, NonZeroPallasBase, NonZeroPallasScalar, PrfExpand,
@@ -188,21 +188,104 @@ impl SpendValidatingKey {
     pub(crate) fn from_bytes(bytes: &[u8]) -> Option<Self> {
         <[u8; 32]>::try_from(bytes)
             .ok()
-            .and_then(|b| {
-                // Structural validity checks for ak_P:
-                // - The point must not be the identity
-                //   (which for Pallas is canonically encoded as all-zeroes).
-                // - The sign of the y-coordinate must be positive.
-                if b != [0; 32] && b[31] & 0x80 == 0 {
-                    <redpallas::VerificationKey<SpendAuth>>::try_from(b).ok()
-                } else {
-                    None
-                }
-            })
+            .and_then(check_structural_validity)
             .map(SpendValidatingKey)
     }
 }
 
+/// An issuer authorizing key, used to create issuer authorization signatures.
+/// This type enforces that the corresponding public point (ik^ℙ) has ỹ = 0.
+///
+/// $\mathsf{isk}$ as defined in
+/// [Zcash Protocol Spec § 4.2.3: Orchard Key Components][orchardkeycomponents].
+///
+/// [orchardkeycomponents]: https://zips.z.cash/protocol/nu5.pdf#orchardkeycomponents
+#[derive(Clone, Debug)]
+pub struct IssuerAuthorizingKey(redpallas::SigningKey<SpendAuth>);
+
+impl IssuerAuthorizingKey {
+    /// Derives isk from sk. Internal use only, does not enforce all constraints.
+    fn derive_inner(sk: &SpendingKey) -> pallas::Scalar {
+        to_scalar(PrfExpand::ZsaIsk.expand(&sk.0))
+    }
+}
+
+impl From<&SpendingKey> for IssuerAuthorizingKey {
+    fn from(sk: &SpendingKey) -> Self {
+        let isk = Self::derive_inner(sk);
+        // IssuerSigningKey cannot be constructed such that this assertion would fail.
+        assert!(!bool::from(isk.is_zero()));
+        let ret = IssuerAuthorizingKey(isk.to_repr().try_into().unwrap());
+        // If the last bit of repr_P(ik) is 1, negate isk.
+        if (<[u8; 32]>::from(IssuerValidatingKey::from(&ret).0)[31] >> 7) == 1 {
+            IssuerAuthorizingKey((-isk).to_repr().try_into().unwrap())
+        } else {
+            ret
+        }
+    }
+}
+
+/// A key used to validate issuer authorization signatures.
+///
+/// Defined in [Zcash Protocol Spec § 4.2.3: Orchard Key Components][orchardkeycomponents].
+/// Note that this is $\mathsf{ik}^\mathbb{P}$, which by construction is equivalent to
+/// $\mathsf{ik}$ but stored here as a RedPallas verification key.
+///
+/// [orchardkeycomponents]: https://zips.z.cash/protocol/nu5.pdf#orchardkeycomponents
+#[derive(Debug, Clone, PartialOrd, Ord)]
+pub struct IssuerValidatingKey(redpallas::VerificationKey<SpendAuth>);
+impl From<&IssuerAuthorizingKey> for IssuerValidatingKey {
+    fn from(isk: &IssuerAuthorizingKey) -> Self {
+        IssuerValidatingKey((&isk.0).into())
+    }
+}
+
+impl From<&IssuerValidatingKey> for pallas::Point {
+    fn from(issuer_validating_key: &IssuerValidatingKey) -> pallas::Point {
+        pallas::Point::from_bytes(&(&issuer_validating_key.0).into()).unwrap()
+    }
+}
+
+impl PartialEq for IssuerValidatingKey {
+    fn eq(&self, other: &Self) -> bool {
+        <[u8; 32]>::from(&self.0).eq(&<[u8; 32]>::from(&other.0))
+    }
+}
+
+impl Eq for IssuerValidatingKey {}
+
+impl IssuerValidatingKey {
+    /// Converts this spend validating key to its serialized form,
+    /// I2LEOSP_256(ik).
+    pub(crate) fn to_bytes(&self) -> [u8; 32] {
+        // This is correct because the wrapped point must have ỹ = 0, and
+        // so the point repr is the same as I2LEOSP of its x-coordinate.
+        <[u8; 32]>::from(&self.0)
+    }
+
+    pub(crate) fn from_bytes(bytes: &[u8]) -> Option<Self> {
+        <[u8; 32]>::try_from(bytes)
+            .ok()
+            .and_then(check_structural_validity)
+            .map(IssuerValidatingKey)
+    }
+}
+
+/// A function to check structural validity of the validating keys for authorizing transfers and
+/// issuing assets
+/// Structural validity checks for ik_P:
+///  - The point must not be the identity (which for Pallas is canonically encoded as all-zeroes).
+///  - The sign of the y-coordinate must be positive.
+fn check_structural_validity(
+    verification_key_bytes: [u8; 32],
+) -> Option<VerificationKey<SpendAuth>> {
+    if verification_key_bytes != [0; 32] && verification_key_bytes[31] & 0x80 == 0 {
+        <redpallas::VerificationKey<SpendAuth>>::try_from(verification_key_bytes).ok()
+    } else {
+        None
+    }
+}
+
 /// A key used to derive [`Nullifier`]s from [`Note`]s.
 ///
 /// $\mathsf{nk}$ as defined in [Zcash Protocol Spec § 4.2.3: Orchard Key Components][orchardkeycomponents].
@@ -1021,9 +1104,15 @@ mod tests {
             let ask: SpendAuthorizingKey = (&sk).into();
             assert_eq!(<[u8; 32]>::from(&ask.0), tv.ask);
 
+            let isk: IssuerAuthorizingKey = (&sk).into();
+            assert_eq!(<[u8; 32]>::from(&isk.0), tv.isk);
+
             let ak: SpendValidatingKey = (&ask).into();
             assert_eq!(<[u8; 32]>::from(ak.0), tv.ak);
 
+            let ik: IssuerValidatingKey = (&isk).into();
+            assert_eq!(<[u8; 32]>::from(ik.0), tv.ik);
+
             let nk: NullifierDerivingKey = (&sk).into();
             assert_eq!(nk.0.to_repr(), tv.nk);
 

src/spec/prf_expand.rs

@@ -10,6 +10,7 @@ pub(crate) enum PrfExpand {
     OrchardNk,
     OrchardRivk,
     Psi,
+    ZsaIsk,
     OrchardZip32Child,
     OrchardDkOvk,
     OrchardRivkInternal,
@@ -24,6 +25,7 @@ impl PrfExpand {
             Self::OrchardNk => 0x07,
             Self::OrchardRivk => 0x08,
             Self::Psi => 0x09,
+            Self::ZsaIsk => 0x0a,
             Self::OrchardZip32Child => 0x81,
             Self::OrchardDkOvk => 0x82,
             Self::OrchardRivkInternal => 0x83,

src/test_vectors/keys.rs

@@ -4,6 +4,8 @@ pub(crate) struct TestVector {
     pub(crate) sk: [u8; 32],
     pub(crate) ask: [u8; 32],
     pub(crate) ak: [u8; 32],
+    pub(crate) isk: [u8; 32],
+    pub(crate) ik: [u8; 32],
     pub(crate) nk: [u8; 32],
     pub(crate) rivk: [u8; 32],
     pub(crate) ivk: [u8; 32],
@@ -41,6 +43,16 @@ pub(crate) fn test_vectors() -> Vec<TestVector> {
                 0x12, 0x8b, 0x9a, 0x14, 0x0d, 0x5e, 0x07, 0xc1, 0x51, 0x72, 0x1d, 0xc1, 0x6d, 0x25,
                 0xd4, 0xe2, 0x0f, 0x15,
             ],
+            isk: [
+                0x95, 0x4a, 0x86, 0xc7, 0xa7, 0x15, 0x53, 0xfa, 0x6c, 0x8b, 0x67, 0x58, 0x54, 0x26,
+                0x8e, 0xa5, 0x4c, 0x51, 0xfb, 0x17, 0xd8, 0x3d, 0x80, 0xee, 0x71, 0xd4, 0xae, 0x42,
+                0xa1, 0xf8, 0xc8, 0x16,
+            ],
+            ik: [
+                0x2e, 0x4f, 0xd4, 0xa6, 0xec, 0x39, 0x94, 0x87, 0xd3, 0x78, 0xb4, 0xc7, 0x25, 0xfb,
+                0x9b, 0xaf, 0xbc, 0x01, 0xa5, 0xe2, 0xb7, 0xf3, 0x68, 0x2e, 0xf4, 0x53, 0x95, 0x91,
+                0xbc, 0xf0, 0x59, 0x02,
+            ],
             nk: [
                 0x9f, 0x2f, 0x82, 0x67, 0x38, 0x94, 0x5a, 0xd0, 0x1f, 0x47, 0xf7, 0x0d, 0xb0, 0xc3,
                 0x67, 0xc2, 0x46, 0xc2, 0x0c, 0x61, 0xff, 0x55, 0x83, 0x94, 0x8c, 0x39, 0xde, 0xa9,
@@ -132,6 +144,16 @@ pub(crate) fn test_vectors() -> Vec<TestVector> {
                 0x2a, 0xd6, 0x43, 0x23, 0x62, 0x9c, 0xfe, 0xd1, 0xe3, 0xaa, 0x24, 0xef, 0x05, 0x2f,
                 0x56, 0xe4, 0x00, 0x2a,
             ],
+            isk: [
+                0xee, 0xf5, 0xe9, 0x1b, 0x36, 0xb8, 0x06, 0x86, 0x72, 0x3d, 0x14, 0xdc, 0xc7, 0x04,
+                0xad, 0x59, 0x67, 0x08, 0x0b, 0x7d, 0x6e, 0x49, 0xaf, 0x97, 0x03, 0x0e, 0x4f, 0xa0,
+                0xbf, 0x5a, 0xd9, 0x0b,
+            ],
+            ik: [
+                0x2e, 0xde, 0xfb, 0x15, 0x8e, 0xa4, 0x48, 0x82, 0x57, 0x2b, 0xcd, 0xb2, 0x35, 0xca,
+                0x36, 0xab, 0x39, 0xc7, 0x47, 0xbb, 0x71, 0xfe, 0x0f, 0x10, 0xfa, 0xa3, 0x9b, 0xfd,
+                0x62, 0x0a, 0xcc, 0x04,
+            ],
             nk: [
                 0xa8, 0xb7, 0x3d, 0x97, 0x9b, 0x6e, 0xaa, 0xda, 0x89, 0x24, 0xbc, 0xbd, 0xc6, 0x3a,
                 0x9e, 0xf4, 0xe8, 0x73, 0x46, 0xf2, 0x30, 0xab, 0xa6, 0xbb, 0xe1, 0xe2, 0xb4, 0x3c,
@@ -223,6 +245,16 @@ pub(crate) fn test_vectors() -> Vec<TestVector> {
                 0xed, 0xb4, 0x26, 0x65, 0x7b, 0x2d, 0x07, 0x40, 0x66, 0x64, 0xd8, 0x95, 0x31, 0x2e,
                 0xa1, 0xc3, 0xb3, 0x34,
             ],
+            isk: [
+                0x5b, 0x1f, 0xc4, 0x57, 0xae, 0x71, 0x38, 0x3c, 0x53, 0xf4, 0x69, 0x41, 0xb7, 0xcb,
+                0x4c, 0xec, 0x3d, 0xea, 0xc0, 0xc6, 0x03, 0xe2, 0xcd, 0xd0, 0xd1, 0x8d, 0x94, 0x01,
+                0x9e, 0x43, 0xe2, 0x07,
+            ],
+            ik: [
+                0x4f, 0x43, 0xeb, 0x7d, 0x9e, 0x03, 0x6f, 0xa6, 0x15, 0xfd, 0x04, 0xa5, 0xef, 0x6a,
+                0xeb, 0x21, 0x6e, 0x06, 0x9b, 0xe9, 0x2d, 0x30, 0xe8, 0xf7, 0x16, 0x3e, 0xe3, 0x15,
+                0x11, 0x6f, 0x18, 0x32,
+            ],
             nk: [
                 0x04, 0x51, 0x4e, 0xa0, 0x48, 0xb9, 0x43, 0x63, 0xde, 0xa7, 0xcb, 0x3b, 0xe8, 0xd6,
                 0x25, 0x82, 0xac, 0x52, 0x92, 0x2e, 0x08, 0x65, 0xf6, 0x62, 0x74, 0x3b, 0x05, 0xea,
@@ -314,6 +346,16 @@ pub(crate) fn test_vectors() -> Vec<TestVector> {
                 0xe7, 0x2c, 0x3b, 0x64, 0x00, 0x06, 0xff, 0x08, 0x50, 0x52, 0x80, 0xe4, 0xf0, 0x0f,
                 0xad, 0xf7, 0x63, 0x28,
             ],
+            isk: [
+                0x71, 0xd0, 0x64, 0xaa, 0xa0, 0x82, 0x63, 0xb8, 0xe4, 0xc3, 0xed, 0x70, 0x3c, 0x6f,
+                0x54, 0x25, 0x4a, 0x88, 0x8c, 0x36, 0xec, 0x69, 0x86, 0x62, 0xf7, 0x1f, 0xbb, 0xf4,
+                0x26, 0xd9, 0x09, 0x28,
+            ],
+            ik: [
+                0x12, 0xb3, 0xab, 0xff, 0x96, 0x75, 0x20, 0x9e, 0x94, 0x54, 0x07, 0x0c, 0x14, 0xac,
+                0x15, 0x54, 0x65, 0xae, 0x83, 0xbe, 0x2c, 0x39, 0x46, 0x63, 0x5e, 0x38, 0x77, 0xba,
+                0x67, 0xdf, 0x49, 0x12,
+            ],
             nk: [
                 0xcf, 0x36, 0xad, 0x6a, 0x06, 0x6c, 0xd2, 0x13, 0xe1, 0xd7, 0x67, 0xab, 0x07, 0x1d,
                 0xc1, 0x16, 0x78, 0x85, 0xc4, 0x16, 0x8b, 0xc2, 0xe2, 0x17, 0x54, 0x48, 0x56, 0x3a,
@@ -405,6 +447,16 @@ pub(crate) fn test_vectors() -> Vec<TestVector> {
                 0xf5, 0x6d, 0x83, 0x20, 0x09, 0xf7, 0x24, 0x2e, 0x1f, 0x7c, 0x77, 0x0a, 0x12, 0x24,
                 0x1d, 0xfa, 0x28, 0x07,
             ],
+            isk: [
+                0x44, 0x36, 0x1a, 0x7b, 0xa6, 0xa1, 0xaa, 0x17, 0x8e, 0x72, 0xaf, 0x47, 0xbd, 0xc1,
+                0x60, 0x40, 0xce, 0x1c, 0x54, 0xdd, 0x4b, 0x56, 0x33, 0x21, 0x55, 0xba, 0x9d, 0x04,
+                0x09, 0x71, 0xd0, 0x07,
+            ],
+            ik: [
+                0x1f, 0x17, 0x2d, 0x79, 0xae, 0xdc, 0xc2, 0x06, 0x8c, 0x3a, 0x09, 0x08, 0x93, 0xe1,
+                0xa1, 0x75, 0xd9, 0xb5, 0x78, 0xf8, 0x91, 0xaf, 0x9a, 0xb6, 0x8d, 0x4f, 0xe1, 0xe9,
+                0x05, 0xa3, 0xb2, 0x11,
+            ],
             nk: [
                 0x51, 0xba, 0xf3, 0x33, 0xcf, 0xf1, 0xf2, 0xd0, 0xc7, 0xe3, 0xcf, 0xf4, 0xd3, 0x01,
                 0x29, 0x9d, 0xc1, 0xef, 0xe9, 0x83, 0x00, 0x31, 0x4a, 0x54, 0x19, 0x38, 0x02, 0x9b,
@@ -496,6 +548,16 @@ pub(crate) fn test_vectors() -> Vec<TestVector> {
                 0xdf, 0x28, 0xbb, 0x0f, 0x10, 0x21, 0xea, 0x84, 0x3f, 0x86, 0x7f, 0x8a, 0x17, 0x0f,
                 0x5c, 0x33, 0x90, 0x1f,
             ],
+            isk: [
+                0xdc, 0x93, 0x72, 0x9f, 0x3f, 0x28, 0x30, 0xed, 0x79, 0x1c, 0x21, 0xbe, 0xbe, 0x45,
+                0x0f, 0xcf, 0x1f, 0x8f, 0xef, 0x49, 0x81, 0x39, 0xc7, 0x99, 0xd1, 0x63, 0x66, 0x5a,
+                0x8c, 0x51, 0xe5, 0x2d,
+            ],
+            ik: [
+                0x1d, 0xb6, 0x1c, 0x29, 0x3e, 0x3a, 0x93, 0x34, 0x5d, 0x06, 0xb9, 0x0b, 0xd7, 0x1f,
+                0xd3, 0x21, 0x5c, 0x2c, 0x1c, 0x29, 0x53, 0x5a, 0x10, 0xde, 0x9d, 0x31, 0x40, 0xb7,
+                0x4d, 0xb6, 0x1d, 0x07,
+            ],
             nk: [
                 0x9e, 0x99, 0x7d, 0x9d, 0x26, 0x97, 0x87, 0x26, 0x8e, 0x09, 0x2a, 0x7c, 0x85, 0x41,
                 0x7d, 0xa5, 0x30, 0xea, 0x42, 0xfa, 0xc6, 0x68, 0xa7, 0x49, 0xaf, 0x55, 0xdf, 0xb7,
@@ -587,6 +649,16 @@ pub(crate) fn test_vectors() -> Vec<TestVector> {
                 0x65, 0x43, 0x46, 0x2a, 0x13, 0x7f, 0xfe, 0xa3, 0x7b, 0xaf, 0x41, 0xef, 0x28, 0x6b,
                 0xb7, 0x32, 0xbe, 0x2c,
             ],
+            isk: [
+                0xf2, 0x34, 0x52, 0x32, 0xc9, 0x19, 0xc1, 0x29, 0xe0, 0x4b, 0x0c, 0x46, 0xac, 0x2c,
+                0xa8, 0x50, 0x65, 0xd9, 0x54, 0x85, 0xb9, 0x02, 0xab, 0x0f, 0x98, 0xf9, 0x3a, 0xee,
+                0x59, 0x4b, 0x5f, 0x02,
+            ],
+            ik: [
+                0x2c, 0x83, 0xd9, 0x20, 0xe9, 0xf6, 0x6d, 0xa5, 0x04, 0x86, 0x37, 0xad, 0x9a, 0xa2,
+                0xcc, 0xe6, 0xe1, 0x6e, 0xf4, 0x8f, 0x86, 0x50, 0xea, 0x00, 0xd8, 0xc2, 0xd7, 0x68,
+                0x61, 0x8a, 0xe3, 0x36,
+            ],
             nk: [
                 0xfd, 0x31, 0x64, 0xc6, 0x32, 0xbe, 0xc9, 0x4c, 0xe9, 0xfb, 0x2f, 0x30, 0x22, 0x63,
                 0xb8, 0x84, 0xab, 0xb9, 0xc1, 0x0e, 0x55, 0xe4, 0x48, 0x64, 0x7f, 0x67, 0x98, 0x49,
@@ -678,6 +750,16 @@ pub(crate) fn test_vectors() -> Vec<TestVector> {
                 0xb3, 0x65, 0x1f, 0xfa, 0x1c, 0x69, 0x69, 0x15, 0xac, 0x00, 0xa2, 0x5e, 0xa3, 0xac,
                 0x7d, 0xff, 0x99, 0x01,
             ],
+            isk: [
+                0x3d, 0xa9, 0x08, 0x88, 0xba, 0x18, 0x24, 0xc8, 0x16, 0x29, 0x2d, 0x7f, 0x17, 0x33,
+                0xac, 0x4c, 0xbe, 0x72, 0x2c, 0x6a, 0x12, 0x1c, 0xc7, 0x80, 0x17, 0x06, 0x26, 0xb7,
+                0x0a, 0x26, 0x95, 0x28,
+            ],
+            ik: [
+                0x16, 0xa8, 0xff, 0x29, 0xb5, 0x17, 0xb5, 0xa8, 0xf7, 0xd0, 0x9b, 0x4e, 0x5e, 0x71,
+                0x3a, 0x9a, 0x78, 0x4c, 0x74, 0x04, 0xd6, 0x1e, 0x3a, 0xf5, 0x30, 0x19, 0xc3, 0x47,
+                0x0e, 0x90, 0x95, 0x22,
+            ],
             nk: [
                 0x02, 0xab, 0x99, 0x5c, 0xe9, 0x8f, 0x63, 0x02, 0x5f, 0xb6, 0x24, 0x28, 0xa0, 0xfb,
                 0xf5, 0x2f, 0x25, 0x22, 0xe6, 0xa2, 0x72, 0x61, 0x07, 0x8a, 0x9f, 0x4d, 0x6a, 0x36,
@@ -769,6 +851,16 @@ pub(crate) fn test_vectors() -> Vec<TestVector> {
                 0x3b, 0x02, 0xd2, 0x5c, 0xc1, 0x0c, 0x90, 0x71, 0xfc, 0x02, 0x19, 0xe9, 0x7f, 0x93,
                 0x92, 0xd0, 0x67, 0x0c,
             ],
+            isk: [
+                0x01, 0x65, 0x33, 0x68, 0x4f, 0xb9, 0x81, 0x15, 0xa4, 0x05, 0xc9, 0xc7, 0xad, 0x47,
+                0x72, 0x76, 0xab, 0x7c, 0x72, 0xfd, 0x67, 0x1a, 0x27, 0xe3, 0x6c, 0x0a, 0x7a, 0xbe,
+                0x0a, 0x76, 0x90, 0x09,
+            ],
+            ik: [
+                0xff, 0xd7, 0x5f, 0x6f, 0x9e, 0xf4, 0x27, 0xf3, 0x26, 0xcd, 0xbf, 0x3a, 0x98, 0xbc,
+                0xb5, 0x93, 0x63, 0x5a, 0x2c, 0x1a, 0xd7, 0x2b, 0x39, 0x99, 0x12, 0x61, 0xe2, 0x75,
+                0xa9, 0xec, 0x6f, 0x10,
+            ],
             nk: [
                 0x25, 0x91, 0xed, 0xf7, 0xef, 0x4c, 0xf2, 0x18, 0x4c, 0x34, 0xbe, 0x93, 0xfc, 0xf6,
                 0x12, 0x91, 0x50, 0x42, 0xf1, 0x5a, 0xb5, 0x08, 0x4b, 0x14, 0xe1, 0x66, 0x79, 0x5b,
@@ -860,6 +952,16 @@ pub(crate) fn test_vectors() -> Vec<TestVector> {
                 0x8d, 0x4b, 0x02, 0x5f, 0x8c, 0xc1, 0x60, 0xe1, 0xf4, 0xe9, 0x5f, 0x0a, 0x85, 0x3e,
                 0xbc, 0x41, 0x6a, 0x2b,
             ],
+            isk: [
+                0x76, 0x08, 0x32, 0x9d, 0xfa, 0x77, 0xc4, 0x2c, 0x4f, 0xc7, 0x6a, 0xc2, 0x95, 0x94,
+                0xa2, 0x72, 0x83, 0x93, 0x4f, 0x5a, 0x93, 0x40, 0x71, 0xb9, 0xf8, 0xcd, 0x34, 0x4e,
+                0x1f, 0x98, 0x45, 0x0e,
+            ],
+            ik: [
+                0x72, 0xa0, 0xac, 0x97, 0x8a, 0x2d, 0xa1, 0x61, 0xf4, 0x1f, 0x5b, 0x7a, 0x40, 0xbd,
+                0x83, 0xc0, 0x58, 0x41, 0xf8, 0x1b, 0xc5, 0x11, 0x40, 0x67, 0xb8, 0x85, 0x98, 0x7f,
+                0x48, 0xca, 0x52, 0x2d,
+            ],
             nk: [
                 0x3e, 0x88, 0xf2, 0x07, 0x1f, 0xd9, 0xa2, 0xbb, 0x26, 0xcd, 0xa2, 0xea, 0x85, 0x6a,
                 0xa0, 0xfb, 0x3a, 0x80, 0xa8, 0x7d, 0x2f, 0xb6, 0x13, 0x6f, 0xab, 0x85, 0xe3, 0x6c,