Merge pull request #1405 from sipa/202301_bip324_update

BIP324 reference code / test vector improvements
2026-10-19 19:29:18 +00:00 · 2023-02-03 12:59:37 +09:00
commit 1a632a3875
@@ -247,19 +247,19 @@ To find encodings of a given X coordinate ''x'', we first need the inverse of ''
 * ''XSwiftECInv(x, u, case)'':
 ** If ''case & 2 = 0'':
 *** If ''lift_x(-x - u)'' succeeds, return ''None''.
-*** Let ''v = x'' if ''case & 1 = 0''; let ''v = -x - u (mod p)'' otherwise.
+*** Let ''v = x''.
 *** Let ''s = -(u<sup>3</sup> + 7)/(u<sup>2</sup> + uv + v<sup>2</sup>) (mod p)''.
 ** If ''case & 2 = 2'':
 *** Let ''s = x - u (mod p)''.
 *** If ''s = 0'', return ''None''.
 *** Let ''r'' be the square root of ''-s(4(u<sup>3</sup> + 7) + 3u<sup>2</sup>s) (mod p).''<ref name="modsqrt">'''How to compute a square root mod ''p''?''' Due to the structure of ''p'', a candidate for the square root of ''a'' mod ''p'' can be computed as ''x = a<sup>(p+1)/4</sup> mod p''. If ''a'' is not a square mod ''p'', this formula returns the square root of ''-a mod p'' instead, so it is necessary to verify that ''x<sup>2</sup> mod p = a''. If that is the case ''-x mod p'' is a solution too, but we define "the" square root to be equal to that expression (the square root will therefore always be a square itself, as ''(p+1)/4'' is even). This algorithm is a specialization of the [https://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm Tonelli-Shanks algorithm].</ref> Return ''None'' if it does not exist.
-*** If ''case & 1 = 1'':
-**** If ''r = 0'', return ''None''.
-**** let ''r = -r (mod p)''.
+** If ''case & 1 = 1'' and ''r = 0'', return ''None''.
 *** Let ''v = (-u + r/s)/2''.
 ** Let ''w'' be the square root of ''s (mod p)''. Return ''None'' if it does not exist.
-** If ''case & 4 = 4'', let ''w = -w (mod p)''.
-** Return ''w(u(c - 1)/2 - v)''.
+** If ''case & 5 = 0'', return ''-w(u(1 - c)/2 + v)''.
+** If ''case & 5 = 1'', return ''w(u(1 + c)/2 + v)''.
+** If ''case & 5 = 4'', return ''w(u(1 - c)/2 + v)''.
+** If ''case & 5 = 5'', return ''-w(u(1 + c)/2 + v)''.

 The overall ''XElligatorSwift'' algorithm, matching the name used in the paper, then uses this inverse to randomly''<ref name="ellswift_helps_parroting">'''Can the ElligatorSwift encoding be used to construct public key encodings that satisfy a certain structure (and not pseudorandom)?''' The algorithm chooses the first 32 bytes (i.e., the value ''u'') and then computes a corresponding ''t'' such that the mapping to the curve point holds. In general, picking ''u'' from a uniformly random distribution provides pseudorandomness. But we can also fix any of the 32 bytes in ''u'', and the algorithm will still find a corresponding ''t''. The fact that it is possible to fix the first 32 bytes, combined with the garbage bytes in the handshake, provides a limited but very simple method of parroting other protocols such as [https://tls13.xargs.org/ TLS 1.3], which can be deployed by one of the peers without explicit support from the other peer. More general methods of parroting, e.g., introduced by defining new protocol or a protocol upgrade, are not precluded.</ref> sample encodings of ''x'':

@@ -586,8 +586,8 @@ Peers supporting the v2 transport protocol signal support by advertising the <co
 == Test Vectors ==

 For development and testing purposes, we provide a collection of test vectors in CSV format, and a naive, highly inefficient, [[bip-0324/reference.py|reference implementation]] of the relevant algorithms. This code is for demonstration purposes only:
-* [[bip-0324/xelligatorswift_test_vectors.csv|XElligatorSwift vectors]] give examples of ElligatorSwift-encoded public keys, and the X coordinate they map to.
-* [[bip-0324/xswiftec_test_vectors.csv|XSwiftEC vectors]] give examples of ''(u, x)'' pairs, and the various ''t'' values that ''xswiftec_inv'' maps them to.
+* [[bip-0324/ellswift_decode_test_vectors.csv|XElligatorSwift decoding vectors]] give examples of ElligatorSwift-encoded public keys, and the X coordinate they map to.
+* [[bip-0324/xswiftec_inv_test_vectors.csv|XSwiftECInv vectors]] give examples of ''(u, x)'' pairs, and the various ''t'' values that ''xswiftec_inv'' maps them to.
 * [[bip-0324/packet_encoding_test_vectors.csv|Packet encoding vectors]] illustrate the lifecycle of the authenticated encryption scheme proposed in this document.

 == Rationale and References ==
@@ -0,0 +1,77 @@
+ellswift,x,comment
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+ffffffffffffffffffffffffffffffffffffffffffffffffffffffff3e91257d932016cbf69c4471bd1f656c6a107f1973de4af7086db897277060e25677f19a,864b3dc902c376709c10a93ad4bbe29fce0012f3dc8672c6286bba28d7d6d6fc,valid_x(x3);u>=p
+ffffffffffffffffffffffffffffffffffffffffffffffffffffffff795d6c1c322cadf599dbb86481522b3cc55f15a67932db2afa0111d9ed6981bcd124bf44,766dfe4a700d9bee288b903ad58870e3d4fe2f0ef780bcac5c823f320d9a9bef,(u'^3+t'^2+7)%p=0;valid_x(x1);u>=p
+ffffffffffffffffffffffffffffffffffffffffffffffffffffffff8e426f0392389078c12b1a89e9542f0593bc96b6bfde8224f8654ef5d5cda935a3582194,faec7bc1987b63233fbc5f956edbf37d54404e7461c58ab8631bc68e451a0478,valid_x(x1);u>=p
+ffffffffffffffffffffffffffffffffffffffffffffffffffffffff91192139ffffffffffffffffffffffffffffffffffffffffffffffffffffffff45f0f1eb,ec29a50bae138dbf7d8e24825006bb5fc1a2cc1243ba335bc6116fb9e498ec1f,valid_x(x2);u>=p;t>=p
+ffffffffffffffffffffffffffffffffffffffffffffffffffffffff98eb9ab76e84499c483b3bf06214abfe065dddf43b8601de596d63b9e45a166a580541fe,1e0ff2dee9b09b136292a9e910f0d6ac3e552a644bba39e64e9dd3e3bbd3d4d4,(u'^3-t'^2+7)%p=0;valid_x(x3);u>=p
+ffffffffffffffffffffffffffffffffffffffffffffffffffffffff9b77b7f2c74d99efceaa550f1ad1c0f43f46e7ff1ee3bd0162b7bf55f2965da9c3450646,8b7dd5c3edba9ee97b70eff438f22dca9849c8254a2f3345a0a572ffeaae0928,valid_x(x2);u>=p
+ffffffffffffffffffffffffffffffffffffffffffffffffffffffff9b77b7f2ffffffffffffffffffffffffffffffffffffffffffffffffffffffff156ca896,0881950c8f51d6b9a6387465d5f12609ef1bb25412a08a74cb2dfb200c74bfbf,valid_x(x3);valid_x(x2);valid_x(x1);u>=p;t>=p
+ffffffffffffffffffffffffffffffffffffffffffffffffffffffffa2f5cd838816c16c4fe8a1661d606fdb13cf9af04b979a2e159a09409ebc8645d58fde02,2f083207b9fd9b550063c31cd62b8746bd543bdc5bbf10e3a35563e927f440c8,(u'^3+t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1);u>=p
+ffffffffffffffffffffffffffffffffffffffffffffffffffffffffb13f75c00000000000000000000000000000000000000000000000000000000000000000,4f51e0be078e0cddab2742156adba7e7a148e73157072fd618cd60942b146bd0,t%p=0;valid_x(x3);u>=p
+ffffffffffffffffffffffffffffffffffffffffffffffffffffffffb13f75c0fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,4f51e0be078e0cddab2742156adba7e7a148e73157072fd618cd60942b146bd0,t%p=0;valid_x(x3);u>=p;t>=p
+ffffffffffffffffffffffffffffffffffffffffffffffffffffffffe7bc1f8d0000000000000000000000000000000000000000000000000000000000000000,16c2ccb54352ff4bd794f6efd613c72197ab7082da5b563bdf9cb3edaafe74c2,t%p=0;valid_x(x2);u>=p
+ffffffffffffffffffffffffffffffffffffffffffffffffffffffffe7bc1f8dfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,16c2ccb54352ff4bd794f6efd613c72197ab7082da5b563bdf9cb3edaafe74c2,t%p=0;valid_x(x2);u>=p;t>=p
+ffffffffffffffffffffffffffffffffffffffffffffffffffffffffef64d162750546ce42b0431361e52d4f5242d8f24f33e6b1f99b591647cbc808f462af51,d41244d11ca4f65240687759f95ca9efbab767ededb38fd18c36e18cd3b6f6a9,(u'^3+t'^2+7)%p=0;valid_x(x3);u>=p
+fffffffffffffffffffffffffffffffffffffffffffffffffffffffff0e5be52372dd6e894b2a326fc3605a6e8f3c69c710bf27d630dfe2004988b78eb6eab36,64bf84dd5e03670fdb24c0f5d3c2c365736f51db6c92d95010716ad2d36134c8,valid_x(x3);valid_x(x2);valid_x(x1);u>=p
+fffffffffffffffffffffffffffffffffffffffffffffffffffffffffefbb982fffffffffffffffffffffffffffffffffffffffffffffffffffffffff6d6db1f,1c92ccdfcf4ac550c28db57cff0c8515cb26936c786584a70114008d6c33a34b,valid_x(x1);u>=p;t>=p
@@ -0,0 +1,418 @@
+"""Generate the BIP-0324 test vectors."""
+
+import csv
+import hashlib
+import os
+import sys
+from reference import (
+    FE,
+    GE,
+    MINUS_3_SQRT,
+    hkdf_sha256,
+    SECP256K1_G,
+    ellswift_decode,
+    ellswift_ecdh_xonly,
+    xswiftec_inv,
+    xswiftec,
+    v2_ecdh,
+    initialize_v2_transport,
+    v2_enc_packet
+)
+
+FILENAME_PACKET_TEST = os.path.join(sys.path[0], 'packet_encoding_test_vectors.csv')
+FILENAME_XSWIFTEC_INV_TEST = os.path.join(sys.path[0], 'xswiftec_inv_test_vectors.csv')
+FILENAME_ELLSWIFT_DECODE_TEST = os.path.join(sys.path[0], 'ellswift_decode_test_vectors.csv')
+
+def xswiftec_flagged(u, t, simplified=False):
+    """A variant of xswiftec which also returns 'flags', describing conditions encountered."""
+    flags = []
+    if u == 0:
+        flags.append("u%p=0")
+        u = FE(1)
+    if t == 0:
+        flags.append("t%p=0")
+        t = FE(1)
+    if u**3 + t**2 + 7 == 0:
+        flags.append("(u'^3+t'^2+7)%p=0")
+        t = 2 * t
+    X = (u**3 + 7 - t**2) / (2 * t)
+    Y = (X + t) / (MINUS_3_SQRT * u)
+    if X == 0:
+        if not simplified:
+            flags.append("(u'^3-t'^2+7)%p=0")
+    x3 = u + 4 * Y**2
+    if GE.is_valid_x(x3):
+        flags.append("valid_x(x3)")
+    x2 = (-X / Y - u) / 2
+    if GE.is_valid_x(x2):
+        flags.append("valid_x(x2)")
+    x1 = (X / Y - u) / 2
+    if GE.is_valid_x(x1):
+        flags.append("valid_x(x1)")
+    for x in (x3, x2, x1):
+        if GE.is_valid_x(x):
+            break
+    return x, flags
+
+
+def ellswift_create_deterministic(seed, features):
+    """This is a variant of ellswift_create which doesn't use randomness.
+
+    features is an integer selecting some properties of the result:
+    - (f & 3) == 0: only x1 is valid on decoding (see xswiftec{_flagged})
+    - (f & 3) == 1: only x2 is valid on decoding
+    - (f & 3) == 2: only x3 is valid on decoding
+    - (f & 3) == 3: x1,x2,x3 are all valid on decoding
+    - (f & 4) == 4: u >= p
+    - (f & 8) == 8: u mod n == 0
+
+    Returns privkey, ellswift
+    """
+
+    cnt = 0
+    while True:
+        sec = hkdf_sha256(32, seed, (cnt).to_bytes(4, 'little'), b"sec")
+        xval = (int.from_bytes(sec, 'big') * SECP256K1_G).x
+        cnt += 1
+        if features & 8:
+            u = 0
+            if features & 4:
+                u += FE.SIZE
+        else:
+            udat = hkdf_sha256(64, seed, (cnt).to_bytes(4, 'little'), b"u")
+            if features & 4:
+                u = FE.SIZE + 1 + int.from_bytes(udat, 'big') % (2**256 - FE.SIZE - 1)
+            else:
+                u = 1 + int.from_bytes(udat, 'big') % (FE.SIZE - 1)
+        case = hkdf_sha256(1, seed, (cnt).to_bytes(4, 'little'), b"case")[0] & 7
+        coru = FE(u) + ((features & 8) == 8)
+        t = xswiftec_inv(xval, coru, case)
+        if t is None:
+            continue
+        assert xswiftec(FE(u), t) == xval
+        x2, flags = xswiftec_flagged(FE(u), t)
+        assert x2 == xval
+        have_x1 = "valid_x(x1)" in flags
+        have_x2 = "valid_x(x2)" in flags
+        have_x3 = "valid_x(x3)" in flags
+        if (features & 4) == 0 and not (have_x1 and not have_x2 and not have_x3):
+            continue
+        if (features & 4) == 1 and not (not have_x1 and have_x2 and not have_x3):
+            continue
+        if (features & 4) == 2 and not (not have_x1 and not have_x2 and have_x3):
+            continue
+        if (features & 4) == 3 and not (have_x1 and have_x2 and have_x3):
+            continue
+        return sec, u.to_bytes(32, 'big') + t.to_bytes()
+
+def ellswift_decode_flagged(ellswift, simplified=False):
+    """Decode a 64-byte ElligatorSwift encoded coordinate, returning byte array + flag string."""
+    uv = int.from_bytes(ellswift[:32], 'big')
+    tv = int.from_bytes(ellswift[32:], 'big')
+    x, flags = xswiftec_flagged(FE(uv), FE(tv))
+    if not simplified:
+        if uv >= FE.SIZE:
+            flags.append("u>=p")
+        if tv >= FE.SIZE:
+            flags.append("t>=p")
+    return int(x).to_bytes(32, 'big'), ";".join(flags)
+
+def random_fe_int(_, seed, i, p):
+    """Function to use in tuple_expand, generating a random integer in 0..p-1."""
+    rng_out = hkdf_sha256(64, seed, i.to_bytes(4, 'little'), b"v%i_fe" % p)
+    return int.from_bytes(rng_out, 'big') % FE.SIZE
+
+def random_fe_int_high(_, seed, i, p):
+    """Function to use in tuple_expand, generating a random integer in p..2^256-1."""
+    rng_out = hkdf_sha256(64, seed, i.to_bytes(4, 'little'), b"v%i_fe_high" % p)
+    return FE.SIZE + int.from_bytes(rng_out, 'big') % (2**256 - FE.SIZE)
+
+def fn_of(p_in, fn):
+    """Function to use in tuple_expand, to pick one variable in function of another."""
+    def inner(vs, _seed, _i, p):
+        assert p != p_in
+        if isinstance(vs[p_in], int):
+            return fn(vs[p_in])
+        return None
+    return inner
+
+def tuple_expand(out, tuplespec, prio, seed=None, cnt=1):
+    """Given a tuple specification, expand it cnt times, and add results to out.
+
+    Expansion is defined recursively:
+    - If any of the spec elements is a list, each element of the list results
+      in an expansion (by replacing the list with its element).
+    - If any of the spec elements is a function, that function is invoked with
+      (spec, seed, expansion count, index in spec) as arguments. If the function
+      needs to wait for other indices to be expanded, it can return None.
+
+    The output consists of (prio, expansion count, SHA256(result), result, seed)
+    tuples."""
+
+    def recurse(vs, seed, i, change_pos=None, change=None):
+        if change_pos is not None:
+            vs = list(vs)
+            vs[change_pos] = change
+        for p, v in enumerate(vs):
+            if v is None:
+                return
+            if isinstance(v, list):
+                for ve in v:
+                    recurse(vs, seed, i, p, ve)
+                return
+            if callable(v):
+                res = v(vs, seed, i, p)
+                if res is not None:
+                    recurse(vs, seed, i, p, res)
+                    return
+        h = hashlib.sha256()
+        for v in vs:
+            h.update(int(v).to_bytes(32, 'big'))
+        out.append((prio, i, h.digest(), vs, seed))
+    for i in range(cnt):
+        recurse(tuplespec, seed, i)
+
+def gen_ellswift_decode_cases(seed, simplified=False):
+    """Generate a set of interesting (ellswift, x, flags) ellswift decoding cases."""
+    inputs = []
+
+    # Aggregate for use in tuple_expand, expanding to int in 0..p-1, and one in p..2^256-1.
+    RANDOM_VAL = [random_fe_int, random_fe_int_high]
+    # Aggregate for use in tuple_expand, expanding to integers which %p equal 0.
+    ZERO_VAL = [0, FE.SIZE]
+    # Helpers for constructing u and t values such that u^3+t^2+7=0 or u^3-t^2+7=0.
+    T_FOR_SUM_ZERO = fn_of(0, lambda u: (-FE(u)**3 - 7).sqrts())
+    T_FOR_DIFF_ZERO = fn_of(0, lambda u: (FE(u)**3 + 7).sqrts())
+    U_FOR_SUM_ZERO = fn_of(1, lambda t: (-FE(t)**2 - 7).cbrts())
+    U_FOR_DIFF_ZERO = fn_of(1, lambda t: (FE(t)**2 - 7).cbrts())
+
+    tuple_expand(inputs, [RANDOM_VAL, RANDOM_VAL], 0, seed + b"random", 64)
+    tuple_expand(inputs, [RANDOM_VAL, T_FOR_SUM_ZERO], 1, seed + b"t=sqrt(-u^3-7)", 64)
+    tuple_expand(inputs, [U_FOR_SUM_ZERO, RANDOM_VAL], 1, seed + b"u=cbrt(-t^2-7)", 64)
+    tuple_expand(inputs, [RANDOM_VAL, T_FOR_DIFF_ZERO], 1, seed + b"t=sqrt(u^3+7)", 64)
+    tuple_expand(inputs, [U_FOR_DIFF_ZERO, RANDOM_VAL], 1, seed + b"u=cbrt(t^2-7)", 64)
+    tuple_expand(inputs, [ZERO_VAL, RANDOM_VAL], 2, seed + b"u=0", 64)
+    tuple_expand(inputs, [RANDOM_VAL, ZERO_VAL], 2, seed + b"t=0", 64)
+    tuple_expand(inputs, [ZERO_VAL, FE(8).sqrts()], 3, seed + b"u=0;t=sqrt(8)")
+    tuple_expand(inputs, [FE(-8).cbrts(), ZERO_VAL], 3, seed + b"t=0;u=cbrt(-8)")
+    tuple_expand(inputs, [FE(-6).cbrts(), ZERO_VAL], 3, seed + b"t=0;u=cbrt(-6)")
+    tuple_expand(inputs, [ZERO_VAL, ZERO_VAL], 3, seed + b"u=0;t=0")
+    # Unused.
+    tuple_expand(inputs, [ZERO_VAL, FE(-8).sqrts()], 4, seed + b"u=0;t=sqrt(-8)")
+
+    seen = set()
+    cases = []
+    for _prio, _cnt, _hash, vs, _seed in sorted(inputs):
+        inp = int(vs[0]).to_bytes(32, 'big') + int(vs[1]).to_bytes(32, 'big')
+        outp, flags = ellswift_decode_flagged(inp, simplified)
+        if flags not in seen:
+            cases.append((inp, outp, flags))
+            seen.add(flags)
+
+    return cases
+
+def gen_all_ellswift_decode_vectors(fil):
+    """Generate all xelligatorswift decoding test vectors."""
+
+    cases = gen_ellswift_decode_cases(b"")
+    writer = csv.DictWriter(fil, ["ellswift", "x", "comment"])
+    writer.writeheader()
+    for val, x, flags in sorted(cases):
+        writer.writerow({"ellswift": val.hex(), "x": x.hex(), "comment": flags})
+
+def xswiftec_inv_flagged(x, u, case):
+    """A variant of xswiftec_inv which also returns flags, describing conditions encountered."""
+
+    flags = []
+
+    if case & 2 == 0:
+        if GE.is_valid_x(-x - u):
+            flags.append("bad[valid_x(-x-u)]")
+            return None, flags
+        v = x if case & 1 == 0 else -x - u
+        if v == 0:
+            flags.append("info[v=0]")
+        s = -(u**3 + 7) / (u**2 + u*v + v**2)
+        assert s != 0 # would imply X=0 on curve
+    else:
+        s = x - u
+        if s == 0:
+            flags.append("bad[s=0]")
+            return None, flags
+        q = (-s * (4 * (u**3 + 7) + 3 * s * u**2))
+        if q == 0:
+            flags.append("info[q=0]")
+        r = q.sqrt()
+        if r is None:
+            flags.append("bad[non_square(q)]")
+            return None, flags
+        if case & 1:
+            if r == 0:
+                flags.append("bad[r=0]")
+                return None, flags
+            r = -r
+        v = (-u + r / s) / 2
+        if v == 0:
+            flags.append("info[v=0]")
+    w = s.sqrt()
+    assert w != 0
+    if w is None:
+        flags.append("bad[non_square(s)]")
+        return None, flags
+    if case & 4:
+        w = -w
+    Y = w / 2
+    assert Y != 0
+    X = 2 * Y * (v + u / 2)
+    if X == 0:
+        flags.append("info[X=0]")
+    flags.append("ok")
+    return w * (u * (MINUS_3_SQRT - 1) / 2 - v), flags
+
+def xswiftec_inv_combo_flagged(x, u):
+    """Compute the aggregate results and flags from xswiftec_inv_flagged for case=0..7."""
+    ts = []
+    allflags = []
+    for case in range(8):
+        t, flags = xswiftec_inv_flagged(x, u, case)
+        if t is not None:
+            assert x == xswiftec(u, t)
+        ts.append(t)
+        allflags.append(f"case{case}:{'&'.join(flags)}")
+    return ts, ";".join(allflags)
+
+def gen_all_xswiftec_inv_vectors(fil):
+    """Generate all xswiftec_inv test vectors."""
+
+    # Two constants used below. Compute them only once.
+    C1 = (FE(MINUS_3_SQRT) - 1) / 2
+    C2 = (-FE(MINUS_3_SQRT) - 1) / 2
+    # Helper functions that pick x and u with special properties.
+    TRIGGER_Q_ZERO = fn_of(1, lambda u: (FE(u)**3 + 28) / (FE(-3) * FE(u)**2))
+    TRIGGER_DIVZERO_A = fn_of(1, lambda u: FE(u) * C1)
+    TRIGGER_DIVZERO_B = fn_of(1, lambda u: FE(u) * C2)
+    TRIGGER_V_ZERO = fn_of(1, lambda u: FE(-7) / FE(u)**2)
+    TRIGGER_X_ZERO = fn_of(0, lambda x: FE(-2) * FE(x))
+
+    inputs = []
+    tuple_expand(inputs, [random_fe_int, random_fe_int], 0, b"uniform", 256)
+    tuple_expand(inputs, [TRIGGER_Q_ZERO, random_fe_int], 1, b"x=-(u^3+28)/(3*u^2)", 64)
+    tuple_expand(inputs, [TRIGGER_V_ZERO, random_fe_int], 1, b"x=-7/u^2", 512)
+    tuple_expand(inputs, [random_fe_int, fn_of(0, lambda x: x)], 2, b"u=x", 64)
+    tuple_expand(inputs, [random_fe_int, fn_of(0, lambda x: -FE(x))], 2, b"u=-x", 64)
+    # Unused.
+    tuple_expand(inputs, [TRIGGER_DIVZERO_A, random_fe_int], 3, b"x=u*(sqrt(-3)-1)/2", 64)
+    tuple_expand(inputs, [TRIGGER_DIVZERO_B, random_fe_int], 3, b"x=u*(-sqrt(-3)-1)/2", 64)
+    tuple_expand(inputs, [random_fe_int, TRIGGER_X_ZERO], 3, b"u=-2x", 64)
+
+    seen = set()
+    cases = []
+    for _prio, _cnt, _hash, vs, _seed in sorted(inputs):
+        x, u = FE(vs[0]), FE(vs[1])
+        if u == 0:
+            continue
+        if not GE.is_valid_x(x):
+            continue
+        ts, flags = xswiftec_inv_combo_flagged(x, u)
+        if flags not in seen:
+            cases.append((int(u), int(x), ts, flags))
+            seen.add(flags)
+
+    writer = csv.DictWriter(fil, ["u", "x"] + [f"case{c}_t" for c in range(8)] + ["comment"])
+    writer.writeheader()
+    for u, x, ts, flags in sorted(cases):
+        row = {"u": FE(u), "x": FE(x), "comment": flags}
+        for c in range(8):
+            if ts[c] is not None:
+                row[f"case{c}_t"] = FE(ts[c])
+        writer.writerow(row)
+
+def gen_packet_encoding_vector(case):
+    """Given a dict case with specs, construct a packet_encoding test vector as a CSV line."""
+    ikm = str(case).encode('utf-8')
+    in_initiating = case["init"]
+    in_ignore = int(case["ignore"])
+    in_priv_ours, in_ellswift_ours = ellswift_create_deterministic(ikm, case["features"])
+    mid_x_ours = (int.from_bytes(in_priv_ours, 'big') * SECP256K1_G).x.to_bytes()
+    assert mid_x_ours == ellswift_decode(in_ellswift_ours)
+    in_ellswift_theirs = case["theirs"]
+    in_contents = hkdf_sha256(case["contentlen"], ikm, b"contents", b"")
+    contents = in_contents * case["multiply"]
+    in_aad = hkdf_sha256(case["aadlen"], ikm, b"aad", b"")
+    mid_shared_secret = v2_ecdh(in_priv_ours, in_ellswift_theirs, in_ellswift_ours, in_initiating)
+
+    peer = initialize_v2_transport(mid_shared_secret, in_initiating)
+    for _ in range(case["idx"]):
+        v2_enc_packet(peer, b"")
+    ciphertext = v2_enc_packet(peer, contents, in_aad, case["ignore"])
+    long_msg = len(ciphertext) > 128
+
+    return {
+        "in_idx": case['idx'],
+        "in_priv_ours": in_priv_ours.hex(),
+        "in_ellswift_ours": in_ellswift_ours.hex(),
+        "in_ellswift_theirs": in_ellswift_theirs.hex(),
+        "in_initiating": int(in_initiating),
+        "in_contents": in_contents.hex(),
+        "in_multiply": case['multiply'],
+        "in_aad": in_aad.hex(),
+        "in_ignore": in_ignore,
+        "mid_x_ours": mid_x_ours.hex(),
+        "mid_x_theirs": ellswift_decode(in_ellswift_theirs).hex(),
+        "mid_x_shared": ellswift_ecdh_xonly(in_ellswift_theirs, in_priv_ours).hex(),
+        "mid_shared_secret": mid_shared_secret.hex(),
+        "mid_initiator_l": peer['initiator_L'].hex(),
+        "mid_initiator_p": peer['initiator_P'].hex(),
+        "mid_responder_l": peer['responder_L'].hex(),
+        "mid_responder_p": peer['responder_P'].hex(),
+        "mid_send_garbage_terminator": peer["send_garbage_terminator"].hex(),
+        "mid_recv_garbage_terminator": peer["recv_garbage_terminator"].hex(),
+        "out_session_id": peer["session_id"].hex(),
+        "out_ciphertext": "" if long_msg else ciphertext.hex(),
+        "out_ciphertext_endswith": ciphertext[-128:].hex() if long_msg else ""
+    }
+
+def gen_all_packet_encoding_vectors(fil):
+    """Return a list of CSV lines, one for each packet encoding vector."""
+
+    ellswift = gen_ellswift_decode_cases(b"simplified_", simplified=True)
+    ellswift.sort(key=lambda x: hashlib.sha256(b"simplified:" + x[0]).digest())
+
+    fields = [
+        "in_idx", "in_priv_ours", "in_ellswift_ours", "in_ellswift_theirs", "in_initiating",
+        "in_contents", "in_multiply", "in_aad", "in_ignore", "mid_x_ours", "mid_x_theirs",
+        "mid_x_shared", "mid_shared_secret", "mid_initiator_l", "mid_initiator_p",
+        "mid_responder_l", "mid_responder_p", "mid_send_garbage_terminator",
+        "mid_recv_garbage_terminator", "out_session_id", "out_ciphertext", "out_ciphertext_endswith"
+    ]
+
+    writer = csv.DictWriter(fil, fields)
+    writer.writeheader()
+    for case in [
+        {"init": True, "contentlen": 1, "multiply": 1, "aadlen": 0, "ignore": False, "idx": 1,
+         "theirs": ellswift[0][0], "features": 0},
+        {"init": False, "contentlen": 17, "multiply": 1, "aadlen": 0, "ignore": False, "idx": 999,
+         "theirs": ellswift[1][0], "features": 1},
+        {"init": True, "contentlen": 63, "multiply": 1, "aadlen": 4095, "ignore": False, "idx": 0,
+         "theirs": ellswift[2][0], "features": 2},
+        {"init": False, "contentlen": 128, "multiply": 1, "aadlen": 0, "ignore": True, "idx": 223,
+         "theirs": ellswift[3][0], "features": 3},
+        {"init": True, "contentlen": 193, "multiply": 1, "aadlen": 0, "ignore": False, "idx": 448,
+         "theirs": ellswift[4][0], "features": 4},
+        {"init": False, "contentlen": 41, "multiply": 97561, "aadlen": 0, "ignore": False,
+         "idx": 673, "theirs": ellswift[5][0], "features": 5},
+        {"init": True, "contentlen": 241, "multiply": 69615, "aadlen": 0, "ignore": True,
+         "idx": 1024, "theirs": ellswift[6][0], "features": 6},
+    ]:
+        writer.writerow(gen_packet_encoding_vector(case))
+
+if __name__ == "__main__":
+    print(f"Generating {FILENAME_PACKET_TEST}...")
+    with open(FILENAME_PACKET_TEST, "w", encoding="utf-8") as fil_packet:
+        gen_all_packet_encoding_vectors(fil_packet)
+    print(f"Generating {FILENAME_XSWIFTEC_INV_TEST}...")
+    with open(FILENAME_XSWIFTEC_INV_TEST, "w", encoding="utf-8") as fil_xswiftec_inv:
+        gen_all_xswiftec_inv_vectors(fil_xswiftec_inv)
+    print(f"Generating {FILENAME_ELLSWIFT_DECODE_TEST}...")
+    with open(FILENAME_ELLSWIFT_DECODE_TEST, "w", encoding="utf-8") as fil_ellswift_decode:
+        gen_all_ellswift_decode_vectors(fil_ellswift_decode)
@@ -1,8 +1,8 @@
-in_idx,in_priv_ours,in_ellswift_ours,in_ellswift_theirs,in_initiating,in_content,in_multiply,in_aad,in_ignore,mid_x_ours,mid_x_shared,mid_shared_secret,mid_initiator_l,mid_initiator_p,mid_responder_l,mid_responder_p,mid_send_garbage_terminator,mid_recv_garbage_terminator,mid_session_id,out_ciphertext,out_ciphertext_endswith
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Line truncated
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+448,a6ec25127ca1aa4cf16b20084ba1e6516baae4d32422288e9b36d8bddd2de35a,ffffffffffffffffffffffffffffffffffffffffffffffffffffffff053d7ecca53e33e185a8b9be4e7699a97c6ff4c795522e5918ab7cd6b6884f67e683f3dc,ffffffffffffffffffffffffffffffffffffffffffffffffffffffffa7730be30000000000000000000000000000000000000000000000000000000000000000,1,00cf68f8f7ac49ffaa02c4864fdf6dfe7bbf2c740b88d98c50ebafe32c92f3427f57601ffcb21a3435979287db8fee6c302926741f9d5e464c647eeb9b7acaeda46e00abd7506fc9a719847e9a7328215801e96198dac141a15c7c2f68e0690dd1176292a0dded04d1f548aad88f1aebdc0a8f87da4bb22df32dd7c160c225b843e83f6525d6d484f502f16d923124fc538794e21da2eb689d18d87406ecced5b9f92137239ed1d37bcfa7836641a83cf5e0a1cf63f51b06f158e499a459ede41c,1,,0,02b225089255f7b02b20276cfe9779144df8fb1957b477bff3239d802d1256e9,5232c4b6bde9d3d45d7b763ebd7495399bb825cc21de51011761cd81a51bdc84,379223d2f1ea7f8a22043c4ce4122623098309e15b1ce58286ebe3d3bf40f4e1,dd210aa6629f20bb328e5d89daa6eb2ac3d1c658a725536ff154f31b536c23b2,393472f85a5cc6b0f02c4bd466db7a2dc5b91fc9dcb15c0dd6dc21116ece8bca,c80b87b793db47320b2795db66d331bd3021cc24e360d59d0fa8974f54687e0c,ef16a43d77e2b270b0a145ee1618d35f3c943cc7877d6cfcff2287d41692be39,20d4b62e2d982c61bb0cc39a93283d98af36530ef12331d44b2477b0e521b490,fead69be77825a23daec377c362aa560,511d4980526c5e64aa7187462faeafdd,acb8f084ea763ddd1b92ac4ed23bf44de20b84ab677d4e4e6666a6090d40353d,,77b4656934a82de1a593d8481f020194ddafd8cac441f9d72aeb8721e6a14f49698ca6d9b2b6d59d07a01aa552fd4d5b68d0d1617574c77dea10bfadbaa31b83885b7ceac2fd45e3e4a331c51a74e7b1698d81b64c87c73c5b9258b4d83297f9debc2e9aa07f8572ff434dc792b83ecf07b3197de8dc9cf7be56acb59c66cff5
+673,0af952659ed76f80f585966b95ab6e6fd68654672827878684c8b547b1b94f5a,ffffffffffffffffffffffffffffffffffffffffffffffffffffffffc81017fd92fd31637c26c906b42092e11cc0d3afae8d9019d2578af22735ce7bc469c72d,9652d78baefc028cd37a6a92625b8b8f85fde1e4c944ad3f20e198bef8c02f19fffffffffffffffffffffffffffffffffffffffffffffffffffffffff2e91870,0,5c6272ee55da855bbbf7b1246d9885aa7aa601a715ab86fa46c50da533badf82b97597c968293ae04e,97561,,0,4b1767466fe2fb8deddf2dc52cc19c7e2032007e19bfb420b30a80152d0f22d6,64c383e0e78ac99476ddff2061683eeefa505e3666673a1371342c3e6c26981d,5bcfeac98d87e87e158bf839f1269705429f7af2a25b566a25811b5f9aef9560,3568f2aea2e14ef4ee4a3c2a8b8d31bc5e3187ba86db10739b4ff8ec92ff6655,c7df866a62b7d404eb530b2be245a7aece0fb4791402a1de8f33530cbf777cc1,8f732e4aae2ba9314e0982492fa47954de9c189d92fbc549763b27b1b47642ce,992085edfecb92c62a3a7f96ea416f853f34d0dfe065b966b6968b8b87a83081,c5ba5eaf9e1c807154ebab3ea472499e815a7be56dfaf0c201cf6e91ffeca8e6,5e2375ac629b8df1e4ff3617c6255a70,70bcbffcb62e4d29d2605d30bceef137,7332e92a3f9d2792c4d444fac5ed888c39a073043a65eefb626318fd649328f8,,657a4a19711ce593c3844cb391b224f60124aba7e04266233bc50cafb971e26c7716b76e98376448f7d214dd11e629ef9a974d60e3770a695810a61c4ba66d78b936ee7892b98f0b48ddae9fcd8b599dca1c9b43e9b95e0226cf8d4459b8a7c2c4e6db80f1d58c7b20dd7208fa5c1057fb78734223ee801dbd851db601fee61e
+1024,f90e080c64b05824c5a24b2501d5aeaf08af3872ee860aa80bdcd430f7b63494,ffffffffffffffffffffffffffffffffffffffffffffffffffffffff115173765dc202cf029ad3f15479735d57697af12b0131dd21430d5772e4ef11474d58b9,12a50f3fafea7c1eeada4cf8d33777704b77361453afc83bda91eef349ae044d20126c6200547ea5a6911776c05dee2a7f1a9ba7dfbabbbd273c3ef29ef46e46,1,5f67d15d22ca9b2804eeab0a66f7f8e3a10fa5de5809a046084348cbc5304e843ef96f59a59c7d7fdfe5946489f3ea297d941bac326225df316a25fc90f0e65b0d31a9c497e960fdbf8c482516bc8a9c1c77b7f6d0e1143810c737f76f9224e6f2c9af5186b4f7259c7e8d165b6e4fe3d38a60bdbdd4d06ecdcaaf62086070dbb68686b802d53dfd7db14b18743832605f5461ad81e2af4b7e8ff0eff0867a25b93cec7becf15c43131895fed09a83bf1ee4a87d44dd0f02a837bf5a1232e201cb882734eb9643dc2dc4d4e8b5690840766212c7ac8f38ad8a9ec47c7a9b3e022ae3eb6a32522128b518bd0d0085dd81c5,69615,,1,8b8de966150bf872b4b695c9983df519c909811954d5d76e99ed0d5f1860247b,eef379db9bd4b1aa90fc347fad33f7d53083389e22e971036f59f4e29d325ac2,0a402d812314646ccc2565c315d1429ec1ed130ff92ff3f48d948f29c3762cf1,e25461fb0e4c162e18123ecde88342d54d449631e9b75a266fd9260c2bb2f41d,97771ce2ce17a25c3d65bf9f8e4acb830dce8d41392be3e4b8ed902a3106681a,2e7022b4eae9152942f68160a93e25d3e197a557385594aa587cb5e431bb470d,613f85a82d783ce450cfd7e91a027fcc4ad5610872f83e4dbe9e2202184c6d6e,cb5de4ed1083222e381401cf88e3167796bc9ab5b8aa1f27b718f39d1e6c0e87,b709dea25e0be287c50e3603482c2e98,1f677e9d7392ebe3633fd82c9efb0f16,889f339285564fd868401fac8380bb9887925122ec8f31c8ae51ce067def103b,,7c4b9e1e6c1ce69da7b01513cdc4588fd93b04dafefaf87f31561763d906c672bac3dfceb751ebd126728ac017d4d580e931b8e5c7d5dfe0123be4dc9b2d2238b655c8a7fadaf8082c31e310909b5b731efc12f0a56e849eae6bfeedcc86dd27ef9b91d159256aa8e8d2b71a311f73350863d70f18d0d7302cf551e4303c7733
@@ -1,3 +1,5 @@
+"""Reference implementation for the cryptographic aspects of BIP-324"""
+
 import sys
 import random
 import hashlib
@@ -70,7 +72,7 @@ class FE:
                self.den = (a.den * b.num) % FE.SIZE
            else:
                self.num = (a * b.den) % FE.SIZE
-                self.den = a.num
+                self.den = b.num
        else:
            b = b % FE.SIZE
            assert b != 0
@@ -85,8 +87,7 @@ class FE:
        """Compute the sum of two field elements (second may be int)."""
        if isinstance(a, FE):
            return FE(self.num * a.den + self.den * a.num, self.den * a.den)
-        else:
-            return FE(self.num + self.den * a, self.den)
+        return FE(self.num + self.den * a, self.den)

    def __radd__(self, a):
        """Compute the sum of an integer and a field element."""
@@ -96,8 +97,7 @@ class FE:
        """Compute the difference of two field elements (second may be int)."""
        if isinstance(a, FE):
            return FE(self.num * a.den - self.den * a.num, self.den * a.den)
-        else:
-            return FE(self.num - self.den * a, self.den)
+        return FE(self.num - self.den * a, self.den)

    def __rsub__(self, a):
        """Compute the difference between an integer and a field element."""
@@ -107,8 +107,7 @@ class FE:
        """Compute the product of two field elements (second may be int)."""
        if isinstance(a, FE):
            return FE(self.num * a.num, self.den * a.den)
-        else:
-            return FE(self.num * a, self.den)
+        return FE(self.num * a, self.den)

    def __rmul__(self, a):
        """Compute the product of an integer with a field element."""
@@ -140,15 +139,57 @@ class FE:
    def sqrt(self):
        """Compute the square root of a field element.

-        Due to the fact that our modulus is of the form (p % 4) == 3, the Tonelli-Shanks
-        algorithm (https://en.wikipedia.org/wiki/Tonelli-Shanks_algorithm) is simply
-        raising the argument to the power (p + 3) / 4."""
+        Due to the fact that our modulus p is of the form p = 3 (mod 4), the
+        Tonelli-Shanks algorithm (https://en.wikipedia.org/wiki/Tonelli-Shanks_algorithm)
+        is simply raising the argument to the power (p + 1) / 4.
+
+        To see why: p-1 = 0 (mod 2), so 2 divides the order of the multiplicative group,
+        and thus only half of the non-zero field elements are squares. An element a is
+        a (nonzero) square when Euler's criterion, a^((p-1)/2) = 1 (mod p), holds. We're
+        looking for x such that x^2 = a (mod p). Given a^((p-1)/2) = 1 (mod p), that is
+        equivalent to x^2 = a^(1 + (p-1)/2) (mod p). As (1 + (p-1)/2) is even, this is
+        equivalent to x = a^((1 + (p-1)/2)/2) (mod p), or x = a^((p+1)/4) (mod p)."""
        v = int(self)
        s = pow(v, (FE.SIZE + 1) // 4, FE.SIZE)
        if s**2 % FE.SIZE == v:
            return FE(s)
        return None

+    def sqrts(self):
+        """Compute all square roots of a field element, if any."""
+        s = self.sqrt()
+        if s is None:
+            return []
+        return [FE(s), -FE(s)]
+
+    # The cube roots of 1 (mod p).
+    CBRT1 = [
+        1,
+        0x851695d49a83f8ef919bb86153cbcb16630fb68aed0a766a3ec693d68e6afa40,
+        0x7ae96a2b657c07106e64479eac3434e99cf0497512f58995c1396c28719501ee
+    ]
+
+
+    def cbrts(self):
+        """Compute all cube roots of a field element, if any.
+
+        Due to the fact that our modulus p is of the form p = 7 (mod 9), one cube root
+        can always be computed by raising to the power (p + 2) / 9. The other roots
+        (if any) can be found by multiplying with the two non-trivial cube roots of 1.
+
+        To see why: p-1 = 0 (mod 3), so 3 divides the order of the multiplicative group,
+        and thus only 1/3 of the non-zero field elements are cubes. An element a is a
+        (nonzero) cube when a^((p-1)/3) = 1 (mod p). We're looking for x such that
+        x^3 = a (mod p). Given a^((p-1)/3) = 1 (mod p), that is equivalent to
+        x^3 = a^(1 + (p-1)/3) (mod p). As (1 + (p-1)/3) is a multiple of 3, this is
+        equivalent to x = a^((1 + (p-1)/3)/3) (mod p), or x = a^((p+2)/9) (mod p)."""
+        v = int(self)
+        c = pow(v, (FE.SIZE + 2) // 9, FE.SIZE)
+
+        if pow(c, 3, FE.SIZE) == v:
+            return [FE(c * f) for f in FE.CBRT1]
+        return []
+
    def is_square(self):
        """Determine if this field element has a square root."""
        # Compute the Jacobi symbol of (self / p). Since our modulus is prime, this
@@ -161,7 +202,7 @@ class FE:
            while n & 1 == 0:
                n >>= 1
                r = k & 7
-                t ^= (r == 3 or r == 5)
+                t ^= (r in (3, 5))
            n, k = k, n
            t ^= (n & k & 3 == 3)
            n = n % k
@@ -172,8 +213,7 @@ class FE:
        """Check whether two field elements are equal (second may be an int)."""
        if isinstance(a, FE):
            return (self.num * a.den - self.den * a.num) % FE.SIZE == 0
-        else:
-            return (self.num - self.den * a) % FE.SIZE == 0
+        return (self.num - self.den * a) % FE.SIZE == 0

    def to_bytes(self):
        """Convert a field element to 32-byte big endian encoding."""
@@ -187,6 +227,16 @@ class FE:
            return None
        return FE(v)

+    def __str__(self):
+        """Convert this field element to a string."""
+        return f"{int(self):064x}"
+
+    def __repr__(self):
+        """Get a string representation of this field element."""
+        return f"FE(0x{int(self):x})"
+
+assert all(pow(c, 3, FE.SIZE) == 1 for c in FE.CBRT1)
+
 class GE:
    """Objects of this class represent points (group elements) on the secp256k1 curve.

@@ -221,12 +271,11 @@ class GE:
            x3 = l**2 - self.x - a.x
            y3 = l * (self.x - x3) - self.y
            return GE(x3, y3)
-        elif self.y == a.y:
+        if self.y == a.y:
            # Adding point to itself
            return self.double()
-        else:
-            # Adding point to its negation
-            return None
+        # Adding point to its negation
+        return None

    def __radd__(self, a):
        """Add infinity to a point."""
@@ -260,13 +309,21 @@ class GE:
        """Determine whether the provided field element is a valid X coordinate."""
        return (FE(x)**3 + 7).is_square()

+    def __str__(self):
+        """Convert this group element to a string."""
+        return f"({self.x},{self.y})"
+
+    def __repr__(self):
+        """Get a string representation for this group element."""
+        return f"GE(0x{int(self.x)},0x{int(self.y)})"
+
 SECP256K1_G = GE(
    0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798,
    0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8)

 ### ElligatorSwift

-# Precomputed constant square root of -3 modulo p.
+# Precomputed constant square root of -3 (mod p).
 MINUS_3_SQRT = FE(-3).sqrt()

 def xswiftec(u, t):
@@ -292,7 +349,7 @@ def xswiftec_inv(x, u, case):
    if case & 2 == 0:
        if GE.is_valid_x(-x - u):
            return None
-        v = x if case & 1 == 0 else -x - u
+        v = x
        s = -(u**3 + 7) / (u**2 + u*v + v**2)
    else:
        s = x - u
@@ -301,17 +358,16 @@ def xswiftec_inv(x, u, case):
        r = (-s * (4 * (u**3 + 7) + 3 * s * u**2)).sqrt()
        if r is None:
            return None
-        if case & 1:
-            if r == 0:
-                return None
-            r = -r
+        if case & 1 and r == 0:
+            return None
        v = (-u + r / s) / 2
    w = s.sqrt()
    if w is None:
        return None
-    if case & 4:
-        w = -w
-    return w * (u * (MINUS_3_SQRT - 1) / 2 - v)
+    if case & 5 == 0: return -w * (u * (1 - MINUS_3_SQRT) / 2 + v)
+    if case & 5 == 1: return  w * (u * (1 + MINUS_3_SQRT) / 2 + v)
+    if case & 5 == 4: return  w * (u * (1 - MINUS_3_SQRT) / 2 + v)
+    if case & 5 == 5: return -w * (u * (1 + MINUS_3_SQRT) / 2 + v)

 def xelligatorswift(x):
    """Given a field element X on the curve, find (u, t) that encode them."""
@@ -328,12 +384,17 @@ def ellswift_create():
    u, t = xelligatorswift((priv * SECP256K1_G).x)
    return priv.to_bytes(32, 'big'), u.to_bytes() + t.to_bytes()

+def ellswift_decode(ellswift):
+    """Convert ellswift encoded X coordinate to 32-byte xonly format."""
+    u = FE(int.from_bytes(ellswift[:32], 'big'))
+    t = FE(int.from_bytes(ellswift[32:], 'big'))
+    return xswiftec(u, t).to_bytes()
+
 def ellswift_ecdh_xonly(pubkey_theirs, privkey):
    """Compute X coordinate of shared ECDH point between elswift pubkey and privkey."""
-    u = FE(int.from_bytes(pubkey_theirs[:32], 'big'))
-    t = FE(int.from_bytes(pubkey_theirs[32:], 'big'))
    d = int.from_bytes(privkey, 'big')
-    return (d * GE.lift_x(xswiftec(u, t))).x.to_bytes()
+    pub = ellswift_decode(pubkey_theirs)
+    return (d * GE.lift_x(FE.from_bytes(pub))).x.to_bytes()

 ### Poly1305

@@ -402,7 +463,7 @@ def chacha20_block(key, nonce, cnt):
    for i in range(3):
        init[13 + i] = int.from_bytes(nonce[4 * i:4 * (i+1)], 'little')
    # Perform 20 rounds.
-    state = [v for v in init]
+    state = list(init)
    for _ in range(10):
        chacha20_doubleround(state)
    # Add initial values back into state.
@@ -459,6 +520,7 @@ class FSChaCha20Poly1305:
        self.packet_counter = 0

    def crypt(self, aad, text, is_decrypt):
+        """Encrypt or decrypt the specified (plain/cipher)text."""
        nonce = ((self.packet_counter % REKEY_INTERVAL).to_bytes(4, 'little') +
                 (self.packet_counter // REKEY_INTERVAL).to_bytes(8, 'little'))
        if is_decrypt:
@@ -474,12 +536,14 @@ class FSChaCha20Poly1305:
        self.packet_counter += 1
        return ret

-    def decrypt(self, aad, ciphertext):
-        return self.crypt(aad, ciphertext, True)
-
    def encrypt(self, aad, plaintext):
+        """Encrypt the specified plaintext with provided AAD."""
        return self.crypt(aad, plaintext, False)

+    def decrypt(self, aad, ciphertext):
+        """Decrypt the specified ciphertext with provided AAD."""
+        return self.crypt(aad, ciphertext, True)
+

 class FSChaCha20:
    """Rekeying wrapper stream cipher around ChaCha20."""
@@ -491,6 +555,7 @@ class FSChaCha20:
        self.keystream = b''

    def get_keystream_bytes(self, nbytes):
+        """Generate nbytes keystream bytes."""
        while len(self.keystream) < nbytes:
            nonce = ((0).to_bytes(4, 'little') +
                     (self.chunk_counter // REKEY_INTERVAL).to_bytes(8, 'little'))
@@ -501,6 +566,7 @@ class FSChaCha20:
        return ret

    def crypt(self, chunk):
+        """Encrypt or decypt chunk."""
        ks = self.get_keystream_bytes(len(chunk))
        ret = bytes([ks[i] ^ chunk[i] for i in range(len(chunk))])
        if ((self.chunk_counter + 1) % REKEY_INTERVAL) == 0:
@@ -509,6 +575,15 @@ class FSChaCha20:
        self.chunk_counter += 1
        return ret

+    def encrypt(self, chunk):
+        """Encrypt chunk."""
+        return self.crypt(chunk)
+
+    def decrypt(self, chunk):
+        """Decrypt chunk."""
+        return self.crypt(chunk)
+
+
 ### Shared secret computation

 def v2_ecdh(priv, ellswift_theirs, ellswift_ours, initiating):
@@ -519,10 +594,9 @@ def v2_ecdh(priv, ellswift_theirs, ellswift_ours, initiating):
        # Initiating, place our public key encoding first.
        return TaggedHash("bip324_ellswift_xonly_ecdh",
            ellswift_ours + ellswift_theirs + ecdh_point_x32)
-    else:
-        # Responding, place their public key encoding first.
-        return TaggedHash("bip324_ellswift_xonly_ecdh",
-            ellswift_theirs + ellswift_ours + ecdh_point_x32)
+    # Responding, place their public key encoding first.
+    return TaggedHash("bip324_ellswift_xonly_ecdh",
+        ellswift_theirs + ellswift_ours + ecdh_point_x32)

 ### Key derivation

@@ -571,5 +645,5 @@ def v2_enc_packet(peer, contents, aad=b'', ignore=False):
    header = (ignore << IGNORE_BIT_POS).to_bytes(HEADER_LEN, 'little')
    plaintext = header + contents
    aead_ciphertext = peer['send_P'].encrypt(aad, plaintext)
-    enc_plaintext_len = peer['send_L'].crypt(len(contents).to_bytes(LENGTH_FIELD_LEN, 'little'))
+    enc_plaintext_len = peer['send_L'].encrypt(len(contents).to_bytes(LENGTH_FIELD_LEN, 'little'))
    return enc_plaintext_len + aead_ciphertext
@@ -1,53 +1,69 @@
+"""Run the BIP-324 test vectors."""
+
 import csv
 import os
 import sys

 import reference

-with open(os.path.join(sys.path[0], 'packet_encoding_test_vectors.csv'), newline='') as csvfile:
-    reader = csv.reader(csvfile)
-    reader.__next__()
+FILENAME_PACKET_TEST = os.path.join(sys.path[0], 'packet_encoding_test_vectors.csv')
+FILENAME_XSWIFTEC_INV_TEST = os.path.join(sys.path[0], 'xswiftec_inv_test_vectors.csv')
+FILENAME_ELLSWIFT_DECODE_TEST = os.path.join(sys.path[0], 'ellswift_decode_test_vectors.csv')
+
+with open(FILENAME_PACKET_TEST, newline='', encoding='utf-8') as csvfile:
+    print(f"Running {FILENAME_PACKET_TEST} tests...")
+    reader = csv.DictReader(csvfile)
    for row in reader:
-        in_idx, in_priv_ours, in_ellswift_ours, in_ellswift_theirs, in_initiating, in_content, in_multiply, in_aad, in_ignore, mid_x_ours, mid_x_shared, mid_shared_secret, mid_initiator_l, mid_initiator_p, mid_responder_l, mid_responder_p, mid_send_garbage_terminator, mid_recv_garbage_terminator, mid_session_id, out_ciphertext, out_ciphertext_endswith = row
+        in_initiating = int(row['in_initiating'])
+        bytes_priv_ours = bytes.fromhex(row['in_priv_ours'])
+        int_priv_ours = int.from_bytes(bytes_priv_ours, 'big')
+        assert row['mid_x_ours'] == (int_priv_ours * reference.SECP256K1_G).x.to_bytes().hex()
+        bytes_ellswift_ours = bytes.fromhex(row['in_ellswift_ours'])
+        assert row['mid_x_ours'] == reference.ellswift_decode(bytes_ellswift_ours).hex()
+        bytes_ellswift_theirs = bytes.fromhex(row['in_ellswift_theirs'])
+        assert row['mid_x_theirs'] == reference.ellswift_decode(bytes_ellswift_theirs).hex()
+        x_shared = reference.ellswift_ecdh_xonly(bytes_ellswift_theirs, bytes_priv_ours)
+        assert row['mid_x_shared'] == x_shared.hex()
+        shared_secret = reference.v2_ecdh(bytes_priv_ours, bytes_ellswift_theirs,
+            bytes_ellswift_ours, in_initiating)
+        assert row['mid_shared_secret'] == shared_secret.hex()

-        assert mid_x_ours == (int.from_bytes(bytes.fromhex(in_priv_ours), 'big') * reference.SECP256K1_G).x.to_bytes().hex()
-        assert mid_x_shared == reference.ellswift_ecdh_xonly(bytes.fromhex(in_ellswift_theirs), bytes.fromhex(in_priv_ours)).hex()
-        assert mid_shared_secret == reference.v2_ecdh(bytes.fromhex(in_priv_ours), bytes.fromhex(in_ellswift_theirs), bytes.fromhex(in_ellswift_ours), int(in_initiating)).hex()
-
-        peer = reference.initialize_v2_transport(bytes.fromhex(mid_shared_secret), int(in_initiating))
-        assert mid_initiator_l == peer['initiator_L'].hex()
-        assert mid_initiator_p == peer['initiator_P'].hex()
-        assert mid_responder_l == peer['responder_L'].hex()
-        assert mid_responder_p == peer['responder_P'].hex()
-        assert mid_send_garbage_terminator == peer['send_garbage_terminator'].hex()
-        assert mid_recv_garbage_terminator == peer['recv_garbage_terminator'].hex()
-        assert mid_session_id == peer['session_id'].hex()
-        for _ in range(int(in_idx)):
+        peer = reference.initialize_v2_transport(shared_secret, in_initiating)
+        assert row['mid_initiator_l'] == peer['initiator_L'].hex()
+        assert row['mid_initiator_p'] == peer['initiator_P'].hex()
+        assert row['mid_responder_l'] == peer['responder_L'].hex()
+        assert row['mid_responder_p'] == peer['responder_P'].hex()
+        assert row['mid_send_garbage_terminator'] == peer['send_garbage_terminator'].hex()
+        assert row['mid_recv_garbage_terminator'] == peer['recv_garbage_terminator'].hex()
+        assert row['out_session_id'] == peer['session_id'].hex()
+        for _ in range(int(row['in_idx'])):
            reference.v2_enc_packet(peer, b"")
-        ciphertext = reference.v2_enc_packet(peer, bytes.fromhex(in_content) * int(in_multiply), bytes.fromhex(in_aad), int(in_ignore))
-        if len(out_ciphertext):
-            assert out_ciphertext == ciphertext.hex()
-        if len(out_ciphertext_endswith):
-            assert ciphertext.hex().endswith(out_ciphertext_endswith)
+        ciphertext = reference.v2_enc_packet(
+            peer,
+            bytes.fromhex(row['in_contents']) * int(row['in_multiply']),
+            bytes.fromhex(row['in_aad']), int(row['in_ignore']))
+        if len(row['out_ciphertext']):
+            assert row['out_ciphertext'] == ciphertext.hex()
+        if len(row['out_ciphertext_endswith']):
+            assert ciphertext.hex().endswith(row['out_ciphertext_endswith'])

-with open(os.path.join(sys.path[0], 'xswiftec_test_vectors.csv'), newline='') as csvfile:
-    reader = csv.reader(csvfile)
-    reader.__next__()
+with open(FILENAME_XSWIFTEC_INV_TEST, newline='', encoding='utf-8') as csvfile:
+    print(f"Running {FILENAME_XSWIFTEC_INV_TEST} tests...")
+    reader = csv.DictReader(csvfile)
    for row in reader:
-        u = reference.FE.from_bytes(bytes.fromhex(row[0]))
-        x = reference.FE.from_bytes(bytes.fromhex(row[1]))
+        u = reference.FE.from_bytes(bytes.fromhex(row['u']))
+        x = reference.FE.from_bytes(bytes.fromhex(row['x']))
        for case in range(8):
            ret = reference.xswiftec_inv(x, u, case)
            if ret is None:
-                assert row[2 + case] == ""
+                assert row[f"case{case}_t"] == ""
            else:
-                assert row[2 + case] == ret.to_bytes().hex()
+                assert row[f"case{case}_t"] == ret.to_bytes().hex()
                assert reference.xswiftec(u, ret) == x

-with open(os.path.join(sys.path[0], 'xelligatorswift_test_vectors.csv'), newline='') as csvfile:
-    reader = csv.reader(csvfile)
-    reader.__next__()
+with open(FILENAME_ELLSWIFT_DECODE_TEST, newline='', encoding='utf-8') as csvfile:
+    print(f"Running {FILENAME_ELLSWIFT_DECODE_TEST} tests...")
+    reader = csv.DictReader(csvfile)
    for row in reader:
-        ellswift = bytes.fromhex(row[0])
-        x = bytes.fromhex(row[1])
-        assert reference.ellswift_ecdh_xonly(ellswift, (1).to_bytes(32, 'big')) == x
+        ellswift = bytes.fromhex(row['ellswift'])
+        assert reference.ellswift_decode(ellswift).hex() == row['x']
@@ -0,0 +1,52 @@
+"""Convert the BIP-324 test vectors to secp256k1 code."""
+
+import csv
+import reference
+import os
+import sys
+
+FILENAME_XSWIFTEC_INV_TEST = os.path.join(sys.path[0], 'xswiftec_inv_test_vectors.csv')
+FILENAME_ELLSWIFT_DECODE_TEST = os.path.join(sys.path[0], 'ellswift_decode_test_vectors.csv')
+
+def format_int(v):
+    """Format 0 as "0", but other integers as 0x%08x."""
+    if v == 0:
+        return "0"
+    return f"0x{v:08x}"
+
+def format_fe(fe):
+    """Format a field element constant as SECP256K1_FE_CONST code."""
+    vals = [(int(fe) >> (32 * (7 - i))) & 0xffffffff for i in range(8)]
+    strs = ", ".join(format_int(v) for v in vals)
+    return f"SECP256K1_FE_CONST({strs})"
+
+def output_xswiftec_inv_cases():
+    """Generate lines corresponding to the xswiftec_inv test cases."""
+    with open(FILENAME_XSWIFTEC_INV_TEST, newline='', encoding='utf-8') as csvfile:
+        reader = csv.DictReader(csvfile)
+        print("xswiftec_inv cases:")
+        for row in reader:
+            u = int.from_bytes(bytes.fromhex(row['u']), 'big')
+            x = int.from_bytes(bytes.fromhex(row['x']), 'big')
+            pat = sum(1<<c for c in range(8) if row[f"case{c}_t"])
+            tstrs = []
+            for c in range(8):
+                tstrs.append(format_fe(int.from_bytes(bytes.fromhex(row[f"case{c}_t"]), 'big')))
+            print(f"    {{0x{pat:02x}, {format_fe(u)}, {format_fe(x)}, {{{', '.join(tstrs)}}}}},")
+    print()
+
+def output_ellswift_decode_cases():
+    """Generate lines corresponding to the ellswift_decode test cases."""
+    with open(FILENAME_ELLSWIFT_DECODE_TEST, newline='', encoding='utf-8') as csvfile:
+        reader = csv.DictReader(csvfile)
+        print("ellswift_decode cases:")
+        for row in reader:
+            enc = bytes.fromhex(row['ellswift'])
+            tval = int.from_bytes(enc[32:], 'big') % reference.FE.SIZE
+            x = int.from_bytes(bytes.fromhex(row['x']), 'big')
+            encstr = ", ".join(f"0x{b:02x}" for b in enc)
+            print(f"    {{{{{encstr}}}, {format_fe(x)}, {tval & 1}}},")
+    print()
+
+output_xswiftec_inv_cases()
+output_ellswift_decode_cases()
@@ -0,0 +1,78 @@
+"""Compare ellswift decoding in the BIP-324 test vectors against the SwiftEC reference code.
+
+Instructions:
+
+* Clone the SwiftEC repository, and enter the directory:
+
+  git clone https://github.com/Jchavezsaab/SwiftEC
+  git checkout 5320a25035d91addde29d14164cce684b56a12ed
+  cd SwiftEC
+
+* Generate parameters for the secp256k1 curve:
+
+  sage --python generate_parameters.py -p secp256k1
+
+* Copy over this file and the CSV test vectors:
+
+  cp PATH_TO_BIPS_REPO/bips/bip-0324/{*.csv,test_sage_decoding.py} .
+
+* Run the tests:
+
+  sage --python test_sage_decoding.py -p secp256k1
+
+No output = good.
+"""
+
+import sys
+import csv
+from config import F
+from Xencoding_0 import Xdecode
+
+
+FILENAME_PACKET_TEST = 'packet_encoding_test_vectors.csv'
+FILENAME_XSWIFTEC_INV_TEST = 'xswiftec_inv_test_vectors.csv'
+FILENAME_ELLSWIFT_DECODE_TEST = 'ellswift_decode_test_vectors.csv'
+
+def ellswift_decode_sage(ellswift):
+    """Given a 64-byte ellswift encoded public key, get the 32-byte X coordinate."""
+
+    u = F(int.from_bytes(ellswift[:32], 'big'))
+    t = F(int.from_bytes(ellswift[32:], 'big'))
+
+    # Reimplement the input correction step.
+    if u == F(0):
+        u = F(1)
+    if t == F(0):
+        t = F(1)
+    if u**3 + t**2 + 7 == F(0):
+        t = F(2) * t
+
+    # Invoke reference code
+    x, z = Xdecode(u, t)
+
+    # Convert to bytes.
+    return int(x / z).to_bytes(32, 'big')
+
+with open(FILENAME_PACKET_TEST, newline='', encoding='utf-8') as csvfile:
+    reader = csv.DictReader(csvfile)
+    for row in reader:
+        bytes_ellswift_ours = bytes.fromhex(row['in_ellswift_ours'])
+        bytes_ellswift_theirs = bytes.fromhex(row['in_ellswift_theirs'])
+        assert row['mid_x_ours'] == ellswift_decode_sage(bytes_ellswift_ours).hex()
+        assert row['mid_x_theirs'] == ellswift_decode_sage(bytes_ellswift_theirs).hex()
+
+with open(FILENAME_XSWIFTEC_INV_TEST, newline='', encoding='utf-8') as csvfile:
+    reader = csv.DictReader(csvfile)
+    for row in reader:
+        udat = bytes.fromhex(row['u'])
+        xdat = bytes.fromhex(row['x'])
+        for case in range(8):
+            tdat = bytes.fromhex(row[f"case{case}_t"])
+            if tdat:
+                assert ellswift_decode_sage(udat + tdat) == xdat
+
+with open(FILENAME_ELLSWIFT_DECODE_TEST, newline='', encoding='utf-8') as csvfile:
+    reader = csv.DictReader(csvfile)
+    for row in reader:
+        ellswift = bytes.fromhex(row['ellswift'])
+        assert ellswift_decode_sage(ellswift).hex() == row['x']
@@ -1,17 +0,0 @@
-ellswift,x
-26b25d457597a7b0463f9620f666dd10aa2c4373a505967c7c8d70922a2d6eceffffffffffffffffffffffffffffffffffffffffffffffffffffffffb2dabde1,240b740607e14d8cb767f53c9dacf5becde98abe73ffa36f096971215280dc58
-5a3e80a37915b1601c363acd1601df7ef257d5d32c664004a2ec0484a4f60628ffffffffffffffffffffffffffffffffffffffffffffffffffffffff15d5f3cd,4deaeb3cfbd2abbc9d57fdd83d825a05c45d773d96e247bda136e154769c1f8b
-6641161dc1faf1293662e9d81dc994fed6a720d6e0e1ab5702c6a866254a9076ffffffffffffffffffffffffffffffffffffffffffffffffffffffff5f44671f,32f5e32639066d09d5184e36cfca82b9f16076666edb2597bf6c8ca0f9423799
-bf5e8ffa51a9e748985800c1d3d7f1a2a6ae7435136593ca8d9637e3f87c699c76cc5805dab9b4eacefdb477f498020fd82bccdbc9c6a2d9ce10586ac85512b4,5579653da55ae6af8c92b0ab623bfede27756fdba141124c72aec43bc5b746e5
-df3f619804a92fdb4057192dc43dd748ea778adc52bc498ce80524c014b81119b40711a88c7039756fb8a73827eabe2c0fe5a0346ca7e0a104adc0fc764f528d,e838221abd40251a45646c40f62550e0acb8ab1ab292df7a9d4f28d72316bd3a
-f0caf11f8aea622a396127c3e7e67a6a854dccb736fcdc1270fc071592083e6da839c820778a009421bb1d1eee17cdea622828bd0d065d5b4adb6c0033570a37,bfae8740fc4926b0387803a8db03fa8b9d8b53ec30713a8227bf421b23b11571
-f1473fa4fb09147ba9d07832c92ccc0bcd651b696ff463931964066a4c849d12ffffffffffffffffffffffffffffffffffffffffffffffffffffffffd7ac238b,de26c723c76ec977f4cf79b3ba3e27800041930000ee7a74337d0e64fe164937
-fd50cee538a3798d17dde484f9d935860a88fe8dd6cd2341254ab5d558b0b67f5c5ec4b2af7c601e0f4b8d3893192292759a5c3b0a760c0589e5337bfb4e8a2d,0969798ab212485d36a0f007f744a17bffbc4fa9c3e73afcb4e7a27fb3580de9
-ffffffffffffffffffffffffffffffffffffffffffffffffffffffff3d60a4a9252c0b6b080fa045acfcd1437f693f3be2be2ac8223ea525d492fa19ab028942,c163b493f047704ba83e241472ebb2a05f95ef47c6bf5feedd8da33504866d68
-ffffffffffffffffffffffffffffffffffffffffffffffffffffffff4f51ebdbafa7518106309c22d325df6d2663249d158d2f36f1976269d6d4104d9198a108,37d7c5665514f85fe6e4cca488e8abdfc6bc4b3e87ff01ac08eb2504180296e9
-ffffffffffffffffffffffffffffffffffffffffffffffffffffffff72a7e655ffffffffffffffffffffffffffffffffffffffffffffffffffffffff1dd15ad9,34818ed876cbbb6710eb832627de9eb7c468846f26bfc336303601593bbb706d
-ffffffffffffffffffffffffffffffffffffffffffffffffffffffff92e6fb5cf32ceb01b42ea21ecdacc88a0e59dfbf72692b68d76924ba59f0a81f373d2cee,438c40e9cc47e577f56932b9bea91433acc7be309c017ff8f45a46046ed5aa9f
-ffffffffffffffffffffffffffffffffffffffffffffffffffffffffba7d0816157655bf7c7eaf74e26c4fb12043675dcce7580ea49d60317a546c3df2e14f9f,11d52804cc52a73185697681ebb8713dfe4204864fb9989b28e5a3696907710c
-ffffffffffffffffffffffffffffffffffffffffffffffffffffffffd4b84378fffffffffffffffffffffffffffffffffffffffffffffffffffffffff69a56d1,523e0758ee088690c9b95c604ef4d143e4fd3f2d1ac9084e3086750a9686f9bc
-ffffffffffffffffffffffffffffffffffffffffffffffffffffffffdac35742ffffffffffffffffffffffffffffffffffffffffffffffffffffffff99d5d507,db7f9b113af6796d460dfc12bef75f947fa1e0102686cf58de6ad5c0af752f82
-ffffffffffffffffffffffffffffffffffffffffffffffffffffffffeeee1f01fffffffffffffffffffffffffffffffffffffffffffffffffffffffff43363e8,c0c123902ec734bde1c6410a93e5d0033e0540120d5be9f555476b842fd2d245
@@ -0,0 +1,33 @@
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+1737a85f4c8d146cec96e3ffdca76d9903dcf3bd53061868d478c78c63c2aa9e,39e48dd150d2f429be088dfd5b61882e7e8407483702ae9a5ab35927b15f85ea,1be8cc0b04be0c681d0c6a68f733f82c6c896e0c8a262fcd392918e303a7abf4,605b5814bf9b8cb066667c9e5480d22dc5b6c92f14b4af3ee0a9eb83b03685e3,,,e41733f4fb41f397e2f3959708cc07d3937691f375d9d032c6d6e71bfc58503b,9fa4a7eb4064734f99998361ab7f2dd23a4936d0eb4b50c11f56147b4fc9764c,,,case0:ok;case1:ok;case2:info[v=0]&bad[non_square(s)];case3:bad[non_square(s)];case4:ok;case5:ok;case6:info[v=0]&bad[non_square(s)];case7:bad[non_square(s)]
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+704cd226e71cb6826a590e80dac90f2d2f5830f0fdf135a3eae3965bff25ff12,138e0afa68936ee670bd2b8db53aedbb7bea2a8597388b24d0518edd22ad66ec,,,,,,,,,case0:bad[non_square(s)];case1:bad[non_square(s)];case2:bad[non_square(q)];case3:bad[non_square(q)];case4:bad[non_square(s)];case5:bad[non_square(s)];case6:bad[non_square(q)];case7:bad[non_square(q)]
+725e914792cb8c8949e7e1168b7cdd8a8094c91c6ec2202ccd53a6a18771edeb,8da16eb86d347376b6181ee9748322757f6b36e3913ddfd332ac595d788e0e44,dd357786b9f6873330391aa5625809654e43116e82a5a5d82ffd1d6624101fc4,a0b7efca01814594c59c9aae8e49700186ca5d95e88bcc80399044d9c2d8613d,,,22ca8879460978cccfc6e55a9da7f69ab1bcee917d5a5a27d002e298dbefdc6b,5f481035fe7eba6b3a63655171b68ffe7935a26a1774337fc66fbb253d279af2,,,case0:ok;case1:info[v=0]&ok;case2:bad[non_square(s)];case3:bad[non_square(s)];case4:ok;case5:info[v=0]&ok;case6:bad[non_square(s)];case7:bad[non_square(s)]
+78fe6b717f2ea4a32708d79c151bf503a5312a18c0963437e865cc6ed3f6ae97,8701948e80d15b5cd8f72863eae40afc5aced5e73f69cbc8179a33902c094d98,,,,,,,,,case0:bad[non_square(s)];case1:info[v=0]&bad[non_square(s)];case2:bad[non_square(q)];case3:bad[non_square(q)];case4:bad[non_square(s)];case5:info[v=0]&bad[non_square(s)];case6:bad[non_square(q)];case7:bad[non_square(q)]
+7c37bb9c5061dc07413f11acd5a34006e64c5c457fdb9a438f217255a961f50d,5c1a76b44568eb59d6789a7442d9ed7cdc6226b7752b4ff8eaf8e1a95736e507,,,b94d30cd7dbff60b64620c17ca0fafaa40b3d1f52d077a60a2e0cafd145086c2,,,,46b2cf32824009f49b9df3e835f05055bf4c2e0ad2f8859f5d1f3501ebaf756d,,case0:bad[non_square(s)];case1:bad[non_square(s)];case2:info[q=0]&info[X=0]&ok;case3:info[q=0]&bad[r=0];case4:bad[non_square(s)];case5:bad[non_square(s)];case6:info[q=0]&info[X=0]&ok;case7:info[q=0]&bad[r=0]
+82388888967f82a6b444438a7d44838e13c0d478b9ca060da95a41fb94303de6,29e9654170628fec8b4972898b113cf98807f4609274f4f3140d0674157c90a0,,,,,,,,,case0:bad[non_square(s)];case1:bad[non_square(s)];case2:bad[non_square(s)];case3:info[v=0]&bad[non_square(s)];case4:bad[non_square(s)];case5:bad[non_square(s)];case6:bad[non_square(s)];case7:info[v=0]&bad[non_square(s)]
+91298f5770af7a27f0a47188d24c3b7bf98ab2990d84b0b898507e3c561d6472,144f4ccbd9a74698a88cbf6fd00ad886d339d29ea19448f2c572cac0a07d5562,e6a0ffa3807f09dadbe71e0f4be4725f2832e76cad8dc1d943ce839375eff248,837b8e68d4917544764ad0903cb11f8615d2823cefbb06d89049dbabc69befda,,,195f005c7f80f6252418e1f0b41b8da0d7cd189352723e26bc317c6b8a1009e7,7c8471972b6e8abb89b52f6fc34ee079ea2d7dc31044f9276fb6245339640c55,,,case0:ok;case1:ok;case2:bad[non_square(s)];case3:info[v=0]&bad[non_square(s)];case4:ok;case5:ok;case6:bad[non_square(s)];case7:info[v=0]&bad[non_square(s)]
+b682f3d03bbb5dee4f54b5ebfba931b4f52f6a191e5c2f483c73c66e9ace97e1,904717bf0bc0cb7873fcdc38aa97f19e3a62630972acff92b24cc6dda197cb96,,,,,,,,,case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:bad[non_square(s)];case3:bad[non_square(s)];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:bad[non_square(s)];case7:bad[non_square(s)]
+c17ec69e665f0fb0dbab48d9c2f94d12ec8a9d7eacb58084833091801eb0b80b,147756e66d96e31c426d3cc85ed0c4cfbef6341dd8b285585aa574ea0204b55e,6f4aea431a0043bdd03134d6d9159119ce034b88c32e50e8e36c4ee45eac7ae9,fd5be16d4ffa2690126c67c3ef7cb9d29b74d397c78b06b3605fda34dc9696a6,5e9c60792a2f000e45c6250f296f875e174efc0e9703e628706103a9dd2d82c7,,90b515bce5ffbc422fcecb2926ea6ee631fcb4773cd1af171c93b11aa1538146,02a41e92b005d96fed93983c1083462d648b2c683874f94c9fa025ca23696589,a1639f86d5d0fff1ba39daf0d69078a1e8b103f168fc19d78f9efc5522d27968,,case0:ok;case1:ok;case2:info[q=0]&info[X=0]&ok;case3:info[q=0]&bad[r=0];case4:ok;case5:ok;case6:info[q=0]&info[X=0]&ok;case7:info[q=0]&bad[r=0]
+c25172fc3f29b6fc4a1155b8575233155486b27464b74b8b260b499a3f53cb14,1ea9cbdb35cf6e0329aa31b0bb0a702a65123ed008655a93b7dcd5280e52e1ab,,,7422edc7843136af0053bb8854448a8299994f9ddcefd3a9a92d45462c59298a,78c7774a266f8b97ea23d05d064f033c77319f923f6b78bce4e20bf05fa5398d,,,8bdd12387bcec950ffac4477abbb757d6666b06223102c5656d2bab8d3a6d2a5,873888b5d990746815dc2fa2f9b0fcc388ce606dc09487431b1df40ea05ac2a2,case0:bad[non_square(s)];case1:bad[non_square(s)];case2:ok;case3:ok;case4:bad[non_square(s)];case5:bad[non_square(s)];case6:ok;case7:ok
+cab6626f832a4b1280ba7add2fc5322ff011caededf7ff4db6735d5026dc0367,2b2bef0852c6f7c95d72ac99a23802b875029cd573b248d1f1b3fc8033788eb6,,,,,,,,,case0:bad[non_square(s)];case1:bad[non_square(s)];case2:info[v=0]&bad[non_square(s)];case3:bad[non_square(s)];case4:bad[non_square(s)];case5:bad[non_square(s)];case6:info[v=0]&bad[non_square(s)];case7:bad[non_square(s)]
+d8621b4ffc85b9ed56e99d8dd1dd24aedcecb14763b861a17112dc771a104fd2,812cabe972a22aa67c7da0c94d8a936296eb9949d70c37cb2b2487574cb3ce58,fbc5febc6fdbc9ae3eb88a93b982196e8b6275a6d5a73c17387e000c711bd0e3,8724c96bd4e5527f2dd195a51c468d2d211ba2fac7cbe0b4b3434253409fb42d,,,043a014390243651c147756c467de691749d8a592a58c3e8c781fff28ee42b4c,78db36942b1aad80d22e6a5ae3b972d2dee45d0538341f4b4cbcbdabbf604802,,,case0:ok;case1:ok;case2:bad[non_square(s)];case3:bad[non_square(s)];case4:ok;case5:ok;case6:bad[non_square(s)];case7:bad[non_square(s)]
+da463164c6f4bf7129ee5f0ec00f65a675a8adf1bd931b39b64806afdcda9a22,25b9ce9b390b408ed611a0f13ff09a598a57520e426ce4c649b7f94f2325620d,,,,,,,,,case0:bad[non_square(s)];case1:info[v=0]&bad[non_square(s)];case2:bad[non_square(s)];case3:bad[non_square(s)];case4:bad[non_square(s)];case5:info[v=0]&bad[non_square(s)];case6:bad[non_square(s)];case7:bad[non_square(s)]
+dafc971e4a3a7b6dcfb42a08d9692d82ad9e7838523fcbda1d4827e14481ae2d,250368e1b5c58492304bd5f72696d27d526187c7adc03425e2b7d81dbb7e4e02,,,370c28f1be665efacde6aa436bf86fe21e6e314c1e53dd040e6c73a46b4c8c49,cd8acee98ffe56531a84d7eb3e48fa4034206ce825ace907d0edf0eaeb5e9ca2,,,c8f3d70e4199a105321955bc9407901de191ceb3e1ac22fbf1938c5a94b36fe6,327531167001a9ace57b2814c1b705bfcbdf9317da5316f82f120f1414a15f8d,case0:bad[non_square(s)];case1:info[v=0]&bad[non_square(s)];case2:ok;case3:ok;case4:bad[non_square(s)];case5:info[v=0]&bad[non_square(s)];case6:ok;case7:ok
+e0294c8bc1a36b4166ee92bfa70a5c34976fa9829405efea8f9cd54dcb29b99e,ae9690d13b8d20a0fbbf37bed8474f67a04e142f56efd78770a76b359165d8a1,,,dcd45d935613916af167b029058ba3a700d37150b9df34728cb05412c16d4182,,,,232ba26ca9ec6e950e984fd6fa745c58ff2c8eaf4620cb8d734fabec3e92baad,,case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:info[q=0]&info[X=0]&ok;case3:info[q=0]&bad[r=0];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:info[q=0]&info[X=0]&ok;case7:info[q=0]&bad[r=0]
+e148441cd7b92b8b0e4fa3bd68712cfd0d709ad198cace611493c10e97f5394e,164a639794d74c53afc4d3294e79cdb3cd25f99f6df45c000f758aba54d699c0,,,,,,,,,case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:bad[non_square(s)];case3:info[v=0]&bad[non_square(s)];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:bad[non_square(s)];case7:info[v=0]&bad[non_square(s)]
+e4b00ec97aadcca97644d3b0c8a931b14ce7bcf7bc8779546d6e35aa5937381c,94e9588d41647b3fcc772dc8d83c67ce3be003538517c834103d2cd49d62ef4d,c88d25f41407376bb2c03a7fffeb3ec7811cc43491a0c3aac0378cdc78357bee,51c02636ce00c2345ecd89adb6089fe4d5e18ac924e3145e6669501cd37a00d4,205b3512db40521cb200952e67b46f67e09e7839e0de44004138329ebd9138c5,58aab390ab6fb55c1d1b80897a207ce94a78fa5b4aa61a33398bcae9adb20d3e,3772da0bebf8c8944d3fc5800014c1387ee33bcb6e5f3c553fc8732287ca8041,ae3fd9c931ff3dcba132765249f7601b2a1e7536db1ceba19996afe22c85fb5b,dfa4caed24bfade34dff6ad1984b90981f6187c61f21bbffbec7cd60426ec36a,a7554c6f54904aa3e2e47f7685df8316b58705a4b559e5ccc6743515524deef1,case0:ok;case1:ok;case2:ok;case3:info[v=0]&ok;case4:ok;case5:ok;case6:ok;case7:info[v=0]&ok
+e5bbb9ef360d0a501618f0067d36dceb75f5be9a620232aa9fd5139d0863fde5,e5bbb9ef360d0a501618f0067d36dceb75f5be9a620232aa9fd5139d0863fde5,,,,,,,,,case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:bad[s=0];case3:bad[s=0];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:bad[s=0];case7:bad[s=0]
+e6bcb5c3d63467d490bfa54fbbc6092a7248c25e11b248dc2964a6e15edb1457,19434a3c29cb982b6f405ab04439f6d58db73da1ee4db723d69b591da124e7d8,67119877832ab8f459a821656d8261f544a553b89ae4f25c52a97134b70f3426,ffee02f5e649c07f0560eff1867ec7b32d0e595e9b1c0ea6e2a4fc70c97cd71f,b5e0c189eb5b4bacd025b7444d74178be8d5246cfa4a9a207964a057ee969992,5746e4591bf7f4c3044609ea372e908603975d279fdef8349f0b08d32f07619d,98ee67887cd5470ba657de9a927d9e0abb5aac47651b0da3ad568eca48f0c809,0011fd0a19b63f80fa9f100e7981384cd2f1a6a164e3f1591d5b038e36832510,4a1f3e7614a4b4532fda48bbb28be874172adb9305b565df869b5fa71169629d,a8b91ba6e4080b3cfbb9f615c8d16f79fc68a2d8602107cb60f4f72bd0f89a92,case0:ok;case1:info[v=0]&ok;case2:ok;case3:ok;case4:ok;case5:info[v=0]&ok;case6:ok;case7:ok
+f28fba64af766845eb2f4302456e2b9f8d80affe57e7aae42738d7cddb1c2ce6,f28fba64af766845eb2f4302456e2b9f8d80affe57e7aae42738d7cddb1c2ce6,4f867ad8bb3d840409d26b67307e62100153273f72fa4b7484becfa14ebe7408,5bbc4f59e452cc5f22a99144b10ce8989a89a995ec3cea1c91ae10e8f721bb5d,,,b079852744c27bfbf62d9498cf819deffeacd8c08d05b48b7b41305db1418827,a443b0a61bad33a0dd566ebb4ef317676576566a13c315e36e51ef1608de40d2,,,case0:ok;case1:ok;case2:bad[s=0];case3:bad[s=0];case4:ok;case5:ok;case6:bad[s=0];case7:bad[s=0]
+f455605bc85bf48e3a908c31023faf98381504c6c6d3aeb9ede55f8dd528924d,d31fbcd5cdb798f6c00db6692f8fe8967fa9c79dd10958f4a194f01374905e99,,,0c00c5715b56fe632d814ad8a77f8e66628ea47a6116834f8c1218f3a03cbd50,df88e44fac84fa52df4d59f48819f18f6a8cd4151d162afaf773166f57c7ff46,,,f3ff3a8ea4a9019cd27eb527588071999d715b859ee97cb073ede70b5fc33edf,20771bb0537b05ad20b2a60b77e60e7095732beae2e9d505088ce98fa837fce9,case0:bad[non_square(s)];case1:bad[non_square(s)];case2:info[v=0]&ok;case3:ok;case4:bad[non_square(s)];case5:bad[non_square(s)];case6:info[v=0]&ok;case7:ok
+f58cd4d9830bad322699035e8246007d4be27e19b6f53621317b4f309b3daa9d,78ec2b3dc0948de560148bbc7c6dc9633ad5df70a5a5750cbed721804f082a3b,6c4c580b76c7594043569f9dae16dc2801c16a1fbe12860881b75f8ef929bce5,94231355e7385c5f25ca436aa64191471aea4393d6e86ab7a35fe2afacaefd0d,dff2a1951ada6db574df834048149da3397a75b829abf58c7e69db1b41ac0989,a52b66d3c907035548028bf804711bf422aba95f1a666fc86f4648e05f29caae,93b3a7f48938a6bfbca9606251e923d7fe3e95e041ed79f77e48a07006d63f4a,6bdcecaa18c7a3a0da35bc9559be6eb8e515bc6c291795485ca01d4f5350ff22,200d5e6ae525924a8b207cbfb7eb625cc6858a47d6540a73819624e3be53f2a6,5ad4992c36f8fcaab7fd7407fb8ee40bdd5456a0e599903790b9b71ea0d63181,case0:ok;case1:ok;case2:info[v=0]&ok;case3:ok;case4:ok;case5:ok;case6:info[v=0]&ok;case7:ok
+fd7d912a40f182a3588800d69ebfb5048766da206fd7ebc8d2436c81cbef6421,8d37c862054debe731694536ff46b273ec122b35a9bf1445ac3c4ff9f262c952,,,,,,,,,case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:info[v=0]&bad[non_square(s)];case3:bad[non_square(s)];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:info[v=0]&bad[non_square(s)];case7:bad[non_square(s)]