[tor-commits] [torspec/master] Add NewHope + X25519 handshake proposal draft.
isis at torproject.org
isis at torproject.org
Fri Jul 22 12:05:35 UTC 2016
commit 0f5ddf6ca863eaa748733557a3a96c44f3361085
Author: Isis Lovecruft <isis at torproject.org>
Date: Fri Apr 22 19:23:43 2016 +0000
Add NewHope + X25519 handshake proposal draft.
---
proposals/XXX-newhope-hybrid-handshake.txt | 768 +++++++++++++++++++++++++++++
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diff --git a/proposals/XXX-newhope-hybrid-handshake.txt b/proposals/XXX-newhope-hybrid-handshake.txt
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+Filename: XXX-newhope-hybrid-handshake.txt
+Title: Post-Quantum Secure Hybrid Handshake Based on NewHope
+Author: Isis Lovecruft, Peter Schwabe
+Created: 16 Apr 2016
+Updated: 4 May 2016
+Status: Draft
+Depends: prop#220 prop#249 prop#264
+
+§0. Introduction
+
+ NewHope is a post-quantum-secure lattice-based key-exchange protocol based
+ on the ring-learning-with-errors (Ring-LWE) problem. We propose a hybrid
+ handshake for Tor, based on a combination of Tor's current NTor handshake
+ and a shared key derived through a NewHope ephemeral key exchange.
+
+ For further details on the NewHope key exchange, the reader is referred to
+ "Post-quantum key exchange - a new hope" by Alkim, Ducas, Pöppelmann, and
+ Schwabe [0][1].
+
+ For the purposes of brevity, we consider that NTor is currently the only
+ handshake protocol in Tor; the older TAP protocol is ignored completely, due
+ to the fact that it is currently deprecated and nearly entirely unused.
+
+
+§1. Motivation
+
+ An attacker currently monitoring and storing circuit-layer NTor handshakes
+ who later has the ability to run Shor's algorithm on a quantum computer will
+ be able to break Tor's current handshake protocol and decrypt previous
+ communications.
+
+ It is unclear if and when such attackers equipped with large quantum
+ computers will exist, but various estimates by researchers in quantum
+ physics and quantum engineering give estimates of only 1 to 2 decades.
+ Clearly, the security requirements of many Tor users include secrecy of
+ their messages beyond this time span, which means that Tor needs to update
+ the key exchange to protect against such attackers as soon as possible.
+
+
+§2. Design
+
+ An initiator and responder, in parallel, conduct two handshakes:
+
+ - An X25519 key exchange, as described in the description of the NTor
+ handshake in Tor proposal #216.
+ - A NewHope key exchange.
+
+ The shared keys derived from these two handshakes are then concatenated and
+ used as input to the SHAKE-256 extendable output function (XOF), as decribed
+ in FIPS-PUB-202 [2], in order to produce a shared key of the desired length.
+ The testvectors in §C assume that this key has a length of 32 bytes, but the
+ use of a XOF allows arbitrary lengths to easily support future updates of
+ the symmetric primitives using the key. See also §3.3.1.
+
+
+§3. Specification
+
+§3.1. Notation
+
+ Let `a || b` be the concatenation of a with b.
+
+ Let `a^b` denote the exponentiation of a to the bth power.
+
+ Let `a == b` denote the equality of a with b, and vice versa.
+
+ Let `a := b` be the assignment of the value of b to the variable a.
+
+ Let `H(x)` be 32-bytes of output of the SHAKE-256 XOF (as described in
+ FIPS-PUB-202) applied to message x.
+
+ Let X25519 refer to the curve25519-based key agreement protocol described
+ in RFC7748 §6.1. [3]
+
+ Let `EXP(a, b) == X25519(., b, a)` with `g == 9`. Let X25519_KEYGEN() do
+ the appropriate manipulations when generating the secret key (clearing the
+ low bits, twidding the high bits).
+
+ [XXX match RFC7748 notation more. --isis]
+
+ Let `X25519_KEYID(B) == B` where B is a valid X25519 public key.
+
+ When representing an element of the Curve25519 subgroup as a byte string,
+ use the standard (32-byte, little-endian, x-coordinate-only) representation
+ for Curve25519 points.
+
+ Let `ID` be a router's identity key taken from the router microdescriptor.
+ In the case for relays possessing Ed25519 identity keys (c.f. Tor proposal
+ #220), this is a 32-byte string representing the public Ed25519 identity key.
+ For backwards and forwards compatibility with routers which do not possess
+ Ed25519 identity keys, this is a 32-byte string created via the output of
+ H(ID).
+
+ We refer to the router as the handshake "responder", and the client (which
+ may be an OR or an OP) as the "initiator".
+
+
+ ID_LENGTH [32 bytes]
+ H_LENGTH [32 bytes]
+ G_LENGTH [32 bytes]
+
+ PROTOID := "pqtor-x25519-newhope-shake256-1"
+ T_MAC := PROTOID || ":mac"
+ T_KEY := PROTOID || ":key_extract"
+ T_VERIFY := PROTOID || ":verify"
+
+ (X25519_SK, X25519_PK) := X25519_KEYGEN()
+
+
+§3.2. Protocol
+
+ ========================================================================================
+ | |
+ | Fig. 1: The NewHope-X25519 Hybrid Handshake. |
+ | |
+ | Before the handshake the Initiator is assumed to know Z, a public X25519 key for |
+ | the Responder, as well as the Responder's ID. |
+ ----------------------------------------------------------------------------------------
+ | |
+ | Initiator Responder |
+ | |
+ | SEED := H(randombytes(32)) |
+ | x, X := X25519_KEYGEN() |
+ | a, A := NEWHOPE_KEYGEN(SEED) |
+ | CLIENT_HDATA := ID || Z || X || A |
+ | |
+ | --- CLIENT_HDATA ---> |
+ | |
+ | y, Y := X25519_KEYGEN() |
+ | NTOR_KEY, AUTH := NTOR_SHAREDB(X,y,Y,z,Z,ID,B) |
+ | M, NEWHOPE_KEY := NEWHOPE_SHAREDB(A) |
+ | SERVER_HDATA := Y || AUTH || M |
+ | sk := SHAKE-256(NTOR_KEY || NEWHOPE_KEY) |
+ | |
+ | <-- SERVER_HDATA ---- |
+ | |
+ | NTOR_KEY := NTOR_SHAREDA(x, X, Y, Z, ID, AUTH) |
+ | NEWHOPE_KEY := NEWHOPE_SHAREDA(M, a) |
+ | sk := SHAKE-256(NTOR_KEY, NEWHOPE_KEY) |
+ | |
+ ========================================================================================
+
+
+§3.2.1. The NTor Handshake
+
+§3.2.1.1. Prologue
+
+ Take a router with identity ID. As setup, the router generates a secret key z,
+ and a public onion key Z with:
+
+ z, Z := X25519_KEYGEN()
+
+ The router publishes Z in its server descriptor in the "ntor-onion-key" entry.
+ Henceforward, we refer to this router as the "responder".
+
+
+§3.2.1.2. Initiator
+
+ To send a create cell, the initiator generates a keypair:
+
+ x, X := X25519_KEYGEN()
+
+ and creates the NTor portion of a CREATE2V cell's HDATA section:
+
+ CLIENT_NTOR := ID || Z || X [96 bytes]
+
+ The initiator includes the responder's ID and Z in the CLIENT_NTOR so that, in
+ the event the responder OR has recently rotated keys, the responder can
+ determine which keypair to use.
+
+ The initiator then concatenates CLIENT_NTOR with CLIENT_NEWHOPE (see §3.2.2),
+ to create CLIENT_HDATA, and creates and sends a CREATE2V cell (see §A.1)
+ to the responder.
+
+ CLIENT_NEWHOPE [1824 bytes] (see §3.2.2)
+ CLIENT_HDATA := CLIENT_NTOR || CLIENT_NEWHOPE [1920 bytes]
+
+ If the responder does not respond with a CREATED2V cell, the initiator SHOULD
+ NOT attempt to extend the circuit through the responder by sending fragmented
+ EXTEND2 cells, since the responder's lack of support for CREATE2V cells is
+ assumed to imply the responder also lacks support for fragmented EXTEND2
+ cells. Alternatively, for initiators with a sufficiently late consensus
+ method, the initiator MUST check that "proto" line in the responder's
+ descriptor (c.f. Tor proposal #264) advertises support for the "Relay"
+ subprotocol version 3 (see §5).
+
+
+§3.2.1.3. Responder
+
+ The responder generates a keypair of y, Y = X25519_KEYGEN(), and does
+ NTOR_SHAREDB() as follows:
+
+ (NTOR_KEY, AUTH) ← NTOR_SHAREDB(X, y, Y, z, Z, ID, B):
+ secret_input := EXP(X, y) || EXP(X, z) || ID || B || Z || Y || PROTOID
+ NTOR_KEY := H(secret_input, T_KEY)
+ verify := H(secret_input, T_VERIFY)
+ auth_input := verify || ID || Z || Y || X || PROTOID || "Server"
+ AUTH := H(auth_input, T_MAC)
+
+ The responder sends a CREATED2V cell containing:
+
+ SERVER_NTOR := Y || AUTH [64 bytes]
+ SERVER_NEWHOPE [2048 bytes] (see §3.2.2)
+ SERVER_HDATA := SERVER_NTOR || SERVER_NEWHOPE [2112 bytes]
+
+ and sends this to the initiator.
+
+
+§3.2.1.4. Finalisation
+
+ The initiator then checks Y is in G^* [see NOTE below], and does
+ NTOR_SHAREDA() as follows:
+
+ (NTOR_KEY) ← NTOR_SHAREDA(x, X, Y, Z, ID, AUTH)
+ secret_input := EXP(Y, x) || EXP(Z, x) || ID || Z || X || Y || PROTOID
+ NTOR_KEY := H(secret_input, T_KEY)
+ verify := H(secret_input, T_VERIFY)
+ auth_input := verify || ID || Z || Y || X || PROTOID || "Server"
+ if AUTH == H(auth_input, T_MAC)
+ return NTOR_KEY
+
+ Both parties check that none of the EXP() operations produced the point at
+ infinity. [NOTE: This is an adequate replacement for checking Y for group
+ membership, if the group is Curve25519.]
+
+ [XXX: This doesn't sound exactly right. You need the scalar tweaking of
+ X25519 for this to work and also, the point at infinity is obviously an
+ element of the group --isis, peter]
+
+ Both parties now have a shared value for NTOR_KEY. They expand this into
+ the keys needed for the Tor relay protocol.
+
+ [XXX We think we want to omit the final hashing in the production of NTOR_KEY
+ here, and instead put all the inputs through SHAKE-256. --isis, peter]
+
+ [XXX We probably want to remove ID and B from the input to the shared key
+ material, since they serve for authentication but, as pre-established
+ "prologue" material to the handshake, they should not be used in attempts to
+ strengthen the cryptographic suitability of the shared key. Also, their
+ inclusion is implicit in the DH exponentiations. I should probably ask Ian
+ about the reasoning for the original design choice. --isis]
+
+
+§3.2.2. The NewHope Handshake
+
+§3.2.2.1. Parameters & Mathematical Structures
+
+ Let ℤ be the ring of rational integers. Let ℤq, for q ≥ 1, denote the quotient
+ ring ℤ/qℤ. We define R = ℤ[X]/((X^n)+1) as the ring of integer polynomials
+ modulo ((X^n)+1), and Rq = ℤq[X]/((X^n)+1) as the ring of integer polynomials
+ modulo ((X^n)+1) where each coefficient is reduced modulo q. When we refer to
+ a polynomial, we mean an element of Rq.
+
+ n := 1024
+ q := 12289
+
+ SEED [32 Bytes]
+ NEWHOPE_POLY [1792 Bytes]
+ NEWHOPE_REC [256 Bytes]
+ NEWHOPE_KEY [32 Bytes]
+
+ NEWHOPE_MSGA := (NEWHOPE_POLY || SEED)
+ NEWHOPE_MSGB := (NEWHOPE_POLY || NEWHOPE_REC)
+
+
+§3.2.2.2. High-level Description of Newhope API Functions
+
+ For a description of internal functions, see §B.
+
+ (NEWHOPE_POLY, NEWHOPE_MSGA) ← NEWHOPE_KEYGEN(SEED):
+ â := gen_a(seed)
+ s := poly_getnoise()
+ e := poly_getnoise()
+ ŝ := poly_ntt(s)
+ ê := poly_ntt(e)
+ b̂ := pointwise(â, ŝ) + ê
+ sp := poly_tobytes(ŝ)
+ bp := poly_tobytes(b̂)
+ return (sp, (bp || seed))
+
+ (NEWHOPE_MSGB, NEWHOPE_KEY) ← NEWHOPE_SHAREDB(NEWHOPE_MSGA):
+ s' := poly_getnoise()
+ e' := poly_getnoise()
+ e" := poly_getnoise()
+ b̂ := poly_frombytes(bp)
+ â := gen_a(seed)
+ ŝ' := poly_ntt(s')
+ ê' := poly_ntt(e')
+ û := poly_pointwise(â, ŝ') + ê'
+ v := poly_invntt(poly_pointwise(b̂,ŝ')) + e"
+ r := helprec(v)
+ up := poly_tobytes(û)
+ k := rec(v, r)
+ return ((up || r), k)
+
+ NEWHOPE_KEY ← NEWHOPE_SHAREDA(NEWHOPE_MSGB, NEWHOPE_POLY):
+ û := poly_frombytes(up)
+ ŝ := poly_frombytes(sp)
+ v' := poly_invntt(poly_pointwise(û, ŝ))
+ k := rec(v', r)
+ return k
+
+ When a client uses a SEED within a CREATE2V cell, the client SHOULD NOT use
+ that SEED in any other CREATE2V or EXTEND2 cells. See §4 for further
+ discussion.
+
+
+§3.3. Key Expansion
+
+ The client and server derive a shared key, SHARED, by:
+
+ HKDFID := "THESE ARENT THE DROIDS YOURE LOOKING FOR"
+ SHARED := SHAKE_256(HKDFID || NTorKey || NewHopeKey)
+
+
+§3.3.1. Note on the Design Choice
+
+ The reader may wonder why one would use SHAKE-256 to produce a 256-bit
+ output, since the security strength in bits for SHAKE-256 is min(d/2,256)
+ for collision resistance and min(d,256) for first- and second-order
+ preimages, where d is the output length.
+
+ The reasoning is that we should be aiming for 256-bit security for all of
+ our symmetric cryptography. One could then argue that we should just use
+ SHA3-256 for the KDF. We choose SHAKE-256 instead in order to provide an
+ easy way to derive longer shared secrets in the future without requiring a
+ new handshake. The construction is odd, but the future is bright.
+ As we are already using SHAKE-256 for the 32-byte output hash, we are also
+ using it for all other 32-byte hashes involved in the protocol. Note that
+ the only difference between SHA3-256 and SHAKE-256 with 32-byte output is
+ one domain-separation byte.
+
+ [XXX why would you want 256-bit security for the symmetric side? Are you
+ talking pre- or post-quantum security? --peter]
+
+
+§4. Security & Anonymity Implications
+
+ This handshake protocol is one-way authenticated. That is, the server is
+ authenticated, while the client remains anonymous.
+
+ The client MUST NOT cache and reuse SEED. Doing so gives non-trivial
+ adversarial advantages w.r.t. all-for-the-price-of-one attacks during the
+ caching period. More importantly, if the SEED used to generate NEWHOPE_MSGA
+ is reused for handshakes along the same circuit or multiple different
+ circuits, an adversary conducting a sybil attack somewhere along the path(s)
+ will be able to correlate the identity of the client across circuits or
+ hops.
+
+
+§5. Compatibility
+
+ Because our proposal requires both the client and server to send more than
+ the 505 bytes possible within a CREATE2 cell's HDATA section, it depends
+ upon the implementation of a mechanism for allowing larger CREATE cells
+ (c.f. Tor proposal #249).
+
+ We reserve the following handshake type for use in CREATE2V/CREATED2V and
+ EXTEND2V/EXTENDED2V cells:
+
+ 0x0003 [NEWHOPE + X25519 HYBRID HANDSHAKE]
+
+ We introduce a new sub-protocol number, "Relay=3", (c.f. Tor proposal #264
+ §5.3) to signify support this handshake, and hence for the CREATE2V and
+ fragmented EXTEND2 cells which it requires.
+
+ There are no additional entries or changes required within either router
+ descriptors or microdescriptors to support this handshake method, due to the
+ NewHope keys being ephemeral and derived on-the-fly, and due to the NTor X25519
+ public keys already being in included within the "ntor-onion-key" entry.
+
+ Add a "UseNewHopeKEX" configuration option and a corresponding consensus
+ parameter to control whether clients prefer using this NewHope hybrid
+ handshake or some previous handshake protocol. If the configuration option
+ is "auto", clients SHOULD obey the consensus parameter. The default
+ configuration SHOULD be "auto" and the consensus value SHOULD initially be "0".
+
+
+§6. Implementation
+
+ The paper by Alkim, Ducas, Pöppelmann and Schwabe describes two software
+ implementations of NewHope, one C reference implementation and an optimized
+ implementation using AVX2 vector instructions. Those implementations are
+ available at [1].
+
+ Additionally, there are implementations in Go by Yawning Angel, available
+ from [4] and in Rust by Isis Lovecruft, available from [5].
+
+ The software used to generate the test vectors in §C is based on the C
+ reference implementation and available from:
+
+ https://code.ciph.re/isis/newhope-tor-testvectors
+ https://github.com/isislovecruft/newhope-tor-testvectors
+
+
+§7. Performance & Scalability
+
+ The computationally expensive part in the current NTor handshake is the
+ X25519 key-pair generation and the X25519 shared-key computation. The
+ current implementation in Tor is a wrapper to support various highly optimized
+ implementations on different architectures. On Intel Haswell processors, the
+ fastest implementation of X25519, as reported by the eBACS benchmarking
+ project [6], takes 169920 cycles for key-pair generation and 161648 cycles
+ for shared-key computation; these add up to a total of 331568 cycles on each
+ side (initiator and responder).
+
+ The C reference implementation of NewHope, also benchmarked on Intel
+ Haswell, takes 358234 cycles for the initiator and 402058 cycles for the
+ Responder. The core computation of the proposed combination of NewHope and
+ X25519 will thus mean a slowdown of about a factor of 2.1 for the Initiator
+ and a slowdown by a factor of 2.2 for the Responder compared to the current
+ NTor handshake. These numbers assume a fully optimized implementation of the
+ NTor handshake and a C reference implementation of NewHope. With optimized
+ implementations of NewHope, such as the one for Intel Haswell described in
+ [0], the computational slowdown will be considerably smaller than a factor
+ of 2.
+
+
+§8. References
+
+[0]: https://cryptojedi.org/papers/newhope-20160328.pdf
+[1]: https://cryptojedi.org/crypto/#newhope
+[2]: http://www.nist.gov/customcf/get_pdf.cfm?pub_id=919061
+[3]: https://tools.ietf.org/html/rfc7748#section-6.1
+[4]: https://github.com/Yawning/newhope
+[5]: https://code.ciph.re/isis/newhopers
+[6]: http://bench.cr.yp.to
+
+
+§A. Cell Formats
+
+§A.1. CREATE2V Cells
+
+ The client portion of the handshake should send CLIENT_HDATA, formatted
+ into a CREATE2V cell as follows:
+
+ CREATE2V { [2114 bytes]
+ HTYPE := 0x0003 [2 bytes]
+ HLEN := 0x0780 [2 bytes]
+ HDATA := CLIENT_HDATA [1920 bytes]
+ IGNORED := 0x00 [194 bytes]
+ }
+
+ [XXX do we really want to pad with IGNORED to make CLIENT_HDATA the
+ same number of bytes as SERVER_HDATA? --isis]
+
+§A.2. CREATED2V Cells
+
+ The server responds to the client's CREATE2V cell with SERVER_HDATA,
+ formatted into a CREATED2V cell as follows:
+
+ CREATED2V { [2114 bytes]
+ HLEN := 0x0800 [2 bytes]
+ HDATA := SERVER_HDATA [2112 bytes]
+ IGNORED := 0x00 [0 bytes]
+ }
+
+§A.3. Fragmented EXTEND2 Cells
+
+ When the client wishes to extend a circuit, the client should fragment
+ CLIENT_HDATA into four EXTEND2 cells:
+
+ EXTEND2 {
+ NSPEC := 0x02 { [1 byte]
+ LINK_ID_SERVER [22 bytes] XXX
+ LINK_ADDRESS_SERVER [8 bytes] XXX
+ }
+ HTYPE := 0x0003 [2 bytes]
+ HLEN := 0x0780 [2 bytes]
+ HDATA := CLIENT_HDATA[0,461] [462 bytes]
+ }
+ EXTEND2 {
+ NSPEC := 0x00 [1 byte]
+ HTYPE := 0xFFFF [2 bytes]
+ HLEN := 0x0000 [2 bytes]
+ HDATA := CLIENT_HDATA[462,954] [492 bytes]
+ }
+ EXTEND2 {
+ NSPEC := 0x00 [1 byte]
+ HTYPE := 0xFFFF [2 bytes]
+ HLEN := 0x0000 [2 bytes]
+ HDATA := CLIENT_HDATA[955,1447] [492 bytes]
+ }
+ EXTEND2 {
+ NSPEC := 0x00 [1 byte]
+ HTYPE := 0xFFFF [2 bytes]
+ HLEN := 0x0000 [2 bytes]
+ HDATA := CLIENT_HDATA[1448,1919] || 0x00[20] [492 bytes]
+ }
+ EXTEND2 {
+ NSPEC := 0x00 [1 byte]
+ HTYPE := 0xFFFF [2 bytes]
+ HLEN := 0x0000 [2 bytes]
+ HDATA := 0x00[172] [172 bytes]
+ }
+
+ The client sends this to the server to extend the circuit from, and that
+ server should format the fragmented EXTEND2 cells into a CREATE2V cell, as
+ described in §A.1.
+
+§A.4. Fragmented EXTENDED2 Cells
+
+ EXTENDED2 {
+ NSPEC := 0x02 { [1 byte]
+ LINK_ID_SERVER [22 bytes] XXX
+ LINK_ADDRESS_SERVER [8 bytes] XXX
+ }
+ HTYPE := 0x0003 [2 bytes]
+ HLEN := 0x0800 [2 bytes]
+ HDATA := SERVER_HDATA[0,461] [462 bytes]
+ }
+ EXTENDED2 {
+ NSPEC := 0x00 [1 byte]
+ HTYPE := 0xFFFF [2 bytes]
+ HLEN := 0x0000 [2 bytes]
+ HDATA := SERVER_HDATA[462,954] [492 bytes]
+ }
+ EXTEND2 {
+ NSPEC := 0x00 [1 byte]
+ HTYPE := 0xFFFF [2 bytes]
+ HLEN := 0x0000 [2 bytes]
+ HDATA := SERVER_HDATA[955,1447] [492 bytes]
+ }
+ EXTEND2 {
+ NSPEC := 0x00 [1 byte]
+ HTYPE := 0xFFFF [2 bytes]
+ HLEN := 0x0000 [2 bytes]
+ HDATA := SERVER_HDATA[1448,1939] [492 bytes]
+ }
+ EXTEND2 {
+ NSPEC := 0x00 [1 byte]
+ HTYPE := 0xFFFF [2 bytes]
+ HLEN := 0x0000 [2 bytes]
+ HDATA := SERVER_HDATA[1940,2112] [172 bytes]
+ }
+
+
+§B. NewHope Internal Functions
+
+ gen_a(SEED): returns a uniformly random poly
+ poly_getnoise(): returns a poly sampled from a centered binomial
+ poly_ntt(poly): number-theoretic transform; returns a poly
+ poly_invntt(poly): inverse number-theoretic transform; returns a poly
+ poly_pointwise(poly, poly): pointwise multiplication; returns a poly
+ poly_tobytes(poly): packs a poly to a NEWHOPE_POLY byte array
+ poly_frombytes(NEWHOPE_POLY): unpacks a NEWHOPE_POLY byte array to a poly
+
+ helprec(poly): returns a NEWHOPE_REC byte array
+ rec(poly, NEWHOPE_REC): returns a NEWHOPE_KEY
+
+
+ --- Description of the Newhope internal functions ---
+
+ gen_a(SEED seed) receives as input a 32-byte (public) seed. It expands
+ this seed through SHAKE-128 from the FIPS202 standard. The output of SHAKE-128
+ is considered a sequence of 16-bit little-endian integers. This sequence is
+ used to initialize the coefficients of the returned polynomial from the least
+ significant (coefficient of X^0) to the most significant (coefficient of
+ X^1023) coefficient. For each of the 16-bit integers first eliminate the
+ highest two bits (to make it a 14-bit integer) and then use it as the next
+ coefficient if it is smaller than q=12289.
+ Note that the amount of output required from SHAKE to initialize all 1024
+ coefficients of the polynomial varies depending on the input seed.
+ Note further that this function does not process any secret data and thus does
+ not need any timing-attack protection.
+
+
+ poly_getnoise() first generates 4096 Bytes of uniformly random data. This can
+ be done by reading these bytes from the system's RNG; efficient
+ implementations will typically only read a 32-byte seed from the system's RNG
+ and expand it through some fast PRNG (for example, ChaCha20 or AES-256 in CTR
+ mode). The output of the PRG is considered an array of 2048 16-bit integers
+ r[0],...,r[2047]. The coefficients of the output polynomial are computed as
+ HW(r[0])-HW(r[1]), HW(r[2])-HW(r[3]),...,HW(r[2046])-HW(r[2047]), where HW
+ stands for Hamming weight.
+ Note that the choice of RNG is a local decision; different implementations are
+ free to use different RNGs.
+ Note further that the output of this function is secret; the PRG (and the
+ computation of HW) need to be protected against timing attacks.
+
+
+ poly_ntt(poly f): For a mathematical description of poly_ntt see the [0]; a
+ pseudocode description of a very naive inplace transformation of an input
+ polynomial f = f[0] + f[1]*X + f[2]*X^2 + ... + f[1023]*X^1023 is the
+ following code (all arithmetic on coefficients performed modulo q):
+
+ psi = 7
+ omega = 49
+
+ for i in range(0,n):
+ t[i] = f[i] * psi^i
+
+ for i in range(0,n):
+ f[i] = 0
+ for j in range(0,n):
+ f[i] += t[j] * omega^((i*j)%n)
+
+ Note that this is not how poly_ntt should be implemented if performance is
+ an issue; in particular, efficient algorithms for the number-theoretic
+ transform take time O(n*log(n)) and not O(n^2)
+ Note further that all arithmetic in poly_ntt has to be protected against
+ timing attacks.
+
+
+ poly_invntt(poly f): For a mathematical description of poly_invntt see the
+ [0]; a pseudocode description of a very naive inplace transformation of an
+ input polynomial f = f[0] + f[1]*X + f[2]*X^2 + ... + f[1023]*X^1023 is the
+ following code (all arithmetic on coefficients performed modulo q):
+
+ invpsi = 8778;
+ invomega = 1254;
+ invn = 12277;
+
+ for i in range(0,n):
+ t[i] = f[i];
+
+ for i in range(0,n):
+ f[i]=0;
+ for j in range(0,n):
+ f[i] += t[j] * invomega^((i*j)%n)
+ f[i] *= invpsi^i
+ f[i] *= invn
+
+ Note that this is not how poly_invntt should be implemented if performance
+ is an issue; in particular, efficient algorithms for the inverse
+ number-theoretic transform take time O(n*log(n)) and not O(n^2)
+ Note further that all arithmetic in poly_invntt has to be protected against
+ timing attacks.
+
+
+ poly_pointwise(poly f, poly g) performs pointwise multiplication of the two
+ polynomials. This means that for f = (f0 + f1*X + f2*X^2 + ... +
+ f1023*X^1023) and g = (g0 + g1*X + g2*X^2 + ... + g1023*X^1023) it computes
+ and returns h = (h0 + h1*X + h2*X^2 + ... + h1023*X^1023) with h0 = f0*g0,
+ h1 = f1*g1,..., h1023 = f1023*g1023.
+
+
+ poly_tobytes(poly f) first reduces all coefficents of f modulo q, i.e.,
+ brings them to the interval [0,q-1]. Denote these reduced coefficients as
+ f0,..., f1023; note that they all fit into 14 bits. The function then packs
+ those coefficients into an array of 1792 bytes r[0],..., r[1792] in "packed
+ little-endian representation", i.e.,
+ r[0] = f[0] & 0xff;
+ r[1] = (f[0] >> 8) & ((f[1] & 0x03) << 6)
+ r[2] = (f[1] >> 2) & 0xff;
+ r[3] = (f[1] >> 10) & ((f[2] & 0x0f) << 4)
+ .
+ .
+ .
+ r[1790] = (f[1022]) >> 12) & ((f[1023] & 0x3f) << 2)
+ r[1791] = f[1023] >> 6
+ Note that this function needs to be protected against timing attacks. In
+ particular, avoid non-constant-time conditional subtractions (or other
+ non-constant-time expressions) in the reduction modulo q of the coefficients.
+
+
+ poly_frombytes(NEWHOPE_POLY b) is the inverse of poly_tobytes; it receives
+ as input an array of 1792 bytes and coverts it into the internal
+ representation of a poly. Note that poly_frombytes does not need to check
+ whether the coefficients are reduced modulo q or reduce coefficients modulo
+ q. Note further that the function must not leak any information about its
+ inputs through timing information, as it is also applied to the secret key
+ of the initiator.
+
+
+ helprec(poly f) computes 256 bytes of reconciliation information from the
+ input poly f. Internally, one byte of reconciliation information is computed
+ from four coefficients of f by a function helprec4. Let the input polynomial f
+ = (f0 + f1*X + f2*X^2 + ... + f1023*X^1023); let the output byte array be
+ r[0],...r[256]. This output byte array is computed as
+ r[0] = helprec4(f0,f256,f512,f768)
+ r[1] = helprec4(f1,f257,f513,f769)
+ r[2] = helprec4(f2,f258,f514,f770)
+ .
+ .
+ .
+ r[255] = helprec4(f255,f511,f767,f1023), where helprec4 does the following:
+
+ helprec4(x0,x1,x2,x3):
+ b = randombit()
+ r0,r1,r2,r3 = CVPD4(8*x0+4*b,8*x1+4*b,8*x2+4*b,8*x3+4*b)
+ r = (r0 & 0x03) | ((r1 & 0x03) << 2) | ((r2 & 0x03) << 4) | ((r3 & 0x03) << 6)
+ return r
+
+ The function CVPD4 does the following:
+
+ CVPD4(y0,y1,y2,y3):
+ v00 = round(y0/2q)
+ v01 = round(y1/2q)
+ v02 = round(y2/2q)
+ v03 = round(y3/2q)
+ v10 = round((y0-1)/2q)
+ v11 = round((y1-1)/2q)
+ v12 = round((y2-1)/2q)
+ v13 = round((y3-1)/2q)
+ t = abs(y0 - 2q*v00)
+ t += abs(y1 - 2q*v01)
+ t += abs(y2 - 2q*v02)
+ t += abs(y3 - 2q*v03)
+ if(t < 2q):
+ v0 = v00
+ v1 = v01
+ v2 = v02
+ v3 = v03
+ k = 0
+ else
+ v0 = v10
+ v1 = v11
+ v2 = v12
+ v3 = v13
+ r = 1
+ return (v0-v3,v1-v3,v2-v3,k+2*v3)
+
+ In this description, round() returns the closest integer and abs() returns the
+ absolute value.
+ Note that all computations involved in helprec operate on secret data and must
+ be protected against timing attacks.
+
+
+ rec(poly f, NEWHOPE_REC r) computes the pre-hash (see paper) Newhope key from
+ f and r. Specifically, it computes one bit of key from 4 coefficients of f and
+ one byte of r. Let f = f0 + f1*X + f2*X^2 + ... + f1023*X^1023 and let r =
+ r[0],r[1],...,r[255]. Let the bytes of the output by k[0],...,k[31] and let
+ the bits of the output by k0,...,k255, where
+ k0 = k[0] & 0x01
+ k1 = (k[0] >> 1) & 0x01
+ k2 = (k[0] >> 2) & 0x01
+ .
+ .
+ .
+ k8 = k[1] & 0x01
+ k9 = (k[1] >> 1) & 0x01
+ .
+ .
+ .
+ k255 = (k[32] >> 7)
+ The function rec computes k0,...,k255 as
+ k0 = rec4(f0,f256,f512,f768,r[0])
+ k1 = rec4(f1,f257,f513,f769,r[1])
+ .
+ .
+ .
+ k255 = rec4(f255,f511,f767,f1023,r[255])
+
+ The function rec4 does the following:
+
+ rec4(y0,y1,y2,y3,r):
+ r0 = r & 0x03
+ r1 = (r >> 2) & 0x03
+ r2 = (r >> 4) & 0x03
+ r3 = (r >> 6) & 0x03
+ Decode(8*y0-2q*r0, 8*y1-2q*r1, 8*y2-2q*r2, 8*y3-q*r3)
+
+ The function Decode does the following:
+
+ Decode(v0,v1,v2,v3):
+ t0 = round(v0/8q)
+ t1 = round(v1/8q)
+ t2 = round(v2/8q)
+ t3 = round(v3/8q)
+ t = abs(v0 - 8q*t0)
+ t += abs(v0 - 8q*t0)
+ t += abs(v0 - 8q*t0)
+ t += abs(v0 - 8q*t0)
+ if(t > 1) return 1
+ else return 0
+
+
+§C. Test Vectors
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