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putty-source/crypto/ntru.c
Jacob Nevins 23572715fd Add IANA kex name sntrup761x25519-sha512. 
draft-ietf-sshm-ntruprime-ssh-00 asserts that it's identical to the
@openssh.com version we already implement:
'[sntrup761x25519-sha512@openssh.com] became the default key exchange
algorithm in OpenSSH during 2022. That is identical to the
"sntrup761x25519-sha512" mechanism described in this document.'
2024-11-09 23:55:57 +00:00

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/*
* Implementation of OpenSSH 9.x's hybrid key exchange protocol
* sntrup761x25519-sha512@openssh.com .
*
* This consists of the 'Streamlined NTRU Prime' quantum-resistant
* cryptosystem, run in parallel with ordinary Curve25519 to generate
* a shared secret combining the output of both systems.
*
* (Hence, even if you don't trust this newfangled NTRU Prime thing at
* all, it's at least no _less_ secure than the kex you were using
* already.)
*
* References for the NTRU Prime cryptosystem, up to and including
* binary encodings of public and private keys and the exact preimages
* of the hashes used in key exchange:
*
* https://ntruprime.cr.yp.to/
* https://ntruprime.cr.yp.to/nist/ntruprime-20201007.pdf
*
* The SSH protocol layer is not documented anywhere I could find (as
* of 2022-04-15, not even in OpenSSH's PROTOCOL.* files). I had to
* read OpenSSH's source code to find out how it worked, and the
* answer is as follows:
*
* This hybrid kex method is treated for SSH purposes as a form of
* elliptic-curve Diffie-Hellman, and shares the same SSH message
* sequence: client sends SSH2_MSG_KEX_ECDH_INIT containing its public
* half, server responds with SSH2_MSG_KEX_ECDH_REPLY containing _its_
* public half plus the host key and signature on the shared secret.
*
* (This is a bit of a fudge, because unlike actual ECDH, this kex
* method is asymmetric: one side sends a public key, and the other
* side encrypts something with it and sends the ciphertext back. So
* while the normal ECDH implementations can compute the two sides
* independently in parallel, this system reusing the same messages
* has to be serial. But the order of the messages _is_ firmly
* specified in SSH ECDH, so it works anyway.)
*
* For this kex method, SSH2_MSG_KEX_ECDH_INIT still contains a single
* SSH 'string', which consists of the concatenation of a Streamlined
* NTRU Prime public key with the Curve25519 public value. (Both of
* these have fixed length in bytes, so there's no ambiguity in the
* concatenation.)
*
* SSH2_MSG_KEX_ECDH_REPLY is mostly the same as usual. The only
* string in the packet that varies is the second one, which would
* normally contain the server's public elliptic curve point. Instead,
* it now contains the concatenation of
*
* - a Streamlined NTRU Prime ciphertext
* - the 'confirmation hash' specified in ntruprime-20201007.pdf,
* hashing the plaintext of that ciphertext together with the
* public key
* - the Curve25519 public point as usual.
*
* Again, all three of those elements have fixed lengths.
*
* The client decrypts the ciphertext, checks the confirmation hash,
* and if successful, generates the 'session hash' specified in
* ntruprime-20201007.pdf, which is 32 bytes long and is the ultimate
* output of the Streamlined NTRU Prime key exchange.
*
* The output of the hybrid kex method as a whole is an SSH 'string'
* of length 64 containing the SHA-512 hash of the concatenatio of
*
* - the Streamlined NTRU Prime session hash (32 bytes)
* - the Curve25519 shared secret (32 bytes).
*
* That string is included directly into the SSH exchange hash and key
* derivation hashes, in place of the mpint that comes out of most
* other kex methods.
*/
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
#include "putty.h"
#include "ssh.h"
#include "mpint.h"
#include "ntru.h"
/* ----------------------------------------------------------------------
* Preliminaries: we're going to need to do modular arithmetic on
* small values (considerably smaller than 2^16), and we need to do it
* without using integer division which might not be time-safe.
*
* The strategy for this is the same as I used in
* mp_mod_known_integer: see there for the proofs. The basic idea is
* that we precompute the reciprocal of our modulus as a fixed-point
* number, and use that to get an approximate quotient which we
* subtract off. For these integer sizes, precomputing a fixed-point
* reciprocal of the form (2^48 / modulus) leaves us at most off by 1
* in the quotient, so there's a single (time-safe) trial subtraction
* at the end.
*
* (It's possible that some speed could be gained by not reducing
* fully at every step. But then you'd have to carefully identify all
* the places in the algorithm where things are compared to zero. This
* was the easiest way to get it all working in the first place.)
*/
/* Precompute the reciprocal */
static uint64_t reciprocal_for_reduction(uint16_t q)
{
return ((uint64_t)1 << 48) / q;
}
/* Reduce x mod q, assuming qrecip == reciprocal_for_reduction(q) */
static uint16_t reduce(uint32_t x, uint16_t q, uint64_t qrecip)
{
uint64_t unshifted_quot = x * qrecip;
uint64_t quot = unshifted_quot >> 48;
uint16_t reduced = x - quot * q;
reduced -= q * (1 & ((q-1 - reduced) >> 15));
return reduced;
}
/* Reduce x mod q as above, but also return the quotient */
static uint16_t reduce_with_quot(uint32_t x, uint32_t *quot_out,
uint16_t q, uint64_t qrecip)
{
uint64_t unshifted_quot = x * qrecip;
uint64_t quot = unshifted_quot >> 48;
uint16_t reduced = x - quot * q;
uint64_t extraquot = (1 & ((q-1 - reduced) >> 15));
reduced -= extraquot * q;
*quot_out = quot + extraquot;
return reduced;
}
/* Invert x mod q, assuming it's nonzero. (For time-safety, no check
* is made for zero; it just returns 0.) */
static uint16_t invert(uint16_t x, uint16_t q, uint64_t qrecip)
{
/* Fermat inversion: compute x^(q-2), since x^(q-1) == 1. */
uint32_t sq = x, bit = 1, acc = 1, exp = q-2;
while (1) {
if (exp & bit) {
acc = reduce(acc * sq, q, qrecip);
exp &= ~bit;
if (!exp)
return acc;
}
sq = reduce(sq * sq, q, qrecip);
bit <<= 1;
}
}
/* Check whether x == 0, time-safely, and return 1 if it is or 0 otherwise. */
static unsigned iszero(uint16_t x)
{
return 1 & ~((x + 0xFFFF) >> 16);
}
/*
* Handy macros to cut down on all those extra function parameters. In
* the common case where a function is working mod the same modulus
* throughout (and has called it q), you can just write 'SETUP;' at
* the top and then call REDUCE(...) and INVERT(...) without having to
* write out q and qrecip every time.
*/
#define SETUP uint64_t qrecip = reciprocal_for_reduction(q)
#define REDUCE(x) reduce(x, q, qrecip)
#define INVERT(x) invert(x, q, qrecip)
/* ----------------------------------------------------------------------
* Quotient-ring functions.
*
* NTRU Prime works with two similar but different quotient rings:
*
* Z_q[x] / <x^p-x-1> where p,q are the prime parameters of the system
* Z_3[x] / <x^p-x-1> with the same p, but coefficients mod 3.
*
* The former is a field (every nonzero element is invertible),
* because the system parameters are chosen such that x^p-x-1 is
* invertible over Z_q. The latter is not a field (or not necessarily,
* and in particular, not for the value of p we use here).
*
* In these core functions, you pass in the modulus you want as the
* parameter q, which is either the 'real' q specified in the system
* parameters, or 3 if you're doing one of the mod-3 parts of the
* algorithm.
*/
/*
* Multiply two elements of a quotient ring.
*
* 'a' and 'b' are arrays of exactly p coefficients, with constant
* term first. 'out' is an array the same size to write the inverse
* into.
*/
void ntru_ring_multiply(uint16_t *out, const uint16_t *a, const uint16_t *b,
unsigned p, unsigned q)
{
SETUP;
/*
* Strategy: just compute the full product with 2p coefficients,
* and then reduce it mod x^p-x-1 by working downwards from the
* top coefficient replacing x^{p+k} with (x+1)x^k for k = ...,1,0.
*
* Possibly some speed could be gained here by doing the recursive
* Karatsuba optimisation for the initial multiplication? But I
* haven't tried it.
*/
uint32_t *unreduced = snewn(2*p, uint32_t);
for (unsigned i = 0; i < 2*p; i++)
unreduced[i] = 0;
for (unsigned i = 0; i < p; i++)
for (unsigned j = 0; j < p; j++)
unreduced[i+j] = REDUCE(unreduced[i+j] + a[i] * b[j]);
for (unsigned i = 2*p - 1; i >= p; i--) {
unreduced[i-p] += unreduced[i];
unreduced[i-p+1] += unreduced[i];
unreduced[i] = 0;
}
for (unsigned i = 0; i < p; i++)
out[i] = REDUCE(unreduced[i]);
smemclr(unreduced, 2*p * sizeof(*unreduced));
sfree(unreduced);
}
/*
* Invert an element of the quotient ring.
*
* 'in' is an array of exactly p coefficients, with constant term
* first. 'out' is an array the same size to write the inverse into.
*
* Method: essentially Stein's gcd algorithm, taking the gcd of the
* input (regarded as an element of Z_q[x] proper) and x^p-x-1. Given
* two polynomials over a field which are not both divisible by x, you
* can find their gcd by iterating the following procedure:
*
* - if one is divisible by x, divide off x
* - otherwise, subtract from the higher-degree one whatever scalar
* multiple of the lower-degree one will make it divisible by x,
* and _then_ divide off x
*
* Neither of these types of step changes the gcd of the two
* polynomials.
*
* Each step reduces the sum of the two polynomials' degree by at
* least one, as long as at least one of the degrees is positive.
* (Maybe more than one if all the stars align in the second case, if
* the subtraction cancels the leading term as well as the constant
* term.) So in at most deg A + deg B steps, we must have reached the
* situation where both polys are constants; in one more step after
* that, one of them will be zero; and in one step after _that_, the
* zero one will reliably be the one we're dividing by x. Or rather,
* that's what happens in the case where A,B are coprime; if not, then
* one hits zero while the other is still nonzero.
*
* In a normal gcd algorithm, you'd track a linear combination of the
* two original polynomials that yields each working value, and end up
* with a linear combination of the inputs that yields the gcd. In
* this algorithm, the 'divide off x' step makes that awkward - but we
* can solve that by instead multiplying by the inverse of x in the
* ring that we want our answer to be valid in! And since the modulus
* polynomial of the ring is x^p-x-1, the inverse of x is easy to
* calculate, because it's always just x^{p-1} - 1, which is also very
* easy to multiply by.
*/
unsigned ntru_ring_invert(uint16_t *out, const uint16_t *in,
unsigned p, unsigned q)
{
SETUP;
/* Size of the polynomial arrays we'll work with */
const size_t SIZE = p+1;
/* Number of steps of the algorithm is the max possible value of
* deg A + deg B + 2, where deg A <= p-1 and deg B = p */
const size_t STEPS = 2*p + 1;
/* Our two working polynomials */
uint16_t *A = snewn(SIZE, uint16_t);
uint16_t *B = snewn(SIZE, uint16_t);
/* Coefficient of the input value in each one */
uint16_t *Ac = snewn(SIZE, uint16_t);
uint16_t *Bc = snewn(SIZE, uint16_t);
/* Initialise A to the input, and Ac correspondingly to 1 */
memcpy(A, in, p*sizeof(uint16_t));
A[p] = 0;
Ac[0] = 1;
for (size_t i = 1; i < SIZE; i++)
Ac[i] = 0;
/* Initialise B to the quotient polynomial of the ring, x^p-x-1
* And Bc = 0 */
B[0] = B[1] = q-1;
for (size_t i = 2; i < p; i++)
B[i] = 0;
B[p] = 1;
for (size_t i = 0; i < SIZE; i++)
Bc[i] = 0;
/* Run the gcd-finding algorithm. */
for (size_t i = 0; i < STEPS; i++) {
/*
* First swap round so that A is the one we'll be dividing by x.
*
* In the case where one of the two polys has a zero constant
* term, it's that one. In the other case, it's the one of
* smaller degree. We must compute both, and choose between
* them in a side-channel-safe way.
*/
unsigned x_divides_A = iszero(A[0]);
unsigned x_divides_B = iszero(B[0]);
unsigned B_is_bigger = 0;
{
unsigned not_seen_top_term_of_A = 1, not_seen_top_term_of_B = 1;
for (size_t j = SIZE; j-- > 0 ;) {
not_seen_top_term_of_A &= iszero(A[j]);
not_seen_top_term_of_B &= iszero(B[j]);
B_is_bigger |= (~not_seen_top_term_of_B &
not_seen_top_term_of_A);
}
}
unsigned need_swap = x_divides_B | (~x_divides_A & B_is_bigger);
uint16_t swap_mask = -need_swap;
for (size_t j = 0; j < SIZE; j++) {
uint16_t diff = (A[j] ^ B[j]) & swap_mask;
A[j] ^= diff;
B[j] ^= diff;
}
for (size_t j = 0; j < SIZE; j++) {
uint16_t diff = (Ac[j] ^ Bc[j]) & swap_mask;
Ac[j] ^= diff;
Bc[j] ^= diff;
}
/*
* Replace A with a linear combination of both A and B that
* has constant term zero, which we do by calculating
*
* (constant term of B) * A - (constant term of A) * B
*
* In one of the two cases, A's constant term is already zero,
* so the coefficient of B will be zero too; hence, this will
* do nothing useful (it will merely scale A by some scalar
* value), but it will take the same length of time as doing
* something, which is just what we want.
*/
uint16_t Amult = B[0], Bmult = q - A[0];
for (size_t j = 0; j < SIZE; j++)
A[j] = REDUCE(Amult * A[j] + Bmult * B[j]);
/* And do the same transformation to Ac */
for (size_t j = 0; j < SIZE; j++)
Ac[j] = REDUCE(Amult * Ac[j] + Bmult * Bc[j]);
/*
* Now divide A by x, and compensate by multiplying Ac by
* x^{p-1}-1 mod x^p-x-1.
*
* That multiplication is particularly easy, precisely because
* x^{p-1}-1 is the multiplicative inverse of x! Each x^n term
* for n>0 just moves down to the x^{n-1} term, and only the
* constant term has to be dealt with in an interesting way.
*/
for (size_t j = 1; j < SIZE; j++)
A[j-1] = A[j];
A[SIZE-1] = 0;
uint16_t Ac0 = Ac[0];
for (size_t j = 1; j < p; j++)
Ac[j-1] = Ac[j];
Ac[p-1] = Ac0;
Ac[0] = REDUCE(Ac[0] + q - Ac0);
}
/*
* Now we expect that A is 0, and B is a constant. If so, then
* they are coprime, and we're going to return success. If not,
* they have a common factor.
*/
unsigned success = iszero(A[0]) & (1 ^ iszero(B[0]));
for (size_t j = 1; j < SIZE; j++)
success &= iszero(A[j]) & iszero(B[j]);
/*
* So we're going to return Bc, but first, scale it by the
* multiplicative inverse of the constant we ended up with in
* B[0].
*/
uint16_t scale = INVERT(B[0]);
for (size_t i = 0; i < p; i++)
out[i] = REDUCE(scale * Bc[i]);
smemclr(A, SIZE * sizeof(*A));
sfree(A);
smemclr(B, SIZE * sizeof(*B));
sfree(B);
smemclr(Ac, SIZE * sizeof(*Ac));
sfree(Ac);
smemclr(Bc, SIZE * sizeof(*Bc));
sfree(Bc);
return success;
}
/*
* Given an array of values mod q, convert each one to its
* minimum-absolute-value representative, and then reduce mod 3.
*
* Output values are 0, 1 and 0xFFFF, representing -1.
*
* (Normally our arrays of uint16_t are in 'minimal non-negative
* residue' form, so the output of this function is unusual. But it's
* useful to have it in this form so that it can be reused by
* ntru_round3. You can put it back to the usual representation using
* ntru_normalise, below.)
*/
void ntru_mod3(uint16_t *out, const uint16_t *in, unsigned p, unsigned q)
{
uint64_t qrecip = reciprocal_for_reduction(q);
uint64_t recip3 = reciprocal_for_reduction(3);
unsigned bias = q/2;
uint16_t adjust = 3 - reduce(bias-1, 3, recip3);
for (unsigned i = 0; i < p; i++) {
uint16_t val = reduce(in[i] + bias, q, qrecip);
uint16_t residue = reduce(val + adjust, 3, recip3);
out[i] = residue - 1;
}
}
/*
* Given an array of values mod q, round each one to the nearest
* multiple of 3 to its minimum-absolute-value representative.
*
* Output values are signed integers coerced to uint16_t, so again,
* use ntru_normalise afterwards to put them back to normal.
*/
void ntru_round3(uint16_t *out, const uint16_t *in, unsigned p, unsigned q)
{
SETUP;
unsigned bias = q/2;
ntru_mod3(out, in, p, q);
for (unsigned i = 0; i < p; i++)
out[i] = REDUCE(in[i] + bias) - bias - out[i];
}
/*
* Given an array of signed integers coerced to uint16_t in the range
* [-q/2,+q/2], normalise them back to mod q values.
*/
static void ntru_normalise(uint16_t *out, const uint16_t *in,
unsigned p, unsigned q)
{
for (unsigned i = 0; i < p; i++)
out[i] = in[i] + q * (in[i] >> 15);
}
/*
* Given an array of values mod q, add a constant to each one.
*/
void ntru_bias(uint16_t *out, const uint16_t *in, unsigned bias,
unsigned p, unsigned q)
{
SETUP;
for (unsigned i = 0; i < p; i++)
out[i] = REDUCE(in[i] + bias);
}
/*
* Given an array of values mod q, multiply each one by a constant.
*/
void ntru_scale(uint16_t *out, const uint16_t *in, uint16_t scale,
unsigned p, unsigned q)
{
SETUP;
for (unsigned i = 0; i < p; i++)
out[i] = REDUCE(in[i] * scale);
}
/*
* Given an array of values mod 3, convert them to values mod q in a
* way that maps -1,0,+1 to -1,0,+1.
*/
static void ntru_expand(
uint16_t *out, const uint16_t *in, unsigned p, unsigned q)
{
for (size_t i = 0; i < p; i++) {
uint16_t v = in[i];
/* Map 2 to q-1, and leave 0 and 1 unchanged */
v += (v >> 1) * (q-3);
out[i] = v;
}
}
/* ----------------------------------------------------------------------
* Implement the binary encoding from ntruprime-20201007.pdf, which is
* used to encode public keys and ciphertexts (though not plaintexts,
* which are done in a much simpler way).
*
* The general idea is that your encoder takes as input a list of
* small non-negative integers (r_i), and a sequence of limits (m_i)
* such that 0 <= r_i < m_i, and emits a sequence of bytes that encode
* all of these as tightly as reasonably possible.
*
* That's more general than is really needed, because in both the
* actual uses of this encoding, the input m_i are all the same! But
* the array of (r_i,m_i) pairs evolves during encoding, so they don't
* _stay_ all the same, so you still have to have all the generality.
*
* The encoding process makes a number of passes along the list of
* inputs. In each step, pairs of adjacent numbers are combined into
* one larger one by turning (r_i,m_i) and (r_{i+1},m_{i+1}) into the
* pair (r_i + m_i r_{i+1}, m_i m_{i+1}), i.e. so that the original
* numbers could be recovered by taking the quotient and remaiinder of
* the new r value by m_i. Then, if the new m_i is at least 2^14, we
* emit the low 8 bits of r_i to the output stream and reduce r_i and
* its limit correspondingly. So at the end of the pass, we've got
* half as many numbers still to encode, they're all still not too
* big, and we've emitted some amount of data into the output. Then do
* another pass, keep going until there's only one number left, and
* emit it little-endian.
*
* That's all very well, but how do you decode it again? DJB exhibits
* a pair of recursive functions that are supposed to be mutually
* inverse, but I didn't have any confidence that I'd be able to debug
* them sensibly if they turned out not to be (or rather, if I
* implemented one of them wrong). So I came up with my own strategy
* instead.
*
* In my strategy, we start by processing just the (m_i) into an
* 'encoding schedule' consisting of a sequence of simple
* instructions. The instructions operate on a FIFO queue of numbers,
* initialised to the original (r_i). The three instruction types are:
*
* - 'COMBINE': consume two numbers a,b from the head of the queue,
* combine them by calculating a + m*b for some specified m, and
* push the result on the tail of the queue.
*
* - 'BYTE': divide the tail element of the queue by 2^8 and emit the
* low bits into the output stream.
*
* - 'COPY': pop a number from the head of the queue and push it
* straight back on the tail. (Used for handling the leftover
* element at the end of a pass if the input to the pass was a list
* of odd length.)
*
* So we effectively implement DJB's encoding process in simulation,
* and instead of actually processing a set of (r_i), we 'compile' the
* process into a sequence of instructions that can be handed just the
* (r_i) later and encode them in the right way. At the end of the
* instructions, the queue is expected to have been reduced to length
* 1 and contain the single integer 0.
*
* The nice thing about this system is that each of those three
* instructions is easy to reverse. So you can also use the same
* instructions for decoding: start with a queue containing 0, and
* process the instructions in reverse order and reverse sense. So
* BYTE means to _consume_ a byte from the encoded data (starting from
* the rightmost end) and use it to make a queue element bigger; and
* COMBINE run in reverse pops a single element from one end of the
* queue, divides it by m, and pushes the quotient and remainder on
* the other end.
*
* (So it's easy to debug, because the queue passes through the exact
* same sequence of states during decoding that it did during
* encoding, just in reverse order.)
*
* Also, the encoding schedule comes with information about the
* expected size of the encoded data, because you can find that out
* easily by just counting the BYTE commands.
*/
enum {
/*
* Command values appearing in the 'ops' array. ENC_COPY and
* ENC_BYTE are single values; values of the form
* (ENC_COMBINE_BASE + m) represent a COMBINE command with
* parameter m.
*/
ENC_COPY, ENC_BYTE, ENC_COMBINE_BASE
};
struct NTRUEncodeSchedule {
/*
* Object representing a compiled set of encoding instructions.
*
* 'nvals' is the number of r_i we expect to encode. 'nops' is the
* number of encoding commands in the 'ops' list; 'opsize' is the
* physical size of the array, used during construction.
*
* 'endpos' is used to avoid a last-minute faff during decoding.
* We implement our FIFO of integers as a ring buffer of size
* 'nvals'. Encoding cycles round it some number of times, and the
* final 0 element ends up at some random location in the array.
* If we know _where_ the 0 ends up during encoding, we can put
* the initial 0 there at the start of decoding, and then when we
* finish reversing all the instructions, we'll end up with the
* output numbers already arranged at their correct positions, so
* that there's no need to rotate the array at the last minute.
*/
size_t nvals, endpos, nops, opsize;
uint32_t *ops;
};
static inline void sched_append(NTRUEncodeSchedule *sched, uint16_t op)
{
/* Helper function to append an operation to the schedule, and
* update endpos. */
sgrowarray(sched->ops, sched->opsize, sched->nops);
sched->ops[sched->nops++] = op;
if (op != ENC_BYTE)
sched->endpos = (sched->endpos + 1) % sched->nvals;
}
/*
* Take in the list of limit values (m_i) and compute the encoding
* schedule.
*/
NTRUEncodeSchedule *ntru_encode_schedule(const uint16_t *ms_in, size_t n)
{
NTRUEncodeSchedule *sched = snew(NTRUEncodeSchedule);
sched->nvals = n;
sched->endpos = n-1;
sched->nops = sched->opsize = 0;
sched->ops = NULL;
assert(n != 0);
/*
* 'ms' is the list of (m_i) on input to the current pass.
* 'ms_new' is the list output from the current pass. After each
* pass we swap the arrays round.
*/
uint32_t *ms = snewn(n, uint32_t);
uint32_t *msnew = snewn(n, uint32_t);
for (size_t i = 0; i < n; i++)
ms[i] = ms_in[i];
while (n > 1) {
size_t nnew = 0;
for (size_t i = 0; i < n; i += 2) {
if (i+1 == n) {
/*
* Odd element at the end of the input list: just copy
* it unchanged to the output.
*/
sched_append(sched, ENC_COPY);
msnew[nnew++] = ms[i];
break;
}
/*
* Normal case: consume two elements from the input list
* and combine them.
*/
uint32_t m1 = ms[i], m2 = ms[i+1], m = m1*m2;
sched_append(sched, ENC_COMBINE_BASE + m1);
/*
* And then, as long as the combined limit is big enough,
* emit an output byte from the bottom of it.
*/
while (m >= (1<<14)) {
sched_append(sched, ENC_BYTE);
m = (m + 0xFF) >> 8;
}
/*
* Whatever is left after that, we emit into the output
* list and append to the fifo.
*/
msnew[nnew++] = m;
}
/*
* End of pass. The output list of (m_i) now becomes the input
* list.
*/
uint32_t *tmp = ms;
ms = msnew;
n = nnew;
msnew = tmp;
}
/*
* When that loop terminates, it's because there's exactly one
* number left to encode. (Or, technically, _at most_ one - but we
* don't support encoding a completely empty list in this
* implementation, because what would be the point?) That number
* is just emitted little-endian until its limit is 1 (meaning its
* only possible actual value is 0).
*/
assert(n == 1);
uint32_t m = ms[0];
while (m > 1) {
sched_append(sched, ENC_BYTE);
m = (m + 0xFF) >> 8;
}
sfree(ms);
sfree(msnew);
return sched;
}
void ntru_encode_schedule_free(NTRUEncodeSchedule *sched)
{
sfree(sched->ops);
sfree(sched);
}
/*
* Calculate the output length of the encoded data in bytes.
*/
size_t ntru_encode_schedule_length(NTRUEncodeSchedule *sched)
{
size_t len = 0;
for (size_t i = 0; i < sched->nops; i++)
if (sched->ops[i] == ENC_BYTE)
len++;
return len;
}
/*
* Retrieve the number of items encoded. (Used by testcrypt.)
*/
size_t ntru_encode_schedule_nvals(NTRUEncodeSchedule *sched)
{
return sched->nvals;
}
/*
* Actually encode a sequence of (r_i), emitting the output bytes to
* an arbitrary BinarySink.
*/
void ntru_encode(NTRUEncodeSchedule *sched, const uint16_t *rs_in,
BinarySink *bs)
{
size_t n = sched->nvals;
uint32_t *rs = snewn(n, uint32_t);
for (size_t i = 0; i < n; i++)
rs[i] = rs_in[i];
/*
* The head and tail pointers of the queue are both 'full'. That
* is, rs[head] is the first element actually in the queue, and
* rs[tail] is the last element.
*
* So you append to the queue by first advancing 'tail' and then
* writing to rs[tail], whereas you consume from the queue by
* first reading rs[head] and _then_ advancing 'head'.
*
* The more normal thing would be to make 'tail' point to the
* first empty slot instead of the last full one. But then you'd
* have to faff about with modular arithmetic to find the last
* full slot for the BYTE command, so in this case, it's easier to
* do it the less usual way.
*/
size_t head = 0, tail = n-1;
for (size_t i = 0; i < sched->nops; i++) {
uint16_t op = sched->ops[i];
switch (op) {
case ENC_BYTE:
put_byte(bs, rs[tail] & 0xFF);
rs[tail] >>= 8;
break;
case ENC_COPY: {
uint32_t r = rs[head];
head = (head + 1) % n;
tail = (tail + 1) % n;
rs[tail] = r;
break;
}
default: {
uint32_t r1 = rs[head];
head = (head + 1) % n;
uint32_t r2 = rs[head];
head = (head + 1) % n;
tail = (tail + 1) % n;
rs[tail] = r1 + (op - ENC_COMBINE_BASE) * r2;
break;
}
}
}
/*
* Expect that we've ended up with a single zero in the queue, at
* exactly the position that the setup-time analysis predicted it.
*/
assert(head == sched->endpos);
assert(tail == sched->endpos);
assert(rs[head] == 0);
smemclr(rs, n * sizeof(*rs));
sfree(rs);
}
/*
* Decode a ptrlen of binary data into a sequence of (r_i). The data
* is expected to be of exactly the right length (on pain of assertion
* failure).
*/
void ntru_decode(NTRUEncodeSchedule *sched, uint16_t *rs_out, ptrlen data)
{
size_t n = sched->nvals;
const uint8_t *base = (const uint8_t *)data.ptr;
const uint8_t *pos = base + data.len;
/*
* Initialise the queue to a single zero, at the 'endpos' position
* that will mean the final output is correctly aligned.
*
* 'head' and 'tail' have the same meanings as in encoding. So
* 'tail' is the location that BYTE modifies and COPY and COMBINE
* consume from, and 'head' is the location that COPY and COMBINE
* push on to. As in encoding, they both point at the extremal
* full slots in the array.
*/
uint32_t *rs = snewn(n, uint32_t);
size_t head = sched->endpos, tail = head;
rs[tail] = 0;
for (size_t i = sched->nops; i-- > 0 ;) {
uint16_t op = sched->ops[i];
switch (op) {
case ENC_BYTE: {
assert(pos > base);
uint8_t byte = *--pos;
rs[tail] = (rs[tail] << 8) | byte;
break;
}
case ENC_COPY: {
uint32_t r = rs[tail];
tail = (tail + n - 1) % n;
head = (head + n - 1) % n;
rs[head] = r;
break;
}
default: {
uint32_t r = rs[tail];
tail = (tail + n - 1) % n;
uint32_t m = op - ENC_COMBINE_BASE;
uint64_t mrecip = reciprocal_for_reduction(m);
uint32_t r1, r2;
r1 = reduce_with_quot(r, &r2, m, mrecip);
head = (head + n - 1) % n;
rs[head] = r2;
head = (head + n - 1) % n;
rs[head] = r1;
break;
}
}
}
assert(pos == base);
assert(head == 0);
assert(tail == n-1);
for (size_t i = 0; i < n; i++)
rs_out[i] = rs[i];
smemclr(rs, n * sizeof(*rs));
sfree(rs);
}
/* ----------------------------------------------------------------------
* The actual public-key cryptosystem.
*/
struct NTRUKeyPair {
unsigned p, q, w;
uint16_t *h; /* public key */
uint16_t *f3, *ginv; /* private key */
uint16_t *rho; /* for implicit rejection */
};
/* Helper function to free an array of uint16_t containing a ring
* element, clearing it on the way since some of them are sensitive. */
static void ring_free(uint16_t *val, unsigned p)
{
smemclr(val, p*sizeof(*val));
sfree(val);
}
void ntru_keypair_free(NTRUKeyPair *keypair)
{
ring_free(keypair->h, keypair->p);
ring_free(keypair->f3, keypair->p);
ring_free(keypair->ginv, keypair->p);
ring_free(keypair->rho, keypair->p);
sfree(keypair);
}
/* Trivial accessors used by test programs. */
unsigned ntru_keypair_p(NTRUKeyPair *keypair) { return keypair->p; }
const uint16_t *ntru_pubkey(NTRUKeyPair *keypair) { return keypair->h; }
/*
* Generate a value of the class DJB describes as 'Short': it consists
* of p terms that are all either 0 or +1 or -1, and exactly w of them
* are not zero.
*
* Values of this kind are used for several purposes: part of the
* private key, a plaintext, and the 'rho' fake-plaintext value used
* for deliberately returning a duff but non-revealing session hash if
* things go wrong.
*
* -1 is represented as 2 in the output array. So if you want these
* numbers mod 3, then they come out already in the right form.
* Otherwise, use ntru_expand.
*/
void ntru_gen_short(uint16_t *v, unsigned p, unsigned w)
{
/*
* Get enough random data to generate a polynomial all of whose p
* terms are in {0,+1,-1}, and exactly w of them are nonzero.
* We'll do this by making up a completely random sequence of
* {+1,-1} and then setting a random subset of them to 0.
*
* So we'll need p random bits to choose the nonzero values, and
* then (doing it the simplest way) log2(p!) bits to shuffle them,
* plus say 128 bits to ensure any fluctuations in uniformity are
* negligible.
*
* log2(p!) is a pain to calculate, so we'll bound it above by
* p*log2(p), which we bound in turn by p*16.
*/
size_t randbitpos = 17 * p + 128;
mp_int *randdata = mp_resize(mp_random_bits(randbitpos), randbitpos + 32);
/*
* Initial value before zeroing out some terms: p randomly chosen
* values in {1,2}.
*/
for (size_t i = 0; i < p; i++)
v[i] = 1 + mp_get_bit(randdata, --randbitpos);
/*
* Hereafter we're going to extract random bits by multiplication,
* treating randdata as a large fixed-point number.
*/
mp_reduce_mod_2to(randdata, randbitpos);
/*
* Zero out some terms, leaving a randomly selected w of them
* nonzero.
*/
uint32_t nonzeros_left = w;
mp_int *x = mp_new(64);
for (size_t i = p; i-- > 0 ;) {
/*
* Pick a random number out of the number of terms remaning.
*/
mp_mul_integer_into(randdata, randdata, i+1);
mp_rshift_fixed_into(x, randdata, randbitpos);
mp_reduce_mod_2to(randdata, randbitpos);
size_t j = mp_get_integer(x);
/*
* If that's less than nonzeros_left, then we're leaving this
* number nonzero. Otherwise we're zeroing it out.
*/
uint32_t keep = (uint32_t)(j - nonzeros_left) >> 31;
v[i] &= -keep; /* clear this field if keep == 0 */
nonzeros_left -= keep; /* decrement counter if keep == 1 */
}
mp_free(x);
mp_free(randdata);
}
/*
* Make a single attempt at generating a key pair. This involves
* inventing random elements of both our quotient rings and hoping
* they're both invertible.
*
* They may not be, if you're unlucky. The element of Z_q/<x^p-x-1>
* will _almost_ certainly be invertible, because that is a field, so
* invertibility can only fail if you were so unlucky as to choose the
* all-0s element. But the element of Z_3/<x^p-x-1> may fail to be
* invertible because it has a common factor with x^p-x-1 (which, over
* Z_3, is not irreducible).
*
* So we can't guarantee to generate a key pair in constant time,
* because there's no predicting how many retries we'll need. However,
* this isn't a failure of side-channel safety, because we completely
* discard all the random numbers and state from each failed attempt.
* So if there were a side-channel leakage from a failure, the only
* thing it would give away would be a bunch of random numbers that
* turned out not to be used anyway.
*
* But a _successful_ call to this function should execute in a
* secret-independent manner, and this 'make a single attempt'
* function is exposed in the API so that 'testsc' can check that.
*/
NTRUKeyPair *ntru_keygen_attempt(unsigned p, unsigned q, unsigned w)
{
/*
* First invent g, which is the one more likely to fail to invert.
* This is simply a uniformly random polynomial with p terms over
* Z_3. So we need p*log2(3) random bits for it, plus 128 for
* uniformity. It's easiest to bound log2(3) above by 2.
*/
size_t randbitpos = 2 * p + 128;
mp_int *randdata = mp_resize(mp_random_bits(randbitpos), randbitpos + 32);
/*
* Select p random values from {0,1,2}.
*/
uint16_t *g = snewn(p, uint16_t);
mp_int *x = mp_new(64);
for (size_t i = 0; i < p; i++) {
mp_mul_integer_into(randdata, randdata, 3);
mp_rshift_fixed_into(x, randdata, randbitpos);
mp_reduce_mod_2to(randdata, randbitpos);
g[i] = mp_get_integer(x);
}
mp_free(x);
mp_free(randdata);
/*
* Try to invert g over Z_3, and fail if it isn't invertible.
*/
uint16_t *ginv = snewn(p, uint16_t);
if (!ntru_ring_invert(ginv, g, p, 3)) {
ring_free(g, p);
ring_free(ginv, p);
return NULL;
}
/*
* Fine; we have g. Now make up an f, and convert it to a
* polynomial over q.
*/
uint16_t *f = snewn(p, uint16_t);
ntru_gen_short(f, p, w);
ntru_expand(f, f, p, q);
/*
* Multiply f by 3.
*/
uint16_t *f3 = snewn(p, uint16_t);
ntru_scale(f3, f, 3, p, q);
/*
* Invert 3*f over Z_q. This is guaranteed to succeed, since
* Z_q/<x^p-x-1> is a field, so the only non-invertible value is
* 0. And f is nonzero because it came from ntru_gen_short (hence,
* w of its components are nonzero), hence so is 3*f.
*/
uint16_t *f3inv = snewn(p, uint16_t);
bool expect_always_success = ntru_ring_invert(f3inv, f3, p, q);
assert(expect_always_success);
/*
* Make the public key, by converting g to a polynomial over q and
* then multiplying by f3inv.
*/
uint16_t *g_q = snewn(p, uint16_t);
ntru_expand(g_q, g, p, q);
uint16_t *h = snewn(p, uint16_t);
ntru_ring_multiply(h, g_q, f3inv, p, q);
/*
* Make up rho, used to substitute for the plaintext in the
* session hash in case of confirmation failure.
*/
uint16_t *rho = snewn(p, uint16_t);
ntru_gen_short(rho, p, w);
/*
* And we're done! Free everything except the pieces we're
* returning.
*/
NTRUKeyPair *keypair = snew(NTRUKeyPair);
keypair->p = p;
keypair->q = q;
keypair->w = w;
keypair->h = h;
keypair->f3 = f3;
keypair->ginv = ginv;
keypair->rho = rho;
ring_free(f, p);
ring_free(f3inv, p);
ring_free(g, p);
ring_free(g_q, p);
return keypair;
}
/*
* The top-level key generation function for real use (as opposed to
* testsc): keep trying to make a key until you succeed.
*/
NTRUKeyPair *ntru_keygen(unsigned p, unsigned q, unsigned w)
{
while (1) {
NTRUKeyPair *keypair = ntru_keygen_attempt(p, q, w);
if (keypair)
return keypair;
}
}
/*
* Public-key encryption.
*/
void ntru_encrypt(uint16_t *ciphertext, const uint16_t *plaintext,
uint16_t *pubkey, unsigned p, unsigned q)
{
uint16_t *r_q = snewn(p, uint16_t);
ntru_expand(r_q, plaintext, p, q);
uint16_t *unrounded = snewn(p, uint16_t);
ntru_ring_multiply(unrounded, r_q, pubkey, p, q);
ntru_round3(ciphertext, unrounded, p, q);
ntru_normalise(ciphertext, ciphertext, p, q);
ring_free(r_q, p);
ring_free(unrounded, p);
}
/*
* Public-key decryption.
*/
void ntru_decrypt(uint16_t *plaintext, const uint16_t *ciphertext,
NTRUKeyPair *keypair)
{
unsigned p = keypair->p, q = keypair->q, w = keypair->w;
uint16_t *tmp = snewn(p, uint16_t);
ntru_ring_multiply(tmp, ciphertext, keypair->f3, p, q);
ntru_mod3(tmp, tmp, p, q);
ntru_normalise(tmp, tmp, p, 3);
ntru_ring_multiply(plaintext, tmp, keypair->ginv, p, 3);
ring_free(tmp, p);
/*
* With luck, this should have recovered exactly the original
* plaintext. But, as per the spec, we check whether it has
* exactly w nonzero coefficients, and if not, then something has
* gone wrong - and in that situation we time-safely substitute a
* different output.
*
* (I don't know exactly why we do this, but I assume it's because
* otherwise the mis-decoded output could be made to disgorge a
* secret about the private key in some way.)
*/
unsigned weight = p;
for (size_t i = 0; i < p; i++)
weight -= iszero(plaintext[i]);
unsigned ok = iszero(weight ^ w);
/*
* The default failure return value consists of w 1s followed by
* 0s.
*/
unsigned mask = ok - 1;
for (size_t i = 0; i < w; i++) {
uint16_t diff = (1 ^ plaintext[i]) & mask;
plaintext[i] ^= diff;
}
for (size_t i = w; i < p; i++) {
uint16_t diff = (0 ^ plaintext[i]) & mask;
plaintext[i] ^= diff;
}
}
/* ----------------------------------------------------------------------
* Encode and decode public keys, ciphertexts and plaintexts.
*
* Public keys and ciphertexts use the complicated binary encoding
* system implemented above. In both cases, the inputs are regarded as
* symmetric about zero, and are first biased to map their most
* negative permitted value to 0, so that they become non-negative and
* hence suitable as inputs to the encoding system. In the case of a
* ciphertext, where the input coefficients have also been coerced to
* be multiples of 3, we divide by 3 as well, saving space by reducing
* the upper bounds (m_i) on all the encoded numbers.
*/
/*
* Compute the encoding schedule for a public key.
*/
static NTRUEncodeSchedule *ntru_encode_pubkey_schedule(unsigned p, unsigned q)
{
uint16_t *ms = snewn(p, uint16_t);
for (size_t i = 0; i < p; i++)
ms[i] = q;
NTRUEncodeSchedule *sched = ntru_encode_schedule(ms, p);
sfree(ms);
return sched;
}
/*
* Encode a public key.
*/
void ntru_encode_pubkey(const uint16_t *pubkey, unsigned p, unsigned q,
BinarySink *bs)
{
/* Compute the biased version for encoding */
uint16_t *biased_pubkey = snewn(p, uint16_t);
ntru_bias(biased_pubkey, pubkey, q / 2, p, q);
/* Encode it */
NTRUEncodeSchedule *sched = ntru_encode_pubkey_schedule(p, q);
ntru_encode(sched, biased_pubkey, bs);
ntru_encode_schedule_free(sched);
ring_free(biased_pubkey, p);
}
/*
* Decode a public key and write it into 'pubkey'. We also return a
* ptrlen pointing at the chunk of data we removed from the
* BinarySource.
*/
ptrlen ntru_decode_pubkey(uint16_t *pubkey, unsigned p, unsigned q,
BinarySource *src)
{
NTRUEncodeSchedule *sched = ntru_encode_pubkey_schedule(p, q);
/* Retrieve the right number of bytes from the source */
size_t len = ntru_encode_schedule_length(sched);
ptrlen encoded = get_data(src, len);
if (get_err(src)) {
/* If there wasn't enough data, give up and return all-zeroes
* purely for determinism. But that value should never be
* used, because the caller will also check get_err(src). */
memset(pubkey, 0, p*sizeof(*pubkey));
} else {
/* Do the decoding */
ntru_decode(sched, pubkey, encoded);
/* Unbias the coefficients */
ntru_bias(pubkey, pubkey, q-q/2, p, q);
}
ntru_encode_schedule_free(sched);
return encoded;
}
/*
* For ciphertext biasing: work out the largest absolute value a
* ciphertext element can take, which is given by taking q/2 and
* rounding it to the nearest multiple of 3.
*/
static inline unsigned ciphertext_bias(unsigned q)
{
return (q/2+1) / 3;
}
/*
* The number of possible values of a ciphertext coefficient (for use
* as the m_i in encoding) ranges from +ciphertext_bias(q) to
* -ciphertext_bias(q) inclusive.
*/
static inline unsigned ciphertext_m(unsigned q)
{
return 1 + 2 * ciphertext_bias(q);
}
/*
* Compute the encoding schedule for a ciphertext.
*/
static NTRUEncodeSchedule *ntru_encode_ciphertext_schedule(
unsigned p, unsigned q)
{
unsigned m = ciphertext_m(q);
uint16_t *ms = snewn(p, uint16_t);
for (size_t i = 0; i < p; i++)
ms[i] = m;
NTRUEncodeSchedule *sched = ntru_encode_schedule(ms, p);
sfree(ms);
return sched;
}
/*
* Encode a ciphertext.
*/
void ntru_encode_ciphertext(const uint16_t *ciphertext, unsigned p, unsigned q,
BinarySink *bs)
{
SETUP;
/*
* Bias the ciphertext, and scale down by 1/3, which we do by
* modular multiplication by the inverse of 3 mod q. (That only
* works if we know the inputs are all _exact_ multiples of 3
* - but we do!)
*/
uint16_t *biased_ciphertext = snewn(p, uint16_t);
ntru_bias(biased_ciphertext, ciphertext, 3 * ciphertext_bias(q), p, q);
ntru_scale(biased_ciphertext, biased_ciphertext, INVERT(3), p, q);
/* Encode. */
NTRUEncodeSchedule *sched = ntru_encode_ciphertext_schedule(p, q);
ntru_encode(sched, biased_ciphertext, bs);
ntru_encode_schedule_free(sched);
ring_free(biased_ciphertext, p);
}
ptrlen ntru_decode_ciphertext(uint16_t *ct, NTRUKeyPair *keypair,
BinarySource *src)
{
unsigned p = keypair->p, q = keypair->q;
NTRUEncodeSchedule *sched = ntru_encode_ciphertext_schedule(p, q);
/* Retrieve the right number of bytes from the source */
size_t len = ntru_encode_schedule_length(sched);
ptrlen encoded = get_data(src, len);
if (get_err(src)) {
/* As above, return deterministic nonsense on failure */
memset(ct, 0, p*sizeof(*ct));
} else {
/* Do the decoding */
ntru_decode(sched, ct, encoded);
/* Undo the scaling and bias */
ntru_scale(ct, ct, 3, p, q);
ntru_bias(ct, ct, q - 3 * ciphertext_bias(q), p, q);
}
ntru_encode_schedule_free(sched);
return encoded; /* also useful to the caller, optionally */
}
/*
* Encode a plaintext.
*
* This is a much simpler encoding than the NTRUEncodeSchedule system:
* since elements of a plaintext are mod 3, we just encode each one in
* 2 bits, applying the usual bias so that {-1,0,+1} map to {0,1,2}
* respectively.
*
* There's no corresponding decode function, because plaintexts are
* never transmitted on the wire (the whole point is that they're too
* secret!). Plaintexts are only encoded in order to put them into
* hash preimages.
*/
void ntru_encode_plaintext(const uint16_t *plaintext, unsigned p,
BinarySink *bs)
{
unsigned byte = 0, bitpos = 0;
for (size_t i = 0; i < p; i++) {
unsigned encoding = (plaintext[i] + 1) * iszero(plaintext[i] >> 1);
byte |= encoding << bitpos;
bitpos += 2;
if (bitpos == 8 || i+1 == p) {
put_byte(bs, byte);
byte = 0;
bitpos = 0;
}
}
}
/* ----------------------------------------------------------------------
* Compute the hashes required by the key exchange layer of NTRU Prime.
*
* There are two of these. The 'confirmation hash' is sent by the
* server along with the ciphertext, and the client can recalculate it
* to check whether the ciphertext was decrypted correctly. Then, the
* 'session hash' is the actual output of key exchange, and if the
* confirmation hash doesn't match, it gets deliberately corrupted.
*/
/*
* Make the confirmation hash, whose inputs are the plaintext and the
* public key.
*
* This is defined as H(2 || H(3 || r) || H(4 || K)), where r is the
* plaintext and K is the public key (as encoded by the above
* functions), and the constants 2,3,4 are single bytes. The choice of
* hash function (H itself) is SHA-512 truncated to 256 bits.
*
* (To be clear: that is _not_ the thing that FIPS 180-4 6.7 defines
* as "SHA-512/256", which varies the initialisation vector of the
* SHA-512 algorithm as well as truncating the output. _This_
* algorithm uses the standard SHA-512 IV, and _just_ truncates the
* output, in the manner suggested by FIPS 180-4 section 7.)
*
* 'out' should therefore expect to receive 32 bytes of data.
*/
static void ntru_confirmation_hash(
uint8_t *out, const uint16_t *plaintext,
const uint16_t *pubkey, unsigned p, unsigned q)
{
/* The outer hash object */
ssh_hash *hconfirm = ssh_hash_new(&ssh_sha512);
put_byte(hconfirm, 2); /* initial byte 2 */
uint8_t hashdata[64];
/* Compute H(3 || r) and add it to the main hash */
ssh_hash *h3r = ssh_hash_new(&ssh_sha512);
put_byte(h3r, 3);
ntru_encode_plaintext(plaintext, p, BinarySink_UPCAST(h3r));
ssh_hash_final(h3r, hashdata);
put_data(hconfirm, hashdata, 32);
/* Compute H(4 || K) and add it to the main hash */
ssh_hash *h4K = ssh_hash_new(&ssh_sha512);
put_byte(h4K, 4);
ntru_encode_pubkey(pubkey, p, q, BinarySink_UPCAST(h4K));
ssh_hash_final(h4K, hashdata);
put_data(hconfirm, hashdata, 32);
/* Compute the full output of the main SHA-512 hash */
ssh_hash_final(hconfirm, hashdata);
/* And copy the first 32 bytes into the caller's output array */
memcpy(out, hashdata, 32);
smemclr(hashdata, sizeof(hashdata));
}
/*
* Make the session hash, whose inputs are the plaintext, the
* ciphertext, and the confirmation hash (hence, transitively, a
* dependence on the public key as well).
*
* As computed by the server, and by the client if the confirmation
* hash matched, this is defined as
*
* H(1 || H(3 || r) || ciphertext || confirmation hash)
*
* but if the confirmation hash _didn't_ match, then the plaintext r
* is replaced with the dummy plaintext-shaped value 'rho' we invented
* during key generation (presumably to avoid leaking any information
* about our secrets), and the initial byte 1 is replaced with 0 (to
* ensure that the resulting hash preimage can't match any legitimate
* preimage). So in that case, you instead get
*
* H(0 || H(3 || rho) || ciphertext || confirmation hash)
*
* The inputs to this function include 'ok', which is the value to use
* as the initial byte (1 on success, 0 on failure), and 'plaintext'
* which should already have been substituted with rho in case of
* failure.
*
* The ciphertext is provided in already-encoded form.
*/
static void ntru_session_hash(
uint8_t *out, unsigned ok, const uint16_t *plaintext,
unsigned p, ptrlen ciphertext, ptrlen confirmation_hash)
{
/* The outer hash object */
ssh_hash *hsession = ssh_hash_new(&ssh_sha512);
put_byte(hsession, ok); /* initial byte 1 or 0 */
uint8_t hashdata[64];
/* Compute H(3 || r), or maybe H(3 || rho), and add it to the main hash */
ssh_hash *h3r = ssh_hash_new(&ssh_sha512);
put_byte(h3r, 3);
ntru_encode_plaintext(plaintext, p, BinarySink_UPCAST(h3r));
ssh_hash_final(h3r, hashdata);
put_data(hsession, hashdata, 32);
/* Put the ciphertext and confirmation hash in */
put_datapl(hsession, ciphertext);
put_datapl(hsession, confirmation_hash);
/* Compute the full output of the main SHA-512 hash */
ssh_hash_final(hsession, hashdata);
/* And copy the first 32 bytes into the caller's output array */
memcpy(out, hashdata, 32);
smemclr(hashdata, sizeof(hashdata));
}
/* ----------------------------------------------------------------------
* Top-level key exchange and SSH integration.
*
* Although this system borrows the ECDH packet structure, it's unlike
* true ECDH in that it is completely asymmetric between client and
* server. So we have two separate vtables of methods for the two
* sides of the system, and a third vtable containing only the class
* methods, in particular a constructor which chooses which one to
* instantiate.
*/
/*
* The parameters p,q,w for the system. There are other choices of
* these, but OpenSSH only specifies this set. (If that ever changes,
* we'll need to turn these into elements of the state structures.)
*/
#define p_LIVE 761
#define q_LIVE 4591
#define w_LIVE 286
static char *ssh_ntru_description(const ssh_kex *kex)
{
return dupprintf("NTRU Prime / Curve25519 hybrid key exchange");
}
/*
* State structure for the client, which takes the role of inventing a
* key pair and decrypting a secret plaintext sent to it by the server.
*/
typedef struct ntru_client_key {
NTRUKeyPair *keypair;
ecdh_key *curve25519;
ecdh_key ek;
} ntru_client_key;
static void ssh_ntru_client_free(ecdh_key *dh);
static void ssh_ntru_client_getpublic(ecdh_key *dh, BinarySink *bs);
static bool ssh_ntru_client_getkey(ecdh_key *dh, ptrlen remoteKey,
BinarySink *bs);
static const ecdh_keyalg ssh_ntru_client_vt = {
/* This vtable has no 'new' method, because it's constructed via
* the selector vt below */
.free = ssh_ntru_client_free,
.getpublic = ssh_ntru_client_getpublic,
.getkey = ssh_ntru_client_getkey,
.description = ssh_ntru_description,
};
static ecdh_key *ssh_ntru_client_new(void)
{
ntru_client_key *nk = snew(ntru_client_key);
nk->ek.vt = &ssh_ntru_client_vt;
nk->keypair = ntru_keygen(p_LIVE, q_LIVE, w_LIVE);
nk->curve25519 = ecdh_key_new(&ssh_ec_kex_curve25519, false);
return &nk->ek;
}
static void ssh_ntru_client_free(ecdh_key *dh)
{
ntru_client_key *nk = container_of(dh, ntru_client_key, ek);
ntru_keypair_free(nk->keypair);
ecdh_key_free(nk->curve25519);
sfree(nk);
}
static void ssh_ntru_client_getpublic(ecdh_key *dh, BinarySink *bs)
{
ntru_client_key *nk = container_of(dh, ntru_client_key, ek);
/*
* The client's public information is a single SSH string
* containing the NTRU public key and the Curve25519 public point
* concatenated. So write both of those into the output
* BinarySink.
*/
ntru_encode_pubkey(nk->keypair->h, p_LIVE, q_LIVE, bs);
ecdh_key_getpublic(nk->curve25519, bs);
}
static bool ssh_ntru_client_getkey(ecdh_key *dh, ptrlen remoteKey,
BinarySink *bs)
{
ntru_client_key *nk = container_of(dh, ntru_client_key, ek);
/*
* We expect the server to have sent us a string containing a
* ciphertext, a confirmation hash, and a Curve25519 public point.
* Extract all three.
*/
BinarySource src[1];
BinarySource_BARE_INIT_PL(src, remoteKey);
uint16_t *ciphertext = snewn(p_LIVE, uint16_t);
ptrlen ciphertext_encoded = ntru_decode_ciphertext(
ciphertext, nk->keypair, src);
ptrlen confirmation_hash = get_data(src, 32);
ptrlen curve25519_remoteKey = get_data(src, 32);
if (get_err(src) || get_avail(src)) {
/* Hard-fail if the input wasn't exactly the right length */
ring_free(ciphertext, p_LIVE);
return false;
}
/*
* Main hash object which will combine the NTRU and Curve25519
* outputs.
*/
ssh_hash *h = ssh_hash_new(&ssh_sha512);
/* Reusable buffer for storing various hash outputs. */
uint8_t hashdata[64];
/*
* NTRU side.
*/
{
/* Decrypt the ciphertext to recover the server's plaintext */
uint16_t *plaintext = snewn(p_LIVE, uint16_t);
ntru_decrypt(plaintext, ciphertext, nk->keypair);
/* Make the confirmation hash */
ntru_confirmation_hash(hashdata, plaintext, nk->keypair->h,
p_LIVE, q_LIVE);
/* Check it matches the one the server sent */
unsigned ok = smemeq(hashdata, confirmation_hash.ptr, 32);
/* If not, substitute in rho for the plaintext in the session hash */
unsigned mask = ok-1;
for (size_t i = 0; i < p_LIVE; i++)
plaintext[i] ^= mask & (plaintext[i] ^ nk->keypair->rho[i]);
/* Compute the session hash, whether or not we did that */
ntru_session_hash(hashdata, ok, plaintext, p_LIVE, ciphertext_encoded,
confirmation_hash);
/* Free temporary values */
ring_free(plaintext, p_LIVE);
ring_free(ciphertext, p_LIVE);
/* And put the NTRU session hash into the main hash object. */
put_data(h, hashdata, 32);
}
/*
* Curve25519 side.
*/
{
strbuf *otherkey = strbuf_new_nm();
/* Call out to Curve25519 to compute the shared secret from that
* kex method */
bool ok = ecdh_key_getkey(nk->curve25519, curve25519_remoteKey,
BinarySink_UPCAST(otherkey));
/* If that failed (which only happens if the other end does
* something wrong, like sending a low-order curve point
* outside the subgroup it's supposed to), we might as well
* just abort and return failure. That's what we'd have done
* in standalone Curve25519. */
if (!ok) {
ssh_hash_free(h);
smemclr(hashdata, sizeof(hashdata));
strbuf_free(otherkey);
return false;
}
/*
* ecdh_key_getkey will have returned us a chunk of data
* containing an encoded mpint, which is how the Curve25519
* output normally goes into the exchange hash. But in this
* context we want to treat it as a fixed big-endian 32 bytes,
* so extract it from its encoding and put it into the main
* hash object in the new format.
*/
BinarySource src[1];
BinarySource_BARE_INIT_PL(src, ptrlen_from_strbuf(otherkey));
mp_int *curvekey = get_mp_ssh2(src);
for (unsigned i = 32; i-- > 0 ;)
put_byte(h, mp_get_byte(curvekey, i));
mp_free(curvekey);
strbuf_free(otherkey);
}
/*
* Finish up: compute the final output hash (full 64 bytes of
* SHA-512 this time), and return it encoded as a string.
*/
ssh_hash_final(h, hashdata);
put_stringpl(bs, make_ptrlen(hashdata, sizeof(hashdata)));
smemclr(hashdata, sizeof(hashdata));
return true;
}
/*
* State structure for the server, which takes the role of inventing a
* secret plaintext and sending it to the client encrypted with the
* public key the client sent.
*/
typedef struct ntru_server_key {
uint16_t *plaintext;
strbuf *ciphertext_encoded, *confirmation_hash;
ecdh_key *curve25519;
ecdh_key ek;
} ntru_server_key;
static void ssh_ntru_server_free(ecdh_key *dh);
static void ssh_ntru_server_getpublic(ecdh_key *dh, BinarySink *bs);
static bool ssh_ntru_server_getkey(ecdh_key *dh, ptrlen remoteKey,
BinarySink *bs);
static const ecdh_keyalg ssh_ntru_server_vt = {
/* This vtable has no 'new' method, because it's constructed via
* the selector vt below */
.free = ssh_ntru_server_free,
.getpublic = ssh_ntru_server_getpublic,
.getkey = ssh_ntru_server_getkey,
.description = ssh_ntru_description,
};
static ecdh_key *ssh_ntru_server_new(void)
{
ntru_server_key *nk = snew(ntru_server_key);
nk->ek.vt = &ssh_ntru_server_vt;
nk->plaintext = snewn(p_LIVE, uint16_t);
nk->ciphertext_encoded = strbuf_new_nm();
nk->confirmation_hash = strbuf_new_nm();
ntru_gen_short(nk->plaintext, p_LIVE, w_LIVE);
nk->curve25519 = ecdh_key_new(&ssh_ec_kex_curve25519, false);
return &nk->ek;
}
static void ssh_ntru_server_free(ecdh_key *dh)
{
ntru_server_key *nk = container_of(dh, ntru_server_key, ek);
ring_free(nk->plaintext, p_LIVE);
strbuf_free(nk->ciphertext_encoded);
strbuf_free(nk->confirmation_hash);
ecdh_key_free(nk->curve25519);
sfree(nk);
}
static bool ssh_ntru_server_getkey(ecdh_key *dh, ptrlen remoteKey,
BinarySink *bs)
{
ntru_server_key *nk = container_of(dh, ntru_server_key, ek);
/*
* In the server, getkey is called first, with the public
* information received from the client. We expect the client to
* have sent us a string containing a public key and a Curve25519
* public point.
*/
BinarySource src[1];
BinarySource_BARE_INIT_PL(src, remoteKey);
uint16_t *pubkey = snewn(p_LIVE, uint16_t);
ntru_decode_pubkey(pubkey, p_LIVE, q_LIVE, src);
ptrlen curve25519_remoteKey = get_data(src, 32);
if (get_err(src) || get_avail(src)) {
/* Hard-fail if the input wasn't exactly the right length */
ring_free(pubkey, p_LIVE);
return false;
}
/*
* Main hash object which will combine the NTRU and Curve25519
* outputs.
*/
ssh_hash *h = ssh_hash_new(&ssh_sha512);
/* Reusable buffer for storing various hash outputs. */
uint8_t hashdata[64];
/*
* NTRU side.
*/
{
/* Encrypt the plaintext we generated at construction time,
* and encode the ciphertext into a strbuf so we can reuse it
* for both the session hash and sending to the client. */
uint16_t *ciphertext = snewn(p_LIVE, uint16_t);
ntru_encrypt(ciphertext, nk->plaintext, pubkey, p_LIVE, q_LIVE);
ntru_encode_ciphertext(ciphertext, p_LIVE, q_LIVE,
BinarySink_UPCAST(nk->ciphertext_encoded));
ring_free(ciphertext, p_LIVE);
/* Compute the confirmation hash, and write it into another
* strbuf. */
ntru_confirmation_hash(hashdata, nk->plaintext, pubkey,
p_LIVE, q_LIVE);
put_data(nk->confirmation_hash, hashdata, 32);
/* Compute the session hash (which is easy on the server side,
* requiring no conditional substitution). */
ntru_session_hash(hashdata, 1, nk->plaintext, p_LIVE,
ptrlen_from_strbuf(nk->ciphertext_encoded),
ptrlen_from_strbuf(nk->confirmation_hash));
/* And put the NTRU session hash into the main hash object. */
put_data(h, hashdata, 32);
/* Now we can free the public key */
ring_free(pubkey, p_LIVE);
}
/*
* Curve25519 side.
*/
{
strbuf *otherkey = strbuf_new_nm();
/* Call out to Curve25519 to compute the shared secret from that
* kex method */
bool ok = ecdh_key_getkey(nk->curve25519, curve25519_remoteKey,
BinarySink_UPCAST(otherkey));
/* As on the client side, abort if Curve25519 reported failure */
if (!ok) {
ssh_hash_free(h);
smemclr(hashdata, sizeof(hashdata));
strbuf_free(otherkey);
return false;
}
/* As on the client side, decode Curve25519's mpint so we can
* re-encode it appropriately for our hash preimage */
BinarySource src[1];
BinarySource_BARE_INIT_PL(src, ptrlen_from_strbuf(otherkey));
mp_int *curvekey = get_mp_ssh2(src);
for (unsigned i = 32; i-- > 0 ;)
put_byte(h, mp_get_byte(curvekey, i));
mp_free(curvekey);
strbuf_free(otherkey);
}
/*
* Finish up: compute the final output hash (full 64 bytes of
* SHA-512 this time), and return it encoded as a string.
*/
ssh_hash_final(h, hashdata);
put_stringpl(bs, make_ptrlen(hashdata, sizeof(hashdata)));
smemclr(hashdata, sizeof(hashdata));
return true;
}
static void ssh_ntru_server_getpublic(ecdh_key *dh, BinarySink *bs)
{
ntru_server_key *nk = container_of(dh, ntru_server_key, ek);
/*
* In the server, this function is called after getkey, so we
* already have all our pieces prepared. Just concatenate them all
* into the 'server's public data' string to go in ECDH_REPLY.
*/
put_datapl(bs, ptrlen_from_strbuf(nk->ciphertext_encoded));
put_datapl(bs, ptrlen_from_strbuf(nk->confirmation_hash));
ecdh_key_getpublic(nk->curve25519, bs);
}
/* ----------------------------------------------------------------------
* Selector vtable that instantiates the appropriate one of the above,
* depending on is_server.
*/
static ecdh_key *ssh_ntru_new(const ssh_kex *kex, bool is_server)
{
if (is_server)
return ssh_ntru_server_new();
else
return ssh_ntru_client_new();
}
static const ecdh_keyalg ssh_ntru_selector_vt = {
/* This is a never-instantiated vtable which only implements the
* functions that don't require an instance. */
.new = ssh_ntru_new,
.description = ssh_ntru_description,
};
static const ssh_kex ssh_ntru_curve25519_openssh = {
.name = "sntrup761x25519-sha512@openssh.com",
.main_type = KEXTYPE_ECDH,
.hash = &ssh_sha512,
.ecdh_vt = &ssh_ntru_selector_vt,
};
static const ssh_kex ssh_ntru_curve25519 = {
/* Same as sntrup761x25519-sha512@openssh.com but with an
* IANA-assigned name */
.name = "sntrup761x25519-sha512",
.main_type = KEXTYPE_ECDH,
.hash = &ssh_sha512,
.ecdh_vt = &ssh_ntru_selector_vt,
};
static const ssh_kex *const hybrid_list[] = {
&ssh_ntru_curve25519,
&ssh_ntru_curve25519_openssh,
};
const ssh_kexes ssh_ntru_hybrid_kex = { lenof(hybrid_list), hybrid_list };
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