CFRG S. Fluhrer
Internet-Draft Cisco Systems
Intended status: Informational S. Prorock
Expires: 23 April 2023 mesur.io
M. Celi
Brave
J. Gray
Entrust
20 October 2022
NTRU Key Encapsulation
draft-fluhrer-cfrg-ntru-00
Abstract
This draft documents NTRU as a post-quantum Key Encapsulation
Mechanism (KEM) scheme. The NTRU method from KEM is believed to be
IPR free and cryptographically sound for both classical and post-
quantum threat environments.
NIST has run a competition to select post-quantum primitives and
preliminary selected Kyber for standarization as a KEM. Kyber
unfortunately has plausible patent claims against it and there are
currently undisclosed agreements with the patent holders and NIST.
It is unknown whether those agreements would be universally
acceptable; if not, there will be organizations for which Kyber is
unusable until the patents expire. This lack of clarity around
licensing or other restrictions on Kyber has provided the motivation
to author this draft.
This document does not define any new cryptography, only describes an
existing cryptographic system.
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
Internet-Drafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
working documents as Internet-Drafts. The list of current Internet-
Drafts is at https://datatracker.ietf.org/drafts/current/.
Internet-Drafts are draft documents valid for a maximum of six months
and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use Internet-Drafts as reference
material or to cite them other than as "work in progress."
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This Internet-Draft will expire on 23 April 2023.
Copyright Notice
Copyright (c) 2022 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents (https://trustee.ietf.org/
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Please review these documents carefully, as they describe your rights
and restrictions with respect to this document. Code Components
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provided without warranty as described in the Revised BSD License.
Table of Contents
1. Conventions and Definitions . . . . . . . . . . . . . . . . . 3
2. Notational Conventions . . . . . . . . . . . . . . . . . . . 3
3. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 3
4. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
5. Parameter sets . . . . . . . . . . . . . . . . . . . . . . . 4
5.1. NTRU-HPS 2048509 . . . . . . . . . . . . . . . . . . . . 4
5.2. NTRU-HPS 2048677 . . . . . . . . . . . . . . . . . . . . 4
5.3. NTRU-HPS 4096821 . . . . . . . . . . . . . . . . . . . . 4
6. Cryptographic Dependencies . . . . . . . . . . . . . . . . . 4
6.1. Polynomials . . . . . . . . . . . . . . . . . . . . . . . 4
6.1.1. Trinary Polynomials . . . . . . . . . . . . . . . . . 5
6.2. Polynomial Addition . . . . . . . . . . . . . . . . . . . 6
6.3. Polynomial Subtraction . . . . . . . . . . . . . . . . . 6
6.4. Polynomial Multiplication . . . . . . . . . . . . . . . . 6
6.5. Polynomial Inversion . . . . . . . . . . . . . . . . . . 6
6.6. Computing a Polynomial Modulo (x^(n-1)/(x-1)) . . . . . . 6
7. Selecting Random Polynomials . . . . . . . . . . . . . . . . 6
7.1. Sample a random trinary polynomial . . . . . . . . . . . 7
7.2. Sample a random balanced trinary polynomial . . . . . . . 7
8. Validating polynomials . . . . . . . . . . . . . . . . . . . 7
8.1. ValidR . . . . . . . . . . . . . . . . . . . . . . . . . 7
8.2. ValidM . . . . . . . . . . . . . . . . . . . . . . . . . 7
9. Converting Between Polynomials and Byte Strings . . . . . . . 8
9.1. Serialize a polynomial base q . . . . . . . . . . . . . . 8
9.2. Serialize a trinary polynomial . . . . . . . . . . . . . 8
10. NTRU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
10.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . 9
10.2. Private and Public Key Generation . . . . . . . . . . . 10
10.3. Key Encapsulation . . . . . . . . . . . . . . . . . . . 10
10.4. Key Decapsulation . . . . . . . . . . . . . . . . . . . 11
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11. Parameter Sets . . . . . . . . . . . . . . . . . . . . . . . 12
12. Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
12.1. Comparison with DH . . . . . . . . . . . . . . . . . . . 13
13. Security Considerations . . . . . . . . . . . . . . . . . . . 13
13.1. Parameter set security . . . . . . . . . . . . . . . . . 13
13.2. Public key reuse . . . . . . . . . . . . . . . . . . . . 14
14. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 14
15. Open Questions . . . . . . . . . . . . . . . . . . . . . . . 14
15.1. Test vectors . . . . . . . . . . . . . . . . . . . . . . 15
16. Normative References . . . . . . . . . . . . . . . . . . . . 15
Appendix A. Acknowledgments . . . . . . . . . . . . . . . . . . 15
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 15
1. Conventions and Definitions
{::boilerplate bcp14-tagged}
2. Notational Conventions
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in [RFC2119].
3. Terminology
The following functions, terminology and notation are used throughout
the document.
* For any object x, we write len(x) to denote its length in bytes.
* For two byte arrays x and y, write x || y to denote their
concatenation.
* I2OSP(x, xLen): Converts a non-negative integer x into a byte
array of specified length xLen as described in {{!RFC8017}}. Note
that this function returns a byte array in big-endian byte order.
* n and q are coprime positive integers. The first defines the size
of the polynomials (treated as a zero indexed arrays), and the
latter refers to the modulus.
4. Introduction
This document describes the key encapsulation mechanism (KEM) scheme
based on Hoffstein, Pipher, and Silverman's NTRU encryption scheme,
commonly referred to as NTRU. NTRU is constructed from a
deterministic public key scheme (correct DPKE) into a KEM (which has
tight proof of IND-CCA2 security in a classical and quantum model).
The method described here is based on a combination of prior
approaches, which eventually merged into the NTRUEncrypt and NTRU-
HRSS-KEM submissions (as submitted to Round 2 of the NIST PQC
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project). The algorithm described here is based on the Round 3
submission and permits the use of three well defined and reviewed
parameter sets.
5. Parameter sets
We define three parameter sets:
5.1. NTRU-HPS 2048509
* Type: HPS
* N: 509
* Q: 2048
* Hash: SHA3-256.
* ID: 0x0001
5.2. NTRU-HPS 2048677
* Type: HPS
* N: 677
* Q: 2048
* Hash: SHA3-256.
* ID: 0x0002
5.3. NTRU-HPS 4096821
* Type: HPS
* N: 821
* Q: 4096
* Hash: SHA3-256.
* ID: 0x0003
6. Cryptographic Dependencies
6.1. Polynomials
NTRU is based on polynomials; these can be viewed as a vector of N
small values (either between 0 and Q-1, or sometimes either 0, 1 or
-1), where the values of both N and Q are specified by the parameter
set. In all parameter sets, Q is less than 65536, hence each small
value fits within a 16 bit value.
Each polynomial is an array of values a(n-1), a(n-2), ..., a(0), with
the implicit polynomial being a(n-1)x^(n-1) + a(n-2)x^(n-2) + ... +
a(2)x^2 + a(1)x + a(0) (where x is an independent variable that
doesn't take a specific value). In this case, we don't think of a
polynomial as a function of x that we can evaluate; instead, it is a
quantity in and of itself.
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When we multiply two polynomials, we first do it as we do in standard
algebra; we multiply each pair of terms (including x exponential),
and then sum the products which have the same resulting x term. For
example, (2x^2 + 3x + 5)(4x + 8) = (2*4)x^3 + (2*8 + 3*4)x^2 + (3*8 +
4*5)x + 5*8 = 8x^3 + 28x^2 + 44x + 40.
For NTRU, however, we do two additional reductions to this
multiplication. First, for each sum of the product, we compute that
sum modulo a constant factor (either 3 or the value Q; NTRU uses both
at times). In the above example, if we were reducing things modulo
3, we would actually get the resulting polynomial (8 mod 3)x^3 + (28
mod 3)x^2 + (44 mod 3)x + (40 mod 3) = 2x^3 + x^2 + 2x + 1.
In addition, we compute the multiplication modulo x^n - 1 (where the
value of n is specified in the parameter set); that is, we subtract
multiples of x^n-1 until the result is a polynomial of degree n-1 or
less. An equivalent way of expressing this is to add the resulting
coefficent to the term x^(i+n) to the coefficent to the term x^i
(modulo the constant factor), and then discard all terms x^n and
above.
In the above example, assuming n=2, the final result would be (2+2
mod 3)x + (1+1 mod 3) = x + 2.
A polynomal can be conveniently represented by an array of n values
(with the x^i factor being implicit in the positions in the array);
16 bits per value are sufficient to represent all the coefficients
that are encountered within NTRU.
For most polynomials A = a(n-1)x^(n-1) + a(n-2)x^(n-2) + ... + a(0),
there is a second polynomial B = b(n-1)x^(n-1) + b(n-2)x^(n-2) + ...
+ b(0), such that when we multiply A and B together (and do the above
reductions), we end up with the polynomial 1 = 0x^(n-1) + 0x^(n-2) +
... + 0x + 1. We state this relationship as B = inv(A).
Inverses can be computed efficiently, and also have the property that
similar polynomials have inverses that are quite different.
6.1.1. Trinary Polynomials
Some of the polynomials that NTRU uses are 'trinary polynomials'.
These are standard polynomials that have all their coefficients being
either 0, 1 or Q-1. The standard operations (including polynomial
multiplication and inversion) can be done the same. However, an
implementation may decide to optimize some operations based on a
specific polynomial being trinary.
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6.2. Polynomial Addition
When NTRU adds two polynomials, it does it by adding each element of
the vector independently modulo Q.
6.3. Polynomial Subtraction
When NTRU subtracts two polynomials, it does it by subtracting each
element of the vector independently modulo Q; that is, if the
subtraction of two elements results in a negative value, it adds Q to
the difference.
6.4. Polynomial Multiplication
When NTRU multiplies two polynomials, it does it by multiplying each
pair of elements from each polynomial, and adding that result to the
element indexed by the sum of the indicies (wrapping around if the
sum is N or more).
Note that this can be optimized; in many cases, one of the
polynomials will be of special form (for example, consists of only 0,
1 and -1), more efficient algorithms may be available.
6.5. Polynomial Inversion
When NTRU 'inverts a polynomial' X, it finds a polynomial Y such that
polynomial_multiply(X, Y) gives the polynomial 0x^(n-1) + 0x^(n-2) +
... + 0x^2 + 0x^1 + 1 = (1, 0, 0, 0, ..., 0).
6.6. Computing a Polynomial Modulo (x^(n-1)/(x-1))
At one point, we need to take a polynomial modulo
x^((n-1)+x)(n-2)+...+1 = (x^n-1)/(x-1). We refer to this operation
as modPhiN, and can be performed by taking the top coefficient, and
subtracting it from all the other coefficients.
7. Selecting Random Polynomials
When running NTRU, we need at times random polynomials with specific
forms; doing this is referred to a sampling.
We need to do this both when generating keys as well as when
encrypting a message. It MUST rely on a cryptographically secure
random number generator to select these values.
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7.1. Sample a random trinary polynomial
This function (called sample_iid in the reference code) selects a
random trinary polynomial, that is, one where all the coefficients
are either 0, 1 or q-1, with the last coefficient 0.
This operation is performed by calling the rng n-1 times to generate
n-1 bytes, and then taking each byte modulo 3, mapping 2 to q-1 (and
setting the last coefficient to be 0) While this operation is not
precisely uniform, it is close enough for the purposes of NTRU.
7.2. Sample a random balanced trinary polynomial
This function (called sample_fixed_type by the reference code)
selects a random trinary sample with a specific weight; it consists
of q/16-1 cofficients which are 1, q/16-1 coefficients which are q-1,
and the remainder (which includes coefficient n) as 0.
This can be done by generating n-1 random values; tagging q/16-1 of
the values as 1; q/16-1 of the values as q-1 and the rest tagged as
0. Then, you can sort (in constant time) the random values; the
resulting tags are in the required random order. You then scan
through the list, and assign the coefficients to the values of the
tags.
8. Validating polynomials
We also need to validate polynomials generated during decapsulation;
that is, whether they were possible outputs of the sample_iid or
sample_fixed_type procedures.
8.1. ValidR
This verifies that R is a possible output of the sample_iid
procedure; that is, that the coefficients of the polynomial R
consists only of 0, 1 and q-1, and that the last coefficient is 0.
8.2. ValidM
This verifies that M is a possible output of the sample_fixed_type
procedure; that is, that the coefficients of the polynomial R
consists only of 0, 1 and q-1, that the last coefficient is 0, and
that there are precisely q/16-1 1 values and q/16-2 q-1 values.
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9. Converting Between Polynomials and Byte Strings
NTRU needs to convert polynomials into byte strings and vice versa,
both to export public keys and ciphertexts, as well as being able to
hash those polynomials. We refer to this process as serialization
and deserialization.
9.1. Serialize a polynomial base q
This function (called pack_Rq0 by the reference code) converts a
polynomial into a byte string.
This function takes the first n-1 coefficients (each a value between
0 and q-1), expresses each as a log_2(q) bit bitstring as a little
endian integer. All n-1 coefficients are of length log_2(q). Then,
it concatinates those n-1 bit strings into a long bit string; the
result is that bit string being parsed into bytes (with any trailing
bits in the last byte being set to 0).
The inverse function (called) unpack_Rq0) converts that byte string
back into a polynomial.
It takes the byte string, parses it into n-1 consecutive log_2(q) bit
strings, takes each such bit string as a little endian integer and
sets the corresponding coefficient of the polynomial to that integer.
Since all bit strings are of equal length, this can be done
efficiently. Then, it adds all those n-1 coefficients together, and
sets the n-th coefficient to the negation of that sum modulo q.
A close reading of the above algorithms will note that the pack_Rq0
doesn't actually depend on the last coefficient. This is because
this code assumes that the polynomial is a multiple of the polynomial
x-1; the unpack_Rq0 code uses that assumption to reconstruct that
last coefficient.
This assumption is true within NTRU because pack_Rq0 will be called
only for polynomials that are a multiple of the polynomial G; we
always sample G values that have an equal number of 1 and -1
coefficients (with the rest 0), and any such polynomial will always
be a multiple of x-1.
9.2. Serialize a trinary polynomial
This function (called pack_S3 by the reference code) converts a
trinary polynomial into a byte string. It works by taking the
coefficients in groups of 5, and packing each such group into a byte.
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This function takes the n-1 coefficients in sets of 5; it converts
the five coefficients c0, c1, c2, c3, c4 into the values 0, 1 or 2.
Then it sums up the coefficients as c0 + 3*c1 + 9*c2 + 27*c3 + 81*c4,
and then stores that value as the next byte in the byte string.
If the last set of 5 is incomplete (which will happen if n-1 is not a
multiple of 5), then the higher missing coefficients are assumed to
be zero.
Now, if the polynomial happens to not be trinary, then it doesn't
matter what byte we store; we need to store some value, and this code
still needs to be constant time. The reason we don't care is this
happens only on decryption failure (someone handed us an invalid
ciphertext); in that case, the value of the hash will end up being
ignored. Of course, no matter what the coefficient is, this still
needs to be done in constant time.
This output of this function will be used only for hashing, hence
there is no need for an inverse function.
10. NTRU
10.1. Overview
This section provides a simplified overview how NTRU works. Minor
details are omitted for clarity reasons, and the Security
Considerations section should be consulted prior to implementation.
To generate a public/private keypair, Alice selects two 'short'
polynomials F and G (where short means that the coefficients are all
0, 1 or q-1). She then multiplies each coefficient of G by 3, and
then computes H = Inv(F) x G; that is the public key. She stores F
in the private key, and computes Inv(F) (with this inverse taken over
the modulo 3 polynomial), and stores that in the private key as well.
She also computes Inv(H), and stores that in the private key.
To generate a KEM key share with the public key H, Bob selects two
short polynomials R and M, and computes C = R x H + M; that is the
ciphertext. Bob also hashes R and M to generate his copy of the
shared secret.
When Alice receives C = R x Inv(F) x G + M, she first multiplies that
by F; this results in C x F = R x G + M x F. Since all the
polynomials R, G, M, F are short, the resulting coefficients are not
large (that is, always less than Q/2 in absolute value), and so the
fact that we computed everything modulo Q can be ignored. Note that
some of the coefficients may be 'negative' (that is, in the range Q/2
to Q-1); those need to be treated as negative values for this next
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step. Then, she take all the coefficients modulo 3 (taking into
account the negative coefficients); because all the coefficients of G
are multiples are 3 (and so is R x G), those drop out, and Alice is
left with M x F (with each coefficient taken modulo 3). She then
multiples that polynomial by Inv(F) (this time over the modulo 3
polynomial), recovering M. She then uses M, the original ciphertext
and the stored value Inv(H) to recover R. She then hashes R and M
together to generate her copy of the shared secret.
Assuming Bob received Alice's public key H correctly, and Alice
recieved Bob's ciphertext C correctly, they will derive the same
shared secret.
10.2. Private and Public Key Generation
To generate a public/private keypair, we can follow this procedure:
* Generate a random polynomial f using using the sample_iid
procedure.
* Generate a random polynomial g using the the sample_fixed_type
procedure
* Multiply each coefficient of g by 3.
* Compute FG_inv = Inverse( f * g mod q) mod q.
* Compute H = FG_inv * g * g (modulo q)
* Compute H_inv = FG_inv * f * f (modulo q)
* Compute F_inv = Inverse( f ) (this computation is done modulo 3)
* Generate a random 32 byte value s randomly
The resulting public key is the value H (serialized by the pack_Rq0
procedure); the resulting private key are the values F, H_inv, F_inv
and S. Any other intermediate values should be securely disposed.
n.b. the initial generation of random polynomials can be combined
into a single procedure
10.3. Key Encapsulation
This takes a public key H, and generates both a ciphertext C as well
as a secret string K. The ciphertext C should be sent to the holder
of the private key; the string K should be used as the secret.
We can follow this procedure:
* Unpack the public key (using the unpack_Rq0 procedure) to obtain
the polynomial H
* Sample a random R using the sample_iid procedure
* Sample a random M using the sample_fixed_type procedure
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* Compute C = R*H + M (perfoming both the polynomial multiplication
and polynomial addition modulo q)
* Serialize both R and M (using the pack_S3 procedure on both) and
use SHA3-256 to hash the concatination; the resulting 32 bytes is
the secret string K
* Return K and C serialized using the pack_Rq0 procedure
10.4. Key Decapsulation
This takes a private key (F, H_inv, F_inv, S) and a ciphertext C, and
produces a secret string K. If the ciphertext is the same as what
was proceduced by the key encapsulation procedure, then this will
generate the same secret string K.
We can follow this procedure:
* Unpack the ciphertext (using the unpack_Rq0 procedure) to obtain
the polynomial C
* Compute A = C*F (modulo q)
* For each coeffient x in A, if it is < q/2, replace it with x mod
3; if it is >= q/2, replace it with 2 - (q-1-x) mod 3
[THIS STEP IS NEEDED BECAUSE WE REPRESENT COEFFICIENTS IN THE RANGE
0..Q-1 - WOULD A BALANCED REPRESENTATION BE CLEARER?]
* Compute M = A*F_inv (modulo 3) - note the change of moduli
* For each 2 coefficient within M, replace it with q-1
* Compute R = (C - M) * H_inv (modulo q)
* Compute R = modPhiN(R) (modulo q)
* Set Success = ValidM(M) AND ValidR(R)
* Serialize both R and M (using the pack_S3 procedure on both) and
use SHA3-256 to hash the concatination; the resulting 32 bytes is
K1
* Use SHA3-256 to hash the concatination of S (from the private key)
and C, the resulting 32 bytes is K2
* If Success, return K=K1; otherwise, return K=K2
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11. Parameter Sets
+================+===================+===========+
| Parameter Set | Polynomial Size N | Modulus Q |
+================+===================+===========+
| ntruhps2048509 | 509 | 2048 |
+----------------+-------------------+-----------+
| ntruhps2048677 | 677 | 2048 |
+----------------+-------------------+-----------+
| ntruhps4096821 | 821 | 4096 |
+----------------+-------------------+-----------+
Table 1
Other parameter sets do exist, such as ntruhrss701, hovewver they are
not supported by this RFC at this time as they introduce complexity
without significant value to security, size or performance.
12. Usage
NTRU solves the problem where two systems (we'll call them Alice and
Bob) wish to establish a common secret string that they can use to
derive keys to protect future communication. They share a
communication path that is authenticated (that is, the problem of
detecting changes to messages between Alice and Bob is solved by
something else), but that communication path may be monitored. What
NTRU tries to achieve is to ensure that someone monitoring the
communication path cannot rederive the common secret string (and
hence cannot derive the communication keys).
To do this, Alice and Bob follow this three step process
* Step 1: Alice follows the 'Private and Public Key Generation'
procedure; this creates a private key (which Alice keeps to
herself) and a public key, which she sends to Bob. Alternatively,
she may decide to reuse a previously generated keypair.
* Step 2: Bob receives Alice's public key, and follows the 'Key
Encapsulation' procedure; this creates a secret string (which Bob
keeps to himself) and a ciphertext, which he sends to Alice
* Step 3: Alice recieves Bob's ciphertext, and follows the 'Key
Decapuslation' procedure; this creates a secret string (which
Alice keeps to herself). Alice can then either destroy her
private key, or keep it around for next time.
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The secret strings that Alice and Bob generate are the same, and can
be used for creating symmetric keys or other key shared material used
to protect future communications.
12.1. Comparison with DH
NTRU at first glace appears as though it may be performing the same
job as Diffie-Hellman. In fact, NTRU can be viewed as a drop-in
replacement for DH (with larger key shares) in some protocols.
However, the equivalence is not exact; with NTRU, Bob cannot compute
the ciphertext until he possesses the public key.
In contrast, in Diffie-Hellman, both sides can generate their key
share g^x mod p independently. Some use cases take advantage of this
property of Diffie-Hellman (for example, everyone publishes their key
shares in a central directory; to generate keys with someone else, we
can download their public key from the directory, and obtain the same
key as they get when they download our key from the directory). A
protocol that does this is known as a NonInteractive Key Exchange
(NIKE). NTRU does NOT provide NIKE capabilities, and if non
interactive use cases are required to be supported, a different
approach should be selected.
13. Security Considerations
Current best practices should be followed, especially in regards to
known plaintext attacks, such as Meet-In-The-Middle (MITM), and known
ciphertext attacks.
Lattice reductions via Lenstra-Lenstra-Lovasz may be possible against
NTRU with weak parameter set selection.
13.1. Parameter set security
In all paramter sets, indistinguishability under adaptive chosen
ciphertext attack (IND-CCA2) is a desired property.
Equivalent bit strengths of the described parameter sets are outlined
in the table below:
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+================+================+==============+
| Parameter Set | Security Model | Bit Strength |
+================+================+==============+
| ntruhps2048509 | IND-CCA2 | 128 |
+----------------+----------------+--------------+
| ntruhps2048677 | IND-CCA2 | 192 |
+----------------+----------------+--------------+
| ntruhps4096821 | IND-CCA2 | 256 |
+----------------+----------------+--------------+
Table 2
13.2. Public key reuse
NTRU public/private keys can be safely reused for certain use cases.
Reusing an NTRU key may be tempting, because the NTRU key generation
process is considerably more costly than the key encapsulation or
decapsulation operations. On the other hand, if you do reuse NTRU
keys, you lose the Perfect Forward Secrecy property. That is, as
long as you don't zeroize the NTRU private key, then an attacker that
can break into the system can extract that private key, and then
recover any symmetric keys that were negotiated with that private
key.
If keys are reused, key revocation mechansims should be considered.
14. IANA Considerations
This document has no IANA actions.
15. Open Questions
* HRSS - currently, we omit that parameter set - it does perform
slightly faster than the HPS parameter set at the same security
level (at the cost of a larger public key/ciphertext). My
expectation is that the larger keyshare size for HRSS is a more
significant cost than the larger computational cost for HPS. It
would also complicate the logic somewhat (as we would need to
specify both the HPS and the HRSS ways of doing things). Is the
decision to leave it out the correct one?
* We don't specify a flattened format for a private key. In my
view, there is no need; systems will generally use ephemerial
public/private key pairs, that is, create them on the fly, use
them for one or a handful of exchanges and then throw them away.
In this use case, there is no need to transfer a private key to
another device. Now, it is possible for NTRU to be used with
static keys - should we try to address that case?
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* There is a tiny chance of failure during key generation (if F
happens to be selected as all 0); this happens with probability <
2^-800 (that is, it'll never happen in practice, unless the random
number generator broke). When this happens, the computation of
the inverse of F will fail; what happens in that case would depend
on the inverter implementation. Should we ignore it or address
it?
* It appears that the parameter set HPS4096821 was added lately to
the NTRU definition, and did not undergo the same vetting that the
other parameter sets did. As such, it is unclear whether that
parameter set gives the claimed level of security. Should we
remove it from this RFC, or just add some warning text within the
security considerations?
15.1. Test vectors
TBD
16. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
.
Appendix A. Acknowledgments
Acknowledge TBD.
Authors' Addresses
Scott Fluhrer
Cisco Systems
Email: sfluhrer@cisco.com
Michael Prorock
mesur.io
Email: mprorock@mesur.io
Sofia Celi
Brave
Email: cherenkov@riseup.net
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John Gray
Entrust
Email: john.gray@entrust.com
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