Internet-Draft NTRU October 2022
Fluhrer, et al. Expires 23 April 2023 [Page]
Intended Status:
S. Fluhrer
Cisco Systems
S. Prorock
M. Celi
J. Gray

NTRU Key Encapsulation


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

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Table of Contents

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.

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 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.

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.

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 (xn-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.

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.

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.

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 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
  • 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


  • 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

11. Parameter Sets

Table 1
Parameter Set Polynomial Size N Modulus Q
ntruhps2048509 509 2048
ntruhps2048677 677 2048
ntruhps4096821 821 4096

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

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:

Table 2
Parameter Set Security Model Bit Strength
ntruhps2048509 IND-CCA2 128
ntruhps2048677 IND-CCA2 192
ntruhps4096821 IND-CCA2 256

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

16. Normative References

Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, , <>.

Appendix A. Acknowledgments

Acknowledge TBD.

Authors' Addresses

Scott Fluhrer
Cisco Systems
Michael Prorock
Sofia Celi
John Gray