New Key Share Extension for Classic McEliece Algorithms
draft-wagner-tls-keysharepqc-07
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| Document | Type | Active Internet-Draft (individual) | |
|---|---|---|---|
| Authors | Jonathan Wagner , Yongge Wang | ||
| Last updated | 2025-12-19 | ||
| RFC stream | (None) | ||
| Intended RFC status | Informational | ||
| Formats | |||
| Stream | Stream state | (No stream defined) | |
| Consensus boilerplate | Unknown | ||
| RFC Editor Note | (None) | ||
| IESG | IESG state | I-D Exists | |
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draft-wagner-tls-keysharepqc-07
Transport Layer Security J. Wagner
Internet-Draft Y. Wang
Intended status: Standards Track UNC Charlotte
Expires: 22 June 2026 19 December 2025
New Key Share Extension for Classic McEliece Algorithms
draft-wagner-tls-keysharepqc-07
Abstract
RFC 8446 is modified to where another key share extension is
introduced to accommodate both public keys and ciphertexts in
ClientHello and ServerHello messages for post-quantum algorithms that
have large public keys, including algorithms of the code-based
cryptographic scheme Classic McEliece.
About This Document
This note is to be removed before publishing as an RFC.
The latest revision of this draft can be found at
https://jwagrunner.github.io/internet-draft/draft-wagner-tls-
keysharepqc.html. Status information for this document may be found
at https://datatracker.ietf.org/doc/draft-wagner-tls-keysharepqc/.
Discussion of this document takes place on the Transport Layer
Security mailing list (mailto:tls@ietf.org), which is archived at
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Source for this draft and an issue tracker can be found at
https://github.com/jwagrunner/internet-draft.
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
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material or to cite them other than as "work in progress."
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This Internet-Draft will expire on 22 June 2026.
Copyright Notice
Copyright (c) 2025 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
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Please review these documents carefully, as they describe your rights
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Conventions and Definitions . . . . . . . . . . . . . . . . . 3
3. New Key Share Extension . . . . . . . . . . . . . . . . . . . 3
4. NamedGroup Addition for Classic McEliece . . . . . . . . . . 7
5. Modification to PskKeyExchangeMode structure . . . . . . . . 8
6. Hello Retry Request using New Key Share Extension . . . . . . 10
7. Other Use Case (RLCE Algorithm) . . . . . . . . . . . . . . . 12
8. Hybrid Combination "x25519classicmceliece348864" . . . . . . 12
9. TLS Implementation . . . . . . . . . . . . . . . . . . . . . 13
10. Summary of Changes from RFC 8446 . . . . . . . . . . . . . . 13
11. Security Considerations . . . . . . . . . . . . . . . . . . . 13
12. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 13
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 14
References . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Normative References . . . . . . . . . . . . . . . . . . . . . 14
Informative References . . . . . . . . . . . . . . . . . . . . 15
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 17
1. Introduction
Large public key algorithms, including the code-based cryptographic
algorithm family Classic McEliece (see [CLASSICMC], [NIST], [DJB25],
[RJM78], and [OQS24]), cannot be easily implemented in Transport
Layer Security (TLS) Protocol Version 1.3 ([RFC8446]) due to the
current key share limitations of 65,535 Bytes. It is important to
consider such uses of algorithms given that Classic McEliece is a
Round 4 algorithm submitted in the National Institute of Standards
and Technology (NIST) standardization process (see [PQC25]).
Therefore, this document proposes a new key share that has a higher
limit and is utilized in ClientHello and ServerHello messages, which
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is a modification of [RFC8446]. For example, if a Classic McEliece
algorithm is requested in a TLS 1.3 key exchange, this new key share
extension will be constructed. However, if a classical algorithm is
requested for key exchange, a normal key share extension is
constructed. Thus, enabling the use of Classic McEliece algorithms
to be used in TLS 1.3 key exchanges and also presenting them as an
alternative option to replace classical algorithms for future
protection against the threat of attackers in possession of powerful
quantum computers that will break classical encryption.
2. Conventions and Definitions
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in
BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
3. New Key Share Extension
Based on the key share extension from [RFC8446] is introduced a new
key share extension in this document, "key_share_pqc". This is
reflected in this document and is represented as KeyShareEntryPQC
below, based on the existing KeyShareEntry from [RFC8446]. However,
this is modified along with the existing KeyShareEntry structure to
test if the key exchange algorithm chosen in a TLS 1.3 connection
belongs from the Classic McEliece family, and if it is, then
KeyShareEntryPQC is constructed. If the opposite is true, where the
key exchange algorithm is not from the Classic McEliece family, then
KeyShareEntry is constructed. Note that the "key_exchange" fields
are expanded in KeyShareEntryPQC to accommodate a large public key
that is greater than 65,535 Bytes:
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struct {
NamedGroup group;
select (KeyShareEntry.group) {
case classicmceliece348864: Empty;
case classicmceliece460896: Empty;
case classicmceliece6688128: Empty;
case classicmceliece6960119: Empty;
case classicmceliece8192128: Empty;
case x25519classicmceliece348864: Empty;
case rlcel5: Empty;
case other large PQ algorithm1: Empty;
case other large PQ algorithm2: Empty;
case etc.: Empty;
default: opaque key_exchange<1..2^16-1>;
}
} KeyShareEntry;
struct {
NamedGroup group;
select (KeyShareEntryPQC.group) {
case classicmceliece348864: opaque key_exchange<1..2^24-1>;
case classicmceliece460896: opaque key_exchange<1..2^24-1>;
case classicmceliece6688128: opaque key_exchange<1..2^24-1>;
case classicmceliece6960119: opaque key_exchange<1..2^24-1>;
case classicmceliece8192128: opaque key_exchange<1..2^24-1>;
case x25519classicmceliece348864: opaque key_exchange<1..2^24-1>;
case rlcel5: opaque key_exchange<1..2^24-1>;
case other large PQ algorithm1: opaque key_exchange<1..2^24-1>;
case other large PQ algorithm2: opaque key_exchange<1..2^24-1>;
case etc.: opaque key_exchange<1..2^24-1>;
default: Empty;
}
} KeyShareEntryPQC;
Note: PQ (Post-Quantum) where "other large PQ algorithm1" and "other
large PQ algorithm2" and "etc." above indicates that one or more
future post-quantum algorithms with large public key sizes can be
added by just defining a constant for each of these post-quantum
algorithms.
Another Note: An additional algorithm is included in the above,
"rlcel5", since it also has a large public key beyond the 65,535 Byte
limit. See Section 7 for more information discussing this RLCE
algorithm.
This is then applied to the existing KeyShareClientHello structure,
which originates from [RFC8446], that now contains an additional
field for KeyShareEntryPQC:
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struct {
KeyShareEntry client_shares<0..2^16-1>;
KeyShareEntryPQC client_shares<0..2^24-1>;
} KeyShareClientHello;
Since the KeyShareClientHello needs to be expanded to accommodate for
the KeyShareEntryPQC struct, the same applies to the existing
Extension struct, originated as well from [RFC8446] but
"extension_data" is now expanded:
struct {
ExtensionType extension_type;
opaque extension_data<0..2^24-1>;
} Extension;
Since there is a new key share extension to accommodate keys larger
than the 65,535 Byte limit (KeyShareEntryPQC), this is reflected in
the existing ExtensionType structure from [RFC8446] where this is the
new type that holds a value of TBD, "key_share_pqc":
enum {
server_name(0), /* RFC 6066 */
max_fragment_length(1), /* RFC 6066 */
status_request(5), /* RFC 6066 */
supported_groups(10), /* RFC 8422, 7919 */
signature_algorithms(13), /* RFC 8446 */
use_srtp(14), /* RFC 5764 */
heartbeat(15), /* RFC 6520 */
application_layer_protocol_negotiation(16), /* RFC 7301 */
signed_certificate_timestamp(18), /* RFC 6962 */
client_certificate_type(19), /* RFC 7250 */
server_certificate_type(20), /* RFC 7250 */
padding(21), /* RFC 7685 */
pre_shared_key(41), /* RFC 8446 */
early_data(42), /* RFC 8446 */
supported_versions(43), /* RFC 8446 */
cookie(44), /* RFC 8446 */
psk_key_exchange_modes(45), /* RFC 8446 */
certificate_authorities(47), /* RFC 8446 */
oid_filters(48), /* RFC 8446 */
post_handshake_auth(49), /* RFC 8446 */
signature_algorithms_cert(50), /* RFC 8446 */
key_share(51), /* RFC 8446 */
key_share_pqc(TBD),
(65535)
} ExtensionType;
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Since the "extension_data" field will be much larger for a
KeyShareClientHello that contains a large public key that is greater
than the previously defined 65,535 Byte limit, an example being a
Classic McEliece public key, the server must be able to handle this
circumstance when receiving the ClientHello message. One way is to
compare the value for a packet that contains extensions including a
large public key from the ClientHello message to a macro constant
(for example, "CLIENT_HELLO_MIN_EXT_LENGTH" as defined in this
introduced TLS implementation in this paper, see [SRVR1650] and
[SRVR1211]) and if this packet value is longer than this constant,
the server will change the way it normally handles all of the
extensions. This constant could be easily modified in the
aforementioned TLS Open Secure Socket Layer (OpenSSL) implementation.
The process of how the server collects the extensions from a
ClientHello message must also be modified, as the server must be able
to process the new key share extension differently than the other
extensions, should the server see this inside a ClientHello message.
For example, see [EXT652].
The ServerHello message is modified as well where the
KeyShareServerHello structure originates from [RFC8446]:
struct {
KeyShareEntry server_share;
KeyShareEntryPQC server_sharePQC;
} KeyShareServerHello;
This new "key_share_pqc" extension is therefore can be implemented in
the full TLS handshake, where Figure 1 from [RFC8446] is modified to
be the following:
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Client Server
Key ^ ClientHello
Exch | + key_share*
| + key_share_pqc*
| + signature_algorithms*
| + psk_key_exchange_modes*
v + pre_shared_key* -------->
ServerHello ^ Key
+ key_share* | Exch
+ key_share_pqc* |
+ pre_shared_key* v
{EncryptedExtensions} ^ Server
{CertificateRequest*} v Params
{Certificate*} ^
{CertificateVerify*} | Auth
{Finished} v
<-------- [Application Data*]
^ {Certificate*}
Auth | {CertificateVerify*}
v {Finished} -------->
[Application Data] <-------> [Application Data]
+ Indicates noteworthy extensions sent in the
previously noted message.
* Indicates optional or situation-dependent
messages/extensions that are not always sent.
{} Indicates messages protected using keys
derived from a [sender]_handshake_traffic_secret.
[] Indicates messages protected using keys
derived from [sender]_application_traffic_secret_N.
Figure 1: Full TLS Handshake with "key_share_pqc" extension.
4. NamedGroup Addition for Classic McEliece
The values for Classic McEliece algorithms and the hybrid combination
"x25519classicmceliece348864" (see Section 8 for more information)
are added below in the NamedGroup struct that originates from
[RFC8446]:
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enum {
/* Elliptic Curve Groups (ECDHE) */
secp256r1(0x0017), secp384r1(0x0018), secp521r1(0x0019),
x25519(0x001D), x448(0x001E),
/* Finite Field Groups (DHE) */
ffdhe2048(0x0100), ffdhe3072(0x0101), ffdhe4096(0x0102),
ffdhe6144(0x0103), ffdhe8192(0x0104),
/* Reserved Code Points */
ffdhe_private_use(0x01FC..0x01FF),
ecdhe_private_use(0xFE00..0xFEFF),
(0xFFFF)
/* Classic McEliece Algorithms */
classicmceliece348864(TBD),
classicmceliece460896(TBD),
classicmceliece6688128(TBD),
classicmceliece6960119(TBD),
classicmceliece8192128(TBD),
/* Hybrid Combination */
x25519classicmceliece348864(TBD),
/* RLCE Algorithm */
rlcel5(TBD),
} NamedGroup;
Note: An RLCE algorithm is also added above. See Section 7 for more
information discussing this RLCE algorithm.
5. Modification to PskKeyExchangeMode structure
There are two key establishments that are considered when examining
the structure of PskKeyExchangeMode from [RFC8446]. Since there is
no Diffie Hellman algorithm in use with a pre-shared key (PSK) when
considering the use of a Classic McEliece algorithm for key exchange,
then there must be another key exchange mode to utilize in this case.
Therefore, this is reflected in the existing [RFC8446]
PskKeyExchangeMode structure below where "psk_pqc_ke(2)" is added:
enum {
psk_ke(0), psk_dhe_ke(1), psk_pqc_ke(2), (255)
} PskKeyExchangeMode;
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When selecting a Classic McEliece algorithm and using an external PSK
or a resumption PSK, "02" will then be listed for the
"psk_key_exchange_modes" extension along with the new "key_share_pqc"
extension in the ClientHello message. At the end of this ClientHello
message is printed the "00 29" extension (pre-shared key extension),
where the PSK identity should be printed and is mapped to the binder
that should proceed it in this pre-shared key extension. The
ServerHello message will also contain the new "key_share_pqc"
extension, and will as well contain the pre-shared key extension,
where it should contain "00 00" at the end which represents the
server selecting the PSK identity of 0 (for example: the Selected
Identity of 0 shown in the pre-shared key extension in a ServerHello
message in this Wireshark example: [RASHOK20]). Overall, this is a
new key exchange selecting a Classic McEliece algorithm using a PSK,
whether its external or resumption, and this can be demonstrated in
the TLS Implementation below.
As stated above, resumption PSK with a Classic McEliece algorithm
chosen as a key exchange algorithm involves the use of the new
"key_share_pqc" extension for both the ClientHello and ServerHello
messages. Thus, the Resumption and PSK Message Flow diagram (which
originates from Figure 3 of [RFC8446]) is derived for this situation
and has been tested with the TLS Implementation mentioned in this
document:
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Client Server
Initial Handshake
ClientHello
key_share_pqc --------->
ServerHello
key_share_pqc
EncryptedExtensions
Certificate
CertificateVerify
<--------- Finished
Finished ---------->
<--------- NewSessionTicket
<--------- NewSessionTicket
Subsequent Handshake
ClientHello
key_share_pqc
pre_shared_key --------->
ServerHello
key_share_pqc
pre_shared_key
EncryptedExtensions
<--------- Finished
Finished ---------->
<--------- NewSessionTicket
Figure 2: A Classic McEliece algorithm used with Resumption PSK.
6. Hello Retry Request using New Key Share Extension
In a Hello Retry Request scenario, the first ClientHello message will
have two algorithms listed in its "supported_groups" extension, where
the numerical identifier (NID) for the algorithm that is no longer
recognized by the server as an acceptable algorithm will first be
listed in this extension, followed by the NID for a Classic McEliece
algorithm. In this same ClientHello message is where "02" will be
listed in the "psk_key_exchange_modes" extension, and the original
"key_share" extension (value 51) is also shown with its public key
for the unacceptable algorithm.
When the server responds with the HelloRetryRequest message, the
random is the same special value for SHA-256 as indicated in
Section 4.1.3 of [RFC8446], and has the same exact fields
("legacy_version", "random", "legacy_session_id_echo",
"cipher_suite", "legacy_compression_method", and "extensions") as in
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the ServerHello structure indicated in [RFC8446] (see section 4.1.3).
The extensions field not only consists of the "supported_versions"
extension, but also the new "key_share_pqc" extension where the
server offers the client the Classic McEliece algorithm NID it shares
with the client.
When the client sends a second ClientHello in response to the
HelloRetryRequest, this will be the same message as the first
ClientHello with one exception: the original "key_share" extension is
replaced with the new "key_share_pqc" extension which contains the
large public key of a Classic McEliece algorithm. Then the
ServerHello message will then respond containing the new
"key_share_pqc" extension.
Therefore, this Hello Retry Request scenario is reflected in Figure 3
below, which is a modification of Figure 2 in [RFC8446], and this can
be demonstrated in the TLS Implementation mentioned in this
documentation:
Client Server
ClientHello
key_share -------->
HelloRetryRequest
<-------- key_share_pqc
ClientHello
key_share_pqc -------->
ServerHello
key_share_pqc
EncryptedExtensions
Certificate
CertificateVerify
<-------- Finished
Finished -------->
<-------- NewSessionTicket
<-------- NewSessionTicket
Figure 3: A Classic McEliece algorithm used in a Hello Retry Request scenario.
Note: When the client processes the HelloRetryRequest message, it
must mark the new "key_share_pqc" extension as an unsolicited
extension, which would be an additional exception to the rule noted
in [RFC8446] regarding extension responses MUST NOT be sent if the
corresponding extension requests were not sent by a remote endpoint
(see section 4.2 in [RFC8446]).
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The following structure would remain intact from [RFC8446], since
support would already be provided for a Classic McEliece algorithm
being in NamedGroup (see Section 4):
struct {
NamedGroup selected_group;
} KeyShareHelloRetryRequest;
When a Hello Retry Request involves a PSK in use with a Classic
McEliece algorithm, both the first and second ClientHello messages
(the second one being sent after a HelloRetryRequest message) will
contain the exact same content except the first ClientHello will have
the original "key_share" extension and the second ClientHello will
have the new "key_share_pqc" extension. Another exception includes
different binders in both ClientHello messages' pre-shared key
extensions. This pre-shared key extension appears as the last
extension in both ClientHello messages as well in the ServerHello
message.
7. Other Use Case (RLCE Algorithm)
The Random Linear Code-based Encryption (RLCE) algorithm group (see
[RLCE17]) is another code-based cryptographic scheme (NIST Round 1
[NIST1]). "rlcel5" is a RLCE algorithm from this group (where the
public key size is 1,232,001 Bytes) that can be used in the new key
share extension, and can be demonstrated for use for TLS key exchange
in the TLS Implementation mentioned in this document.
8. Hybrid Combination "x25519classicmceliece348864"
"x25519classicmceliece348864" is a hybrid mechanism introduced in
this document that combines both classicmceliece348864 and x25519
[curve25519], [RFC7748] in TLS key exchanges. The experiment TLS
implementation presented in this document, which uses the fork
[JWYWPROV] of the oqs-provider [OQSPROV], is one example of using
x25519classicmceliece348864 in a hybrid key exchange; when
x25519classicmceliece348864 is chosen in this circumstance, it uses
the "concatenating" method mentioned in [HYBRIDTLS] in the new
key_share_pqc extension. In the ClientHello message, this new key
share extension contains both the Classic McEliece public key and
X25519 key concatenated together. In the ServerHello message, this
new key share extension then contains the classicmceliece348864
ciphertext and X25519 key concatenated together.
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9. TLS Implementation
A TLS implementation exists that tests the use of a new key share
extension for both the ClientHello and ServerHello messages that is
implemented for OpenSSL, and also where the mentioned Classic
McEliece algorithms can be chosen for key exchange when initiating
TLS connections. It can be accessed here: [JWYW25].
10. Summary of Changes from RFC 8446
A new structure is introduced of KeyShareEntryPQC along with
modifications of existing structures including KeyShareEntry,
NamedGroup, Extension, ExtensionType, KeyShareClientHello, and
KeyShareServerHello. Adding a new ExtensionType of "key_share_pqc"
allows for the addition of this new structure of KeyShareEntryPQC,
which is based on the existing KeyShareEntry, but "key_exchange" has
been expanded and select statements are added to both structures
which depend on the KeyShareEntry.group or KeyShareEntryPQC.group
being called in a TLS connection for key exchange. This new
KeyShareEntryPQC will now also appear in existing structures of
KeyShareClientHello and KeyShareServerHello. Thus, the
"extension_data" is expanded in the existing Extension structure.
11. Security Considerations
There appears to be no security considerations at this time.
12. IANA Considerations
Probable request for the new key share proposed in this document
"key_share_pqc" to have a value in the registry specified for TLS
ExtensionType Values (see [TLSE]):
Extension Name: key_share_pqc
Value: TBD
Probable request for the registry for TLS Supported Groups to have
the proper values assigned to the Classic McEliece and the RLCE
algorithms mentioned in this document; also requested is the hybrid
combination "x25519classicmceliece348864" (see [TLSP]):
Description: x25519classicmceliece348864
Value: TBD
Description: classicmceliece348864
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Value: TBD
Description: classicmceliece460896
Value: TBD
Description: classicmceliece6688128
Value: TBD
Description: classicmceliece6960119
Value: TBD
Description: classicmceliece8192128
Value: TBD
Description: rlcel5
Value: TBD
Acknowledgements
Thank you D. J. Bernstein and Simon Josefsson as they advised to
have at least one reference for the description of Classic McEliece.
Thank you also to Eliot Lear for his feedback on other fields
regarding the next algorithm needed.
Thank you as well to Martin Thomson and David Schinazi, as their
Internet Draft template was used to generate this document, before
the authors' information was added. The authors also want to thank
the contributors of the kramdown-rfc GitHub repository, as their
examples helped with the format of the figures, references, and
authors' information presented in this document. Thank you also to
Joyce Reynolds and Robert Braden, as their Internet Draft [JR04] was
helpful as a guide on how to write the citations in this document
(i.e., using citation brackets with author's initials, year, etc.).
References
Normative References
[CLASSICMC]
Josefsson, S., "Classic McEliece", July 2025,
<https://datatracker.ietf.org/doc/draft-josefsson-
mceliece/>.
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[curve25519]
Bernstein, D. J., "Curve25519: new Diffie-Hellman speed
records", 2006,
<https://www.iacr.org/cryptodb/archive/2006/
PKC/3351/3351.pdf>.
[HYBRIDTLS]
Stebila, D., Fluhrer, S., and S. Gueron, "Hybrid key
exchange in TLS 1.3", September 2025,
<https://datatracker.ietf.org/doc/draft-ietf-tls-hybrid-
design/>.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
<https://www.rfc-editor.org/rfc/rfc2119>.
[RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
for Security", n.d.,
<https://datatracker.ietf.org/doc/html/rfc7748>.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, <https://www.rfc-editor.org/rfc/rfc8174>.
[RFC8446] Rescorla, E., "The Transport Layer Security (TLS) Protocol
Version 1.3", 2018,
<https://datatracker.ietf.org/doc/html/rfc8446>.
[TLSE] Internet Assigned Numbers Authority, "Transport Layer
Security (TLS) Extensions", 2025,
<https://www.iana.org/assignments/tls-extensiontype-
values/tls-extensiontype-values.xhtml>.
[TLSP] Internet Assigned Numbers Authority, "Transport Layer
Security (TLS) Parameters", 2025,
<https://www.iana.org/assignments/tls-parameters/tls-
parameters.xhtml>.
Informative References
[DJB25] Bernstein, D., Chou, T., Cid, C., Gilcher, J., Lange, T.,
Maram, V., von Maurich, I., Misoczki, R., Niederhagen, R.,
Persichetti, E., Peters, C., Sendrier, N., Szefer, J.,
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Implementation", 2024,
<https://classic.mceliece.org/impl.html>.
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[EXT652] Wagner, J., "ssl/statem/extensions.c#L652C9-L663C9", 2024,
<https://github.com/jwagrunner/openssl/blob/master/ssl/
statem/extensions.c#L652C9-L663C9>.
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[RLCE17] Wang, Y., "Quantum Resistant Public Key Encryption Scheme
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[SRVR1211] Wagner, J., "ssl/statem/statem_srvr.c#L1211", 2024,
<https://github.com/jwagrunner/openssl/blob/master/ssl/
statem/statem_srvr.c#L1211>.
[SRVR1650] Wagner, J., "ssl/statem/statem_srvr.c#L1650", 2024,
<https://github.com/jwagrunner/openssl/blob/master/ssl/
statem/statem_srvr.c#L1650>.
Authors' Addresses
Jonathan Wagner
UNC Charlotte
9201 University City Blvd
Charlotte, NC, 28223
United States of America
Email: jwagne31@charlotte.edu
Yongge Wang
UNC Charlotte
9201 University City Blvd
Charlotte, NC, 28223
United States of America
Email: yongge.wang@charlotte.edu
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