CFRG S. Smyshlyaev, Ed.
Internet-Draft CryptoPro
Intended status: Informational October 7, 2017
Expires: April 10, 2018

Re-keying Mechanisms for Symmetric Keys
draft-irtf-cfrg-re-keying-07

Abstract

A certain maximum amount of data can be safely encrypted when encryption is performed under a single key. This amount is called "key lifetime". This specification describes a variety of methods to increase the lifetime of symmetric keys. It provides external and internal re-keying mechanisms based on hash functions and on block ciphers, that can be used with modes of operations such as CTR, GCM, CBC, CFB and OMAC.

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Table of Contents

1. Introduction

A certain maximum amount of data can be safely encrypted when encryption is performed under a single key. This amount is called "key lifetime" and can be calculated from the following considerations:

  1. Methods based on the combinatorial properties of the used block cipher mode of operation

  2. Methods based on side-channel analysis issues

  3. Methods based on the properties of the used block cipher

As a result, it is important to replace a key as soon as the total size of the processed plaintext under that key reaches the lifetime limitation. A specific value of the key lifetime should be determined in accordance with some safety margin for protocol security and the methods outlined above.

Suppose L is a key lifetime limitation in some protocol P. For simplicity, assume that all messages have the same length m. Hence, the number of messages q that can be processed with a single key K should be such that m * q <= L. This can be depicted graphically as a rectangle with sides m and q which is enclosed by area L (see Figure 1).

                            
          +------------------------+
          |                      L | 
          | +--------m---------+   |
          | |==================|   |
          | |==================|   |
          | q==================|   |       m * q <= L
          | |==================|   |
          | |==================|   | 
          | +------------------+   |
          +------------------------+
 
Figure 1: Graphic display of the key lifetime limitation

                
                    

In practice, such amount of data that corresponds to limitation L may not be enough. The simplest and obvious way in this situation is a regular renegotiation of an initial key after processing this threshold amount of data L. However, this reduces the total performance, since it usually entails termination of application data transmission, additional service messages, the use of random number generator and many other additional calculations, including resource-intensive public key cryptography.

This specification presents two approaches to extend the lifetime of a key while avoiding renegotiation: external and internal re-keying.

External re-keying is performed by a protocol, and it is independent of the underlying block cipher and the mode of operation. External re-keying can use parallel and serial constructions. In the parallel case, data processing keys K^1, K^2, ... are generated directly from the initial key K independently of each other. In the serial case, every data processing key depends on the state that is updated after the generation of each new data processing key.

Internal re-keying is built into the mode, and it depends heavily on the properties of the mode of operation and the block size.

The re-keying approaches extend the key lifetime for a single initial key by providing the possibility to limit the leakages (via side channels) and by improving combinatorial properties of the used block cipher mode of operation.

In practical applications, re-keying can be useful for protocols that need to operate in hostile environments or under restricted resource conditions (e.g., that require lightweight cryptography, where ciphers have a small block size, that imposes strict combinatorial limitations). Moreover, mechanisms that use external and internal re-keying may provide some properties of forward security and potentially some protection against future attacks (by limiting the number of plaintext-ciphertext pairs that an adversary can collect).

2. Conventions Used in This Document

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. Basic Terms and Definitions

This document uses the following terms and definitions for the sets and operations on the elements of these sets:

V*
the set of all bit strings of a finite length (hereinafter referred to as strings), including the empty string; substrings and string components are enumerated from right to left starting from one;
V_s
the set of all bit strings of length s, where s is a non-negative integer;
|X|
the bit length of the bit string X;
A | B
concatenation of strings A and B both belonging to V*, i.e., a string in V_{|A|+|B|}, where the left substring in V_|A| is equal to A, and the right substring in V_|B| is equal to B;
(xor)
exclusive-or of two bit strings of the same length;
Z_{2^n}
ring of residues modulo 2^n;
Int_s: V_s -> Z_{2^s}
the transformation that maps a string a = (a_s, ... , a_1), a in V_s, into the integer Int_s(a) = 2^{s-1} * a_s + ... + 2 * a_2 + a_1;
Vec_s: Z_{2^s} -> V_s
the transformation inverse to the mapping Int_s;
MSB_i: V_s -> V_i
the transformation that maps the string a = (a_s, ... , a_1) in V_s, into the string MSB_i(a) = (a_s, ... , a_{s-i+1}) in V_i;
LSB_i: V_s -> V_i
the transformation that maps the string a = (a_s, ... , a_1) in V_s, into the string LSB_i(a) = (a_i, ... , a_1) in V_i;
Inc_c: V_s -> V_s
the transformation that maps the string a = (a_s, ... , a_1) in V_s, into the string Inc_c(a) = MSB_{|a|-c}(a) | Vec_c(Int_c(LSB_c(a)) + 1(mod 2^c)) in V_s;
a^s
denotes the string in V_s that consists of s 'a' bits;
E_{K}: V_n -> V_n
the block cipher permutation under the key K in V_k;
ceil(x)
the smallest integer that is greater than or equal to x;
floor(x)
the biggest integer that is less than or equal to x;
k
the bit-length of the K; k is assumed to be divisible by 8;
n
the block size of the block cipher (in bits); n is assumed to be divisible by 8;
b
the number of data blocks in the plaintext P (b = ceil(|P|/n));
N
the section size (the number of bits that are processed with one section key before this key is transformed);

A plaintext message P and the corresponding ciphertext C are divided into b = ceil(|P|/n) blocks, denoted P = P_1 | P_2 | ... | P_b and C = C_1 | C_2 | ... | C_b, respectively. The first b-1 blocks P_i and C_i are in V_n, for i = 1, 2, ... , b-1. The b-th block P_b, C_b may be an incomplete block, i.e., in V_r, where r <= n if not otherwise specified.

4. Choosing Constructions and Security Parameters

External re-keying is an approach assuming that a key is transformed after encrypting a limited number of entire messages. External re-keying method is chosen at the protocol level, regardless of the underlying block cipher or the encryption mode. External re-keying is recommended for protocols that process relatively short messages or for protocols that have a way to divide a long message into manageable pieces. Through external re-keying the number of messages that can be securely processed with a single initial key K is substantially increased without loss in message length.

External re-keying has the following advantages:

  1. it increases the lifetime of an initial key by increasing the number of messages processed with this key;
  2. it has negligible affect on the performance, when the number of messages processed under one initial key is sufficiently large;
  3. it provides forward and backward security of data processing keys.

However, the use of external re-keying has the following disadvantage: in case of restrictive key lifetime limitations the message sizes can become inconvenient due to impossibility of processing sufficiently large messages, so it could be necessary to perform additional fragmentation at the protocol level. E.g. if the key lifetime L is 1 GB and the message length m = 3 GB, then this message cannot be processed as a whole and it should be divided into three fragments that will be processed separately.

Internal re-keying is an approach assuming that a key is transformed during each separate message processing. Such procedures are integrated into the base modes of operations, so every internal re-keying mechanism is defined for the particular operation mode and the block size of the used cipher. Internal re-keying is recommended for protocols that process long messages: the size of each single message can be substantially increased without loss in number of messages that can be securely processed with a single initial key.

Internal re-keying has the following advantages:

  1. it increases the lifetime of an initial key by increasing the size of the messages processed with one initial key;
  2. it has minimal impact on performance;
  3. internal re-keying mechanisms without a master key does not affect short messages transformation at all;
  4. it is transparent (works like any mode of operation): does not require changes of IV’s and restarting MACing.

However, the use of internal re-keying has the following disadvantages:

  1. a specific method must not be chosen independently of a mode of operation;
  2. internal re-keying mechanisms without a master key do not provide backward security of data processing keys.

Any block cipher modes of operations with internal re-keying can be jointly used with any external re-keying mechanisms. Such joint usage increases both the number of messages processed with one initial key and their maximum possible size.

If the adversary has access to the data processing interface the use of the same cryptographic primitives both for data processing and re-keying transformation decreases the code size but can lead to some possible vulnerabilities. This vulnerability can be eliminated by using different primitives for data processing and re-keying, however, in this case the security of the whole scheme cannot be reduced to standard notions like PRF or PRP, so security estimations become more difficult and unclear.

Summing up the above-mentioned issues briefly:

  1. If a protocol assumes processing long records (e.g., [CMS]), internal re-keying should be used. If a protocol assumes processing a significant amount of ordered records, which can be considered as a single data stream (e.g., [TLS], [SSH]), internal re-keying may also be used.
  2. For protocols which allow out-of-order delivery and lost records (e.g., [DTLS], [ESP]) external re-keying should be used. If at the same time records are long enough, internal re-keying should be additionally used during each separate message processing.

For external re-keying:

  1. If it is desirable to separate transformations used for data processing and for key update, hash function based re-keying should be used.
  2. If parallel data processing is required, then parallel external re-keying should be used.
  3. If key lifetime limitations are very strict, then key hierarchy external re-keying should be used.

For internal re-keying:

  1. If the property of forward and backward security is desirable for data processing keys and if additional key material can be easily obtained for the data processing stage, internal re-keying with a master key should be used.

5. External Re-keying Mechanisms

This section presents an approach to increase the initial key lifetime by using a transformation of a data processing key (frame key) after processing a limited number of entire messages (frame). It provides external parallel and serial re-keying mechanisms (see [AbBell]). These mechanisms use initial key K only for frame key generation and never use it directly for data processing. Such mechanisms operate outside of the base modes of operations and do not change them at all, therefore they are called "external re-keying" mechanisms in this document.

External re-keying mechanisms are recommended for usage in protocols that process quite small messages, since the maximum gain in increasing the initial key lifetime is achieved by increasing the number of messages.

External re-keying increases the initial key lifetime through the following approach. Suppose there is a protocol P with some mode of operation (base encryption or authentication mode). Let L1 be a key lifetime limitation induced by side-channel analysis methods (side-channel limitation), let L2 be a key lifetime limitation induced by methods based on the combinatorial properties of a used mode of operation (combinatorial limitation) and let q1, q2 be the total numbers of messages of length m, that can be safely processed with an initial key K according to these limitations.

Let L = min(L1, L2), q = min (q1, q2), q * m <= L. As L1 limitation is usually much stronger than L2 limitation (L1 < L2), the final key lifetime restriction is equal to the most restrictive limitation L1. Thus, as displayed in Figure 2, without re-keying only q1 (q1 * m <= L1) messages can be safely processed.

                        
           <--------m------->
           +----------------+ ^ ^  
           |================| | | 
           |================| | | 
       K-->|================| q1|
           |================| | | 
           |==============L1| | | 
           +----------------+ v |  
           |                |   | 
           |                |   | 
           |                |   q2
           |                |   |
           |                |   | 
           |                |   | 
           |                |   | 
           |                |   | 
           |                |   | 
           |                |   | 
           |                |   | 
           |              L2|   | 
           +----------------+   v 

Figure 2: Basic principles of message processing without external re-keying

                    

Suppose that the safety margin for the protocol P is fixed and the external re-keying approach is applied to the initial key K to generate the sequence of frame keys. The frame keys are generated in such a way that the leakage of a previous frame key does not have any impact on the following one, so the side channel limitation L1 goes off. Thus, the resulting key lifetime limitation of the initial key K can be calculated on the basis of a new combinatorial limitation L2'. It is proven (see [AbBell]) that the security of the mode of operation that uses external re-keying leads to an increase when compared to base mode without re-keying (thus, L2 < L2'). Hence, as displayed in Figure 3, the resulting key lifetime limitation in case of using external re-keying can be increased up to L2'.

                        
 
                 <--------m------->
           K     +----------------+ 
           |     |================|
           v     |================|
          K^1--> |================|
           |     |================|
           |     |==============L1|
           |     +----------------+ 
           |     |================|
           v     |================|
          K^2--> |================|
           |     |================|
           |     |==============L1|
           |     +----------------+
           |     |================|
           v     |================|
          ...    |      . . .     |
                 |                |
                 |                |
                 |              L2|
                 +----------------+
                 |                |
                ...              ...
                 |             L2'|
                 +----------------+  

Figure 3: Basic principles of message processing with external re-keying
                
                    

Note: the key transformation process is depicted in a simplified form. A specific approach (parallel and serial) is described below.

Consider an example. Let the message size in a protocol P be equal to 1 KB. Suppose L1 = 128 MB and L2 = 1 TB. Thus, if an external re-keying mechanism is not used, the initial key K must be renegotiated after processing 128 MB / 1 KB = 131072 messages.

If an external re-keying mechanism is used, the key lifetime limitation L1 goes off. Hence the resulting key lifetime limitation L2' can be set to more then 1 TB. Thus if an external re-keying mechanism is used, more then 1 TB / 1 KB = 2^30 messages can be processed before the initial key K is renegotiated. This is 8192 times greater than the number of messages that can be processed, when external re-keying mechanism is not used.

5.1. Methods of Key Lifetime Control

Suppose L is an amount of data that can be safely processed with one section key. For i in {1, 2, ... , t} the frame key K^i (see Figure 4 and Figure 5) should be transformed after processing q_i messages, where q_i can be calculated in accordance with one of the following two approaches:

5.2. Parallel Constructions

The main idea behind external re-keying with a parallel construction is presented in Figure 4:

                            
Maximum message size = m_max.
_____________________________________________________________

                                m_max
                          <---------------->
                M^{1,1}   |===             |
                M^{1,2}   |=============== |
      +->K^1-->   ...            ...       
      |         M^{1,q_1} |========        |
      |
      | 
      |         M^{2,1}   |================|  
      |         M^{2,2}   |=====           | 
K-----|->K^2-->   ...            ...        
      |         M^{2,q_2} |==========      | 
      |                                       
     ... 
      |         M^{t,1}   |============    |
      |         M^{t,2}   |=============   |
      +->K^t-->   ...            ...       
                M^{t,q_t} |==========      |
      
_____________________________________________________________

       Figure 4: External parallel re-keying mechanisms
                
                        

The frame key K^i, i = 1, ... , t-1, is updated after processing a certain amount of messages (see Section 5.1).

5.2.1. Parallel Construction Based on a KDF on a Block Cipher

ExtParallelC re-keying mechanism is based on the key derivation function on a block cipher and is used to generate t frame keys as follows:

where R = ceil(t * k/n).

5.2.2. Parallel Construction Based on HKDF

ExtParallelH re-keying mechanism is based on the key derivation function HKDF-Expand, described in [RFC5869], and is used to generate t frame keys as follows:

where label is a string (may be a zero-length string) that is defined by a specific protocol.

5.2.3. Key Hierarchy Construction

The application of external key hierarchy mechanism leads to the construction of the key tree with the initial key K (root key) at the 0-level and the frame keys K^1, K^2, ... at the last level as described in Figure 6.

                            

                             K = K_root
                     ___________|___________
                    |          ...          |
                    V                       V
                   K{1,1}                K{1,W1}
              ______|______           ______|______
             |     ...     |         |     ...     |
             V             V         V             V
          K{2,1}       K{2,W2}  K{2,(W1-1)*W2+1} K{2,W1*W2}
           __|__         __|__     __|__         __|__
          | ... |       | ... |   | ... |       | ... |
          V     V       V     V   V     V       V     V
       K{3,1}  ...     ...   ... ...   ...     ...   K{3,W1*W2*W3}

      ...                                             ...   
     __|__                     ...                   __|__      
    | ... |                                         | ... |   
    V     V                                         V     V   
K{h,1}   K{h,Wh}            K{h,(W1*...*W{h-1}-1)*Wh+1}  K{h,W1*...*Wh}
  //       \\                                     //       \\      
K^1       K^{Wh}           K^{(W1*...*W{h-1}-1)*Wh+1}     K^{W1*...*Wh}
_______________________________________________________________________

               Figure 6: External Key Hierarchy Mechanism
                
                        

Each j-level key K{j,w}, where j in {1, ... , t}, w in {1, ... , W1 * ... * Wj}, is derived from the (j-1)-level "parent" key K{j-1,ceil(w/Wi)} (and other appropriate input data) using j-th level derivation function that can be based on the block cipher function or on the HMAC-Hash function, described in [RFC5869], and that is defined in accordance with a specific protocol.

The i-th frame K^i can be calculated as follows:

where Der_j is a j-th level derivation function that takes two arguments (the parent key value and the integer in range from 1 to Wj) and outputs the j-th level key value.

The height of tree h and the number of keys Wj, j in {1, ... , h}, which can be partitioned from "parent" key, are defined in accordance with a specific protocol and key lifetime limitations for a used derivation functions.

The frame key K^i, i = 1, ... , t - 1, is updated after processing a certain amount of messages (see Section 5.1).

In order to create an effective implementation, the derivation functions Der_j, j in {1, ... , h}, should be used only in case when ceil(i / (W{h-j+2} * ... * Wh)) != ceil((i - 1) / (W{h-j+2} * ... * Wh)); otherwise it is necessary to use previously generated value. This approach also makes it possible to take countermeasures against side channels attacks.

An example of the external key hierarchy mechanism can be found in Section 6 of [NISTSP800-108].

5.3. Serial Constructions

The main idea behind external re-keying with a serial construction is presented in Figure 5:

                            
Maximum message size = m_max.
_____________________________________________________________
                                     m_max
                               <---------------->           
                     M^{1,1}   |===             |
                     M^{1,2}   |=============== |
K*_1 = K --->K^1-->    ...            ...
  |                  M^{1,q_1} |========        |
  |
  |
  |                  M^{2,1}   |================| 
  v                  M^{2,2}   |=====           | 
K*_2 ------->K^2-->    ...            ...                 
  |                  M^{2,q_2} |==========      | 
  |                                               
 ...
  |                  M^{t,1}   |============    |
  v                  M^{t,2}   |=============   |
K*_t ------->K^t-->    ...            ...
                     M^{t,q_t} |==========      |
            

_____________________________________________________________

       Figure 5: External serial re-keying mechanisms
                
                        

The frame key K^i, i = 1, ... , t - 1, is updated after processing a certain amount of messages (see Section 5.1).

5.3.1. Serial Construction Based on a KDF on a Block Cipher

The frame key K^i is calculated using ExtSerialC transformation as follows:

where J = ceil(k / n), i = 1, ... , t, K*_i is calculated as follows:

where j = 1, ... , t - 1.

5.3.2. Serial Construction Based on HKDF

The frame key K^i is calculated using ExtSerialH transformation as follows:

where i = 1, ... , t, HKDF-Expand is the HMAC-based key derivation function, described in [RFC5869], K*_i is calculated as follows:

where label1 and label2 are different strings from V* that are defined by a specific protocol (see, for example, TLS 1.3 updating traffic keys algorithm [TLSDraft]).

6. Internal Re-keying Mechanisms

This section presents an approach to increase the key lifetime by using a transformation of a data processing key (section key) during each separate message processing. Each message is processed starting with the same key (the first section key) and each section key is updated after processing N bits of message (section).

This section provides internal re-keying mechanisms called ACPKM (Advanced Cryptographic Prolongation of Key Material) and ACPKM-Master that do not use a master key and use a master key respectively. Such mechanisms are integrated into the base modes of operation and actually form new modes of operation, therefore they are called "internal re-keying" mechanisms in this document.

Internal re-keying mechanisms are recommended to be used in protocols that process large single messages (e.g., CMS messages), since the maximum gain in increasing the key lifetime is achieved by increasing the length of a message, while it provides almost no increase in the number of messages that can be processed with one initial key.

Internal re-keying increases the key lifetime through the following approach. Suppose protocol P uses some base mode of operation. Let L1 and L2 be a side channel and combinatorial limitations respectively and for some fixed amount of messages q let m1, m2 be the lengths of messages, that can be safely processed with a single initial key K according to these limitations.

Thus, by analogy with the Section 5 without re-keying the final key lifetime restriction, as displayed in Figure 7, is equal to L1 and only q messages of the length m1 can be safely processed.

                        
             K
             |
             v
   ^ +----------------+------------------------------------+ 
   | |==============L1|                                  L2|
   | |================|                                    |
   q |================|                                    |
   | |================|                                    |
   | |================|                                    |
   v +----------------+------------------------------------+
     <-------m1------->
     <----------------------------m2----------------------->
   
Figure 7: Basic principles of message processing without internal re-keying

                
                    

Suppose that the safety margin for the protocol P is fixed and internal re-keying approach is applied to the base mode of operation. Suppose further that every message is processed with a section key, which is transformed after processing N bits of data, where N is a parameter. If q * N does not exceed L1 then the side channel limitation L1 goes off and the resulting key lifetime limitation of the initial key K can be calculated on the basis of a new combinatorial limitation L2'. The security of the mode of operation that uses internal re-keying increases when compared to base mode of operation without re-keying (thus, L2 < L2'). Hence, as displayed in Figure 8, the resulting key lifetime limitation in case of using internal re-keying can be increased up to L2'.

                        
  K-----> K^1-------------> K^2 -----------> . . .
          |                 |
          v                 v
^ +----------------+----------------+-------------------+--...--+
| |==============L1|==============L1|======           L2|    L2’|
| |================|================|======             |       |
q |================|================|====== . . .       |       |
| |================|================|======             |       |
| |================|================|======             |       |
v +----------------+----------------+-------------------+--...--+
  <-------N-------->
  
Figure 8: Basic principles of message processing with internal re-keying

                
                    

Note: the key transformation process is depicted in a simplified form. A specific approach (ACPKM and ACPKM-Master re-keying mechanisms) is described below.

Since the performance of encryption can slightly decrease for rather small values of N, the parameter N should be selected for a particular protocol as maximum possible to provide necessary key lifetime for the security models that are considered.

Consider an example. Suppose L1 = 128 MB and L2 = 10 TB. Let the message size in the protocol be large/unlimited (may exhaust the whole key lifetime L2). The most restrictive resulting key lifetime limitation is equal to 128 MB.

Thus, there is a need to put a limit on the maximum message size m_max. For example, if m_max = 32 MB, it may happen that the renegotiation of initial key K would be required after processing only four messages.

If an internal re-keying mechanism with section size N = 1 MB is used, more than L1 / N = 128 MB / 1 MB = 128 messages can be processed before the renegotiation of initial key K (instead of 4 messages in case when an internal re-keying mechanism is not used). Note that only one section of each message is processed with the section key K^i, and, consequently, the key lifetime limitation L1 goes off. Hence the resulting key lifetime limitation L2' can be set to more then 10 TB (in the case when a single large message is processed using the initial key K).

6.1. Methods of Key Lifetime Control

Suppose L is an amount of data that can be safely processed with one section key, N is a section size (fixed parameter). Suppose M^{i}_1 is the first section of message M^{i}, i = 1, ... , q (see Figure 9 and Figure 10), then the parameter q can be calculated in accordance with one of the following two approaches:

6.2. Constructions that Do Not Require Master Key

This section describes the block cipher modes that use the ACPKM re-keying mechanism, which does not use a master key: an initial key is used directly for the encryption of the data.

6.2.1. ACPKM Re-keying Mechanisms

This section defines periodical key transformation without a master key, which is called ACPKM re-keying mechanism. This mechanism can be applied to one of the basic encryption modes (CTR and GCM block cipher modes) for getting an extension of this encryption mode that uses periodical key transformation without a master key. This extension can be considered as a new encryption mode.

An additional parameter that defines functioning of base encryption modes with the ACPKM re-keying mechanism is the section size N. The value of N is measured in bits and is fixed within a specific protocol based on the requirements of the system capacity and the key lifetime. The section size N MUST be divisible by the block size n.

The main idea behind internal re-keying without a master key is presented in Figure 9:

                                
Section size = const = N,
maximum message size = m_max.
____________________________________________________________________

              ACPKM       ACPKM              ACPKM
       K^1 = K ---> K^2 ---...-> K^{l_max-1} ----> K^{l_max}
           |          |                |           |
           |          |                |           |
           v          v                v           v
M^{1} |==========|==========| ... |==========|=======:  |
M^{2} |==========|==========| ... |===       |       :  |
  .        .          .        .       .          .  :  
  :        :          :        :       :          :  :  
M^{q} |==========|==========| ... |==========|=====  :  |
                   section                           :  
                 <---------->                      m_max  
                    N bit                            
___________________________________________________________________
l_max = ceil(m_max/N).

            Figure 9: Internal re-keying without a master key
                
                            

During the processing of the input message M with the length m in some encryption mode that uses ACPKM key transformation of the initial key K the message is divided into l = ceil(m / N) sections (denoted as M = M_1 | M_2 | ... | M_l, where M_i is in V_N for i in {1, 2, ... , l - 1} and M_l is in V_r, r <= N). The first section of each message is processed with the section key K^1 = K. To process the (i + 1)-th section of each message the section key K^{i+1} is calculated using ACPKM transformation as follows:

where J = ceil(k/n) and D_1, D_2, ... , D_J are in V_n and are calculated as follows:

where D is the following constant in V_{1024}:

D = ( 80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 
    | 88 | 89 | 8a | 8b | 8c | 8d | 8e | 8f 
    | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 
    | 98 | 99 | 9a | 9b | 9c | 9d | 9e | 9f 
    | a0 | a1 | a2 | a3 | a4 | a5 | a6 | a7 
    | a8 | a9 | aa | ab | ac | ad | ae | af 
    | b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 
    | b8 | b9 | ba | bb | bc | bd | be | bf 
    | c0 | c1 | c2 | c3 | c4 | c5 | c6 | c7 
    | c8 | c9 | ca | cb | cc | cd | ce | cf 
    | d0 | d1 | d2 | d3 | d4 | d5 | d6 | d7 
    | d8 | d9 | da | db | dc | dd | de | df 
    | e0 | e1 | e2 | e3 | e4 | e5 | e6 | e7 
    | e8 | e9 | ea | eb | ec | ed | ee | ef 
    | f0 | f1 | f2 | f3 | f4 | f5 | f6 | f7 
    | f8 | f9 | fa | fb | fc | fd | fe | ff )  
                            

N o t e : The constant D is such that D_1, ... , D_J are pairwise different for any allowed n and k values.

N o t e : The constant D is such that the highest bit of its each octet is equal to 1. This condition is important, as in conjunction with message length limitation it allows to prevent collisions of block cipher permutation inputs in cases of key transformation and message processing.

6.2.2. CTR-ACPKM Encryption Mode

This section defines a CTR-ACPKM encryption mode that uses internal ACPKM re-keying mechanism for the periodical key transformation.

The CTR-ACPKM mode can be considered as the basic encryption mode CTR (see [MODES]) extended by the ACPKM re-keying mechanism.

The CTR-ACPKM encryption mode can be used with the following parameters:

The CTR-ACPKM mode encryption and decryption procedures are defined as follows:

                                
+----------------------------------------------------------------+
|  CTR-ACPKM-Encrypt(N, K, ICN, P)                               |
|----------------------------------------------------------------|
|  Input:                                                        |
|  - section size N,                                             |
|  - initial key K,                                              |
|  - initial counter nonce ICN in V_{n-c},                       |
|  - plaintext P = P_1 | ... | P_b, |P| <= m_max.                |
|  Output:                                                       |
|  - ciphertext C.                                               |
|----------------------------------------------------------------|
|  1. CTR_1 = ICN | 0^c                                          |
|  2. For j = 2, 3, ... , b do                                   |
|         CTR_{j} = Inc_c(CTR_{j-1})                             |
|  3. K^1 = K                                                    |
|  4. For i = 2, 3, ... , ceil(|P| / N)                          |
|         K^i = ACPKM(K^{i-1})                                   |
|  5. For j = 1, 2, ... , b do                                   |
|         i = ceil(j * n / N),                                   |
|         G_j = E_{K^i}(CTR_j)                                   |
|  6. C = P (xor) MSB_{|P|}(G_1 | ... | G_b)                     |
|  7. Return C                                                   |
+----------------------------------------------------------------+

+----------------------------------------------------------------+
|  CTR-ACPKM-Decrypt(N, K, ICN, C)                               |
|----------------------------------------------------------------|
|  Input:                                                        |
|  - section size N,                                             |
|  - initial key K,                                              |
|  - initial counter nonce ICN in V_{n-c},                       |
|  - ciphertext C = C_1 | ... | C_b, |C| <= m_max.               |
|  Output:                                                       |
|  - plaintext P.                                                |
|----------------------------------------------------------------|
|  1. P = CTR-ACPKM-Encrypt(N, K, ICN, C)                        |
|  2. Return P                                                   |  
+----------------------------------------------------------------+
                            

The initial counter nonce ICN value for each message that is encrypted under the given initial key K must be chosen in a unique manner.

6.2.3. GCM-ACPKM Authenticated Encryption Mode

This section defines GCM-ACPKM authenticated encryption mode that uses internal ACPKM re-keying mechanism for the periodical key transformation.

The GCM-ACPKM mode can be considered as the basic authenticated encryption mode GCM (see [GCM]) extended by the ACPKM re-keying mechanism.

The GCM-ACPKM authenticated encryption mode can be used with the following parameters:

The GCM-ACPKM mode encryption and decryption procedures are defined as follows:

                                
+-------------------------------------------------------------------+
|  GHASH(X, H)                                                      |
|-------------------------------------------------------------------|
|  Input:                                                           |  
|  - bit string X = X_1 | ... | X_m, X_1, ... , X_m in V_n.         |
|  Output:                                                          |
|  - block GHASH(X, H) in V_n.                                      | 
|-------------------------------------------------------------------|
|  1. Y_0 = 0^n                                                     |
|  2. For i = 1, ... , m do                                         |
|         Y_i = (Y_{i-1} (xor) X_i) * H                             |
|  3. Return Y_m                                                    |
+-------------------------------------------------------------------+

+-------------------------------------------------------------------+
|  GCTR(N, K, ICB, X)                                               |
|-------------------------------------------------------------------|
|  Input:                                                           |
|  - section size N,                                                |
|  - initial key K,                                                 |
|  - initial counter block ICB,                                     |
|  - X = X_1 | ... | X_b.                                           |
|  Output:                                                          |
|  - Y in V_{|X|}.                                                  |
|-------------------------------------------------------------------|
|  1. If X in V_0 then return Y, where Y in V_0                     |
|  2. GCTR_1 = ICB                                                  |
|  3. For i = 2, ... , b do                                         |
|         GCTR_i = Inc_c(GCTR_{i-1})                                |
|  4. K^1 = K                                                       |
|  5. For j = 2, ... , ceil(|X| / N)                                |
|         K^j = ACPKM(K^{j-1})                                      |
|  6. For i = 1, ... , b do                                         |
|         j = ceil(i * n / N),                                      |
|         G_i = E_{K_j}(GCTR_i)                                     |
|  7. Y = X (xor) MSB_{|X|}(G_1 | ... | G_b)                        |
|  8. Return Y                                                      |
+-------------------------------------------------------------------+

+-------------------------------------------------------------------+
|  GCM-ACPKM-Encrypt(N, K, ICN, P, A)                               |
|-------------------------------------------------------------------|
|  Input:                                                           |
|  - section size N,                                                |
|  - initial key K,                                                 |
|  - initial counter nonce ICN in V_{n-c},                          |
|  - plaintext P = P_1 | ... | P_b, |P| <= m_max,                   |
|  - additional authenticated data A.                               |
|  Output:                                                          |
|  - ciphertext C,                                                  |
|  - authentication tag T.                                          |
|-------------------------------------------------------------------|
|  1. H = E_{K}(0^n)                                                |
|  2. ICB_0 = ICN | 0^{c-1} | 1                                     |
|  3. C = GCTR(N, K, Inc_c(ICB_0), P)                               |
|  4. u = n * ceil(|C| / n) - |C|                                   |
|     v = n * ceil(|A| / n) - |A|                                   |
|  5. S = GHASH(A | 0^v | C | 0^u | Vec_{n/2}(|A|) |                | 
|               | Vec_{n/2}(|C|), H)                                |
|  6. T = MSB_t(E_{K}(ICB_0) (xor) S)                               |
|  7. Return C | T                                                  |
+-------------------------------------------------------------------+

+-------------------------------------------------------------------+
|  GCM-ACPKM-Decrypt(N, K, ICN, A, C, T)                            |
|-------------------------------------------------------------------|
|  Input:                                                           |
|  - section size N,                                                |
|  - initial key K,                                                 |
|  - initial counter block ICN,                                     |
|  - additional authenticated data A,                               |
|  - ciphertext C = C_1 | ... | C_b, |C| <= m_max,                  |
|  - authentication tag T.                                          |
|  Output:                                                          |
|  - plaintext P or FAIL.                                           |
|-------------------------------------------------------------------|
|  1. H = E_{K}(0^n)                                                |
|  2. ICB_0 = ICN | 0^{c-1} | 1                                     |
|  3. P = GCTR(N, K, Inc_c(ICB_0), C)                               |
|  4. u = n * ceil(|C| / n) - |C|                                   |
|     v = n * ceil(|A| / n) - |A|                                   |
|  5. S = GHASH(A | 0^v | C | 0^u | Vec_{n/2}(|A|) |                |
|               | Vec_{n/2}(|C|), H)                                |
|  6. T' = MSB_t(E_{K}(ICB_0) (xor) S)                              |
|  7. If T = T' then return P; else return FAIL                     |
+-------------------------------------------------------------------+
                            

The * operation on (pairs of) the 2^n possible blocks corresponds to the multiplication operation for the binary Galois (finite) field of 2^n elements defined by the polynomial f as follows (by analogy with [GCM]):

n = 128:
f = a^128 + a^7 + a^2 + a^1 + 1,
n = 256:
f = a^256 + a^10 + a^5 + a^2 + 1.

The initial vector IV value for each message that is encrypted under the given initial key K must be chosen in a unique manner.

The key for computing values E_{K}(ICB_0) and H is not updated and is equal to the initial key K.

6.3. Constructions that Require Master Key

This section describes the block cipher modes that use the ACPKM-Master re-keying mechanism, which use the initial key K as a master key, so K is never used directly for data processing but is used for key derivation.

6.3.1. ACPKM-Master Key Derivation from the Master Key

This section defines periodical key transformation with a master key, which is called ACPKM-Master re-keying mechanism. This mechanism can be applied to one of the basic modes of operation (CTR, GCM, CBC, CFB, OMAC modes) for getting an extension that uses periodical key transformation with a master key. This extension can be considered as a new mode of operation.

Additional parameters that define the functioning of modes of operation that use the ACPKM-Master re-keying mechanism are the section size N, the change frequency T* of the master keys K*_1, K*_2, ... (see Figure 10) and the size d of the section key material. The values of N and T* are measured in bits and are fixed within a specific protocol, based on the requirements of the system capacity and the key lifetime. The section size N MUST also be divisible by the block size n. The master key frequency T* MUST be divisible by d and by n.

The main idea behind internal re-keying with a master key is presented in Figure 10:

                                
                                
Master key frequency T*,
section size N,
maximum message size = m_max.
__________________________________________________________________________________

                            ACPKM                   ACPKM
               K*_1 = K--------------> K*_2 ---------...---------> K*_l_max
              ___|___                ___|___                     ___|___
             |       |              |       |                   |       |
             v  ...  v              v  ...  v                   v  ...  v
            K[1]     K[t]        K[t+1]   K[2t]      K[(l_max-1)t+1]   K[l_max*t]
             |       |              |       |                   |       |
             |       |              |       |                   |       |
             v       v              v       v                   v       v
M^{1}||========|...|========||========|...|========||...||========|...|==    : ||
M^{2}||========|...|========||========|...|========||...||========|...|======: ||
 ... ||        |   |        ||        |   |        ||   ||        |   |      : ||
M^{q}||========|...|========||====    |...|        ||...||        |...|      : ||
       section                                                               :
      <-------->                                                             :
         N bit                                                             m_max
__________________________________________________________________________________
|K[i]| = d,
t = T* / d,
l_max = ceil(m_max / (N * t)).  

                   Figure 10: Internal re-keying with a master key

                            

During the processing of the input message M with the length m in some mode of operation that uses ACPKM-Master key transformation with the initial key K and the master key frequency T* the message M is divided into l = ceil(m / N) sections (denoted as M = M_1 | M_2 | ... | M_l, where M_i is in V_N for i in {1, 2, ... , l - 1} and M_l is in V_r, r <= N). The j-th section of each message is processed with the key material K[j], j in {1, ... , l}, |K[j]| = d, that is calculated with the ACPKM-Master algorithm as follows:

Note: the parameters d and l MUST be such that d * l <= n * 2^{n/2-1}.

6.3.2. CTR-ACPKM-Master Encryption Mode

This section defines a CTR-ACPKM-Master encryption mode that uses internal ACPKM-Master re-keying mechanism for the periodical key transformation.

The CTR-ACPKM-Master encryption mode can be considered as the basic encryption mode CTR (see [MODES]) extended by the ACPKM-Master re-keying mechanism.

The CTR-ACPKM-Master encryption mode can be used with the following parameters:

The key material K[j] that is used for one section processing is equal to K^j, |K^j| = k bits.

The CTR-ACPKM-Master mode encryption and decryption procedures are defined as follows:

                               
+----------------------------------------------------------------+
|  CTR-ACPKM-Master-Encrypt(N, K, T*, ICN, P)                    |
|----------------------------------------------------------------|
|  Input:                                                        |
|  - section size N,                                             |
|  - initial key K,                                              |
|  - master key frequency T*,                                    |
|  - initial counter nonce ICN in V_{n-c},                       |
|  - plaintext P = P_1 | ... | P_b, |P| <= m_max.                |
|  Output:                                                       |
|  - ciphertext C.                                               |
|----------------------------------------------------------------|
|  1. CTR_1 = ICN | 0^c                                          |
|  2. For j = 2, 3, ... , b do                                   |
|         CTR_{j} = Inc_c(CTR_{j-1})                             |
|  3. l = ceil(|P| / N)                                          |
|  4. K^1 | ... | K^l = ACPKM-Master(T*, K, k, l)                |
|  5. For j = 1, 2, ... , b do                                   |
|         i = ceil(j * n / N),                                   |
|         G_j = E_{K^i}(CTR_j)                                   |
|  6. C = P (xor) MSB_{|P|}(G_1 | ... |G_b)                      |
|  7. Return C                                                   |
|----------------------------------------------------------------+

+----------------------------------------------------------------+
|  CTR-ACPKM-Master-Decrypt(N, K, T*, ICN, C)                    |
|----------------------------------------------------------------|
|  Input:                                                        |
|  - section size N,                                             |
|  - initial key K,                                              |
|  - master key frequency T*,                                    |
|  - initial counter nonce ICN in V_{n-c},                       |
|  - ciphertext C = C_1 | ... | C_b, |C| <= m_max.               |
|  Output:                                                       |
|  - plaintext P.                                                |
|----------------------------------------------------------------|
|  1. P = CTR-ACPKM-Master-Encrypt(N, K, T*, ICN, C)             |
|  1. Return P                                                   |
+----------------------------------------------------------------+
                            

The initial counter nonce ICN value for each message that is encrypted under the given initial key must be chosen in a unique manner.

6.3.3. GCM-ACPKM-Master Authenticated Encryption Mode

This section defines a GCM-ACPKM-Master authenticated encryption mode that uses internal ACPKM-Master re-keying mechanism for the periodical key transformation.

The GCM-ACPKM-Master authenticated encryption mode can be considered as the basic authenticated encryption mode GCM (see [GCM]) extended by the ACPKM-Master re-keying mechanism.

The GCM-ACPKM-Master authenticated encryption mode can be used with the following parameters:

The key material K[j] that is used for the j-th section processing is equal to K^j, |K^j| = k bits.

The GCM-ACPKM-Master mode encryption and decryption procedures are defined as follows:

                                
+-------------------------------------------------------------------+
|  GHASH(X, H)                                                      |
|-------------------------------------------------------------------|
|  Input:                                                           |  
|  - bit string X = X_1 | ... | X_m, X_i in V_n for i in {1, ... ,m}|
|  Output:                                                          |
|  - block GHASH(X, H) in V_n                                       | 
|-------------------------------------------------------------------|
|  1. Y_0 = 0^n                                                     |
|  2. For i = 1, ... , m do                                         |
|         Y_i = (Y_{i-1} (xor) X_i) * H                             |
|  3. Return Y_m                                                    |
+-------------------------------------------------------------------+

+-------------------------------------------------------------------+
|  GCTR(N, K, T*, ICB, X)                                           |
|-------------------------------------------------------------------|
|  Input:                                                           |
|  - section size N,                                                |
|  - initial key K,                                                 |
|  - master key frequency T*,                                       |
|  - initial counter block ICB,                                     |
|  - X = X_1 | ... | X_b.                                           |
|  Output:                                                          |
|  - Y in V_{|X|}.                                                  |
|-------------------------------------------------------------------|
|  1. If X in V_0 then return Y, where Y in V_0                     |
|  2. GCTR_1 = ICB                                                  |
|  3. For i = 2, ... , b do                                         |
|         GCTR_i = Inc_c(GCTR_{i-1})                                |
|  4. l = ceil(|X| / N)                                             |
|  5. K^1 | ... | K^l = ACPKM-Master(T*, K, k, l)                   |
|  6. For j = 1, ... , b do                                         |
|         i = ceil(j * n / N),                                      |
|         G_j = E_{K^i}(GCTR_j)                                     |
|  7. Y = X (xor) MSB_{|X|}(G_1 | ... | G_b)                        |
|  8. Return Y                                                      |
+-------------------------------------------------------------------+

+-------------------------------------------------------------------+
|  GCM-ACPKM-Master-Encrypt(N, K, T*, ICN, P, A)                    |
|-------------------------------------------------------------------|
|  Input:                                                           |
|  - section size N,                                                |
|  - initial key K,                                                 |
|  - master key frequency T*,                                       |
|  - initial counter nonce ICN in V_{n-c},                          |
|  - plaintext P = P_1 | ... | P_b, |P| <= m_max.                   |
|  - additional authenticated data A.                               |
|  Output:                                                          |
|  - ciphertext C,                                                  |
|  - authentication tag T.                                          |
|-------------------------------------------------------------------|
|  1. K^1 = ACPKM-Master(T*, K, k, 1)                               |
|  2. H = E_{K^1}(0^n)                                              |
|  3. ICB_0 = ICN | 0^{c-1} | 1                                     |
|  4. C = GCTR(N, K, T*, Inc_c(ICB_0), P)                           |
|  5. u = n * ceil(|C| / n) - |C|                                   |
|     v = n * ceil(|A| / n) - |A|                                   |
|  6. S = GHASH(A | 0^v | C | 0^u | Vec_{n/2}(|A|) |                |
|               | Vec_{n/2}(|C|), H)                                |
|  7. T = MSB_t(E_{K^1}(ICB_0) (xor) S)                             |
|  8. Return C | T                                                  |
+-------------------------------------------------------------------+

+-------------------------------------------------------------------+
|  GCM-ACPKM-Master-Decrypt(N, K, T*, ICN, A, C, T)                 |
|-------------------------------------------------------------------|
|  Input:                                                           |
|  - section size N,                                                |
|  - initial key K,                                                 |
|  - master key frequency T*,                                       |
|  - initial counter nonce ICN in V_{n-c},                          |
|  - additional authenticated data A.                               |
|  - ciphertext C = C_1 | ... | C_b, |C| <= m_max,                  |
|  - authentication tag T.                                          |
|  Output:                                                          |
|  - plaintext P or FAIL.                                           |
|-------------------------------------------------------------------|
|  1. K^1 = ACPKM-Master(T*, K, k, 1)                               |
|  2. H = E_{K^1}(0^n)                                              |
|  3. ICB_0 = ICN | 0^{c-1} | 1                                     |
|  4. P = GCTR(N, K, T*, Inc_c(ICB_0), C)                           |
|  5. u = n * ceil(|C| / n) - |C|                                   |
|     v = n * ceil(|A| / n) - |A|                                   |
|  6. S = GHASH(A | 0^v | C | 0^u | Vec_{n/2}(|A|) |                |
|               | Vec_{n/2}(|C|), H)                                |
|  7. T' = MSB_t(E_{K^1}(ICB_0) (xor) S)                            |
|  8. IF T = T' then return P; else return FAIL.                    |
+-------------------------------------------------------------------+
                            

The * operation on (pairs of) the 2^n possible blocks corresponds to the multiplication operation for the binary Galois (finite) field of 2^n elements defined by the polynomial f as follows (by analogy with [GCM]):

n = 128:
f = a^128 + a^7 + a^2 + a^1 + 1,
n = 256:
f = a^256 + a^10 + a^5 + a^2 + 1.

The initial vector IV value for each message that is encrypted under the given initial key must be chosen in a unique manner.

6.3.4. CBC-ACPKM-Master Encryption Mode

This section defines a CBC-ACPKM-Master encryption mode that uses internal ACPKM-Master re-keying mechanism for the periodical key transformation.

The CBC-ACPKM-Master encryption mode can be considered as the basic encryption mode CBC (see [MODES]) extended by the ACPKM-Master re-keying mechanism.

The CBC-ACPKM-Master encryption mode can be used with the following parameters:

In the specification of the CBC-ACPKM-Master mode the plaintext and ciphertext must be a sequence of one or more complete data blocks. If the data string to be encrypted does not initially satisfy this property, then it MUST be padded to form complete data blocks. The padding methods are out of the scope of this document. An example of a padding method can be found in Appendix A of [MODES].

The key material K[j] that is used for the j-th section processing is equal to K^j, |K^j| = k bits.

We will denote by D_{K} the decryption function which is a permutation inverse to the E_{K}.

The CBC-ACPKM-Master mode encryption and decryption procedures are defined as follows:

                               
+----------------------------------------------------------------+
|  CBC-ACPKM-Master-Encrypt(N, K, T*, IV, P)                     |
|----------------------------------------------------------------|
|  Input:                                                        |
|  - section size N,                                             |
|  - initial key K,                                              |
|  - master key frequency T*,                                    |
|  - initialization vector IV in V_n,                            |
|  - plaintext P = P_1 | ... | P_b, |P_b| = n, |P| <= m_max.     |
|  Output:                                                       |
|  - ciphertext C.                                               |
|----------------------------------------------------------------|
|  1. l = ceil(|P| / N)                                          |
|  2. K^1 | ... | K^l = ACPKM-Master(T*, K, k, l)                |
|  3. C_0 = IV                                                   |
|  4. For j = 1, 2, ... , b do                                   |
|         i = ceil(j * n / N),                                   |
|         C_j = E_{K^i}(P_j (xor) C_{j-1})                       |
|  5. Return C = C_1 | ... | C_b                                 |
|----------------------------------------------------------------+

+----------------------------------------------------------------+
|  CBC-ACPKM-Master-Decrypt(N, K, T*, IV, C)                     |
|----------------------------------------------------------------|
|  Input:                                                        |
|  - section size N,                                             |
|  - initial key K,                                              |
|  - master key frequency T*,                                    |
|  - initialization vector IV in V_n,                            |
|  - ciphertext C = C_1 | ... | C_b, |C_b| = n, |C| <= m_max.    |
|  Output:                                                       |
|  - plaintext P.                                                |
|----------------------------------------------------------------|
|  1. l = ceil(|C| / N)                                          |
|  2. K^1 | ... | K^l = ACPKM-Master(T*, K, k, l)                |
|  3. C_0 = IV                                                   |
|  4. For j = 1, 2, ... , b do                                   |
|         i = ceil(j * n / N)                                    |
|         P_j = D_{K^i}(C_j) (xor) C_{j-1}                       |
|  5. Return P = P_1 | ... | P_b                                 |
+----------------------------------------------------------------+
                            

The initialization vector IV for each message that is encrypted under the given initial key does not need to be secret, but must be unpredictable.

6.3.5. CFB-ACPKM-Master Encryption Mode

This section defines a CFB-ACPKM-Master encryption mode that uses internal ACPKM-Master re-keying mechanism for the periodical key transformation.

The CFB-ACPKM-Master encryption mode can be considered as the basic encryption mode CFB (see [MODES]) extended by the ACPKM-Master re-keying mechanism.

The CFB-ACPKM-Master encryption mode can be used with the following parameters:

The key material K[j] that is used for the j-th section processing is equal to K^j, |K^j| = k bits.

The CFB-ACPKM-Master mode encryption and decryption procedures are defined as follows:

                               
+-------------------------------------------------------------+
|  CFB-ACPKM-Master-Encrypt(N, K, T*, IV, P)                  |
|-------------------------------------------------------------|
|  Input:                                                     |
|  - section size N,                                          |
|  - initial key K,                                           |
|  - master key frequency T*,                                 |
|  - initialization vector IV in V_n,                         |
|  - plaintext P = P_1 | ... | P_b, |P| <= m_max.             |
|  Output:                                                    |
|  - ciphertext C.                                            |
|-------------------------------------------------------------|
|  1. l = ceil(|P| / N)                                       |
|  2. K^1 | ... | K^l = ACPKM-Master(T*, K, k, l)             |
|  3. C_0 = IV                                                |
|  4. For j = 1, 2, ... , b - 1 do                            |
|         i = ceil(j * n / N),                                |
|         C_j = E_{K^i}(C_{j-1}) (xor) P_j                    |
|  5. C_b = MSB_{|P_b|}(E_{K^l}(C_{b-1})) (xor) P_b           |
|  6. Return C = C_1 | ... | C_b                              |
|-------------------------------------------------------------+

+-------------------------------------------------------------+
|  CFB-ACPKM-Master-Decrypt(N, K, T*, IV, C)                  |
|-------------------------------------------------------------|
|  Input:                                                     |
|  - section size N,                                          |
|  - initial key K,                                           |
|  - master key frequency T*,                                 |
|  - initialization vector IV in V_n,                         |
|  - ciphertext C = C_1 | ... | C_b, |C| <= m_max.            |
|  Output:                                                    |
|  - plaintext P.                                             |
|-------------------------------------------------------------|
|  1. l = ceil(|C| / N)                                       |
|  2. K^1 | ... | K^l = ACPKM-Master(T*, K, k, l)             |
|  3. C_0 = IV                                                |
|  4. For j = 1, 2, ... , b - 1 do                            |
|         i = ceil(j * n / N),                                |
|         P_j = E_{K^i}(C_{j-1}) (xor) C_j                    |
|  5. P_b = MSB_{|C_b|}(E_{K^l}(C_{b-1})) (xor) C_b           |
|  6. Return P = P_1 | ... | P_b                              |
+-------------------------------------------------------------+
                            

The initialization vector IV for each message that is encrypted under the given initial key need not to be secret, but must be unpredictable.

6.3.6. OMAC-ACPKM-Master Authentication Mode

This section defines an OMAC-ACPKM-Master message authentication code calculation mode that uses internal ACPKM-Master re-keying mechanism for the periodical key transformation.

The OMAC-ACPKM-Master mode can be considered as the basic message authentication code calculation mode OMAC, which is also known as CMAC (see [RFC4493]), extended by the ACPKM-Master re-keying mechanism.

The OMAC-ACPKM-Master message authentication code calculation mode can be used with the following parameters:

The key material K[j] that is used for one section processing is equal to K^j | K^j_1, where |K^j| = k and |K^j_1| = n.

The following is a specification of the subkey generation process of OMAC:

                               
+-------------------------------------------------------------------+
| Generate_Subkey(K1, r)                                            |
|-------------------------------------------------------------------|
| Input:                                                            |
|  - key K1.                                                        |
|  Output:                                                          |
|  - key SK.                                                        |
|-------------------------------------------------------------------|
|   1. If r = n then return K1                                      |
|   2. If r < n then                                                |
|          if MSB_1(K1) = 0                                         | 
|              return K1 << 1                                       |
|          else                                                     |
|              return (K1 << 1) (xor) R_n                           |
|                                                                   |
+-------------------------------------------------------------------+
                            

Where R_n takes the following values:

The OMAC-ACPKM-Master message authentication code calculation mode is defined as follows:

                               
+----------------------------------------------------------------------+
| OMAC-ACPKM-Master(K, N, T*, M)                                       |
|----------------------------------------------------------------------|
| Input:                                                               |
|  - section size N,                                                   |
|  - initial key K,                                                    |
|  - master key frequency T*,                                          |
|  - plaintext M = M_1 | ... | M_b, |M| <= m_max.                      |
|  Output:                                                             |
|  - message authentication code T.                                    |
|----------------------------------------------------------------------|
| 1. C_0 = 0^n                                                         |
| 2. l = ceil(|M| / N)                                                 |
| 3. K^1 | K^1_1 | ... | K^l | K^l_1 = ACPKM-Master(T*, K, (k + n), l) |
| 4. For j = 1, 2, ... , b - 1 do                                      |
|        i = ceil(j * n / N),                                          |
|        C_j = E_{K^i}(M_j (xor) C_{j-1})                              |
| 5. SK = Generate_Subkey(K^l_1, |M_b|)                                |
| 6. If |M_b| = n then M*_b = M_b                                      |
|                 else M*_b = M_b | 1 | 0^{n - 1 -|M_b|}               |
| 7. T = E_{K^l}(M*_b (xor) C_{b-1} (xor) SK)                          |
| 8. Return T                                                          |
+----------------------------------------------------------------------+
                            

7. Joint Usage of External and Internal Re-keying

Any mechanism described in Section 5 can be used with any mechanism described in Section 6.

8. Security Considerations

Re-keying should be used to increase "a priori" security properties of ciphers in hostile environments (e.g., with side-channel adversaries). If some efficient attacks are known for a cipher, it must not be used. So re-keying cannot be used as a patch for vulnerable ciphers. Base cipher properties must be well analyzed, because the security of re-keying mechanisms is based on the security of a block cipher as a pseudorandom function.

Re-keying is not intended to solve any post-quantum security issues for symmetric crypto, since the reduction of security caused by Grover's algorithm is not connected with a size of plaintext transformed by a cipher - only a negligible (sufficient for key uniqueness) material is needed; and the aim of re-keying is to limit a size of plaintext transformed on one initial key.

Re-keying can provide backward security only if previous traffic keys are securely deleted by all parties that have the keys.

9. References

9.1. Normative References

[CMS] Housley, R., "Cryptographic Message Syntax (CMS)", STD 70, RFC 5652, DOI 10.17487/RFC5652, September 2009.
[DTLS] Rescorla, E. and N. Modadugu, "Datagram Transport Layer Security Version 1.2", RFC 6347, DOI 10.17487/RFC6347, January 2012.
[ESP] Kent, S., "IP Encapsulating Security Payload (ESP)", RFC 4303, DOI 10.17487/RFC4303, December 2005.
[GCM] McGrew, D. and J. Viega, "The Galois/Counter Mode of Operation (GCM)", Submission to NIST http://csrc.nist.gov/CryptoToolkit/modes/proposedmodes/gcm/gcm-spec.pdf, January 2004.
[MODES] Dworkin, M., "Recommendation for Block Cipher Modes of Operation: Methods and Techniques", NIST Special Publication 800-38A, December 2001.
[NISTSP800-108] National Institute of Standards and Technology, "Recommendation for Key Derivation Using Pseudorandom Functions", NIST Special Publication 800-108, November 2008.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997.
[RFC4493] Song, JH., Poovendran, R., Lee, J. and T. Iwata, "The AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June 2006.
[RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand Key Derivation Function (HKDF)", RFC 5869, DOI 10.17487/RFC5869, May 2010.
[SSH] Ylonen, T. and C. Lonvick, "The Secure Shell (SSH) Transport Layer Protocol", RFC 4253, DOI 10.17487/RFC4253, January 2006.
[TLS] Dierks, T. and E. Rescorla, "The Transport Layer Security (TLS) Protocol Version 1.2", RFC 5246, DOI 10.17487/RFC5246, August 2008.
[TLSDraft] Rescorla, E., "The Transport Layer Security (TLS) Protocol Version 1.3", 2017.

9.2. Informative References

[AbBell] Michel Abdalla and Mihir Bellare, "Increasing the Lifetime of a Key: A Comparative Analysis of the Security of Re-keying Techniques", ASIACRYPT2000, LNCS 1976, pp. 546–559, 2000.
[LDC] Howard M. Heys, "A Tutorial on Linear and Differential Cryptanalysis", 2017.
[Sweet32] Karthikeyan Bhargavan, Gaëtan Leurent, "On the Practical (In-)Security of 64-bit Block Ciphers: Collision Attacks on HTTP over TLS and OpenVPN", Cryptology ePrint Archive Report 2016/798, 2016.
[TEMPEST] By Craig Ramsay, Jasper Lohuis, "TEMPEST attacks against AES. Covertly stealing keys for 200 euro", 2017.

Appendix A. Test examples

                        
External re-keying with a parallel construction based on AES-256                       
****************************************************************
k = 256 
t = 128

Initial key:
00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 
0F 0E 0D 0C 0B 0A 09 08 07 06 05 04 03 02 01 00 

K^1:
51 16 8A B6 C8 A8 38 65 54 85 31 A5 D2 BA C3 86 
64 7D 5C D5 1C 3D 62 98 BC 09 B1 D8 64 EC D9 B1 

K^2:
6F ED F5 D3 77 57 48 75 35 2B 5F 4D B6 5B E0 15 
B8 02 92 32 D8 D3 8D 73 FE DC DD C6 C8 36 78 BD 

K^3:
B6 40 24 85 A4 24 BD 35 B4 26 43 13 76 26 70 B6 
5B F3 30 3D 3B 20 EB 14 D1 3B B7 91 74 E3 DB EC 

...

K^126:
2F 3F 15 1B 53 88 23 CD 7D 03 FC 3D FD B3 57 5E 
23 E4 1C 4E 46 FF 6B 33 34 12 27 84 EF 5D 82 23 

K^127:
8E 51 31 FB 0B 64 BB D0 BC D4 C5 7B 1C 66 EF FD 
97 43 75 10 6C AF 5D 5E 41 E0 17 F4 05 63 05 ED 

K^128:
77 4F BF B3 22 60 C5 3B A3 8E FE B1 96 46 76 41 
94 49 AF 84 2D 84 65 A7 F4 F7 2C DC A4 9D 84 F9                 

                    

                        
External re-keying with a serial construction based on SHA-256                       
**************************************************************
k = 256 
t = 128

Initial key:
00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 
0F 0E 0D 0C 0B 0A 09 08 07 06 05 04 03 02 01 00 

label1:
SHA2label1

label2:
SHA2label2

K*_1:
00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 
0F 0E 0D 0C 0B 0A 09 08 07 06 05 04 03 02 01 00 

K^1:
2D A8 D1 37 6C FD 52 7F F7 36 A4 E2 81 C6 0A 9B 
F3 8E 66 97 ED 70 4F B5 FB 10 33 CC EC EE D5 EC 

K*_2:
14 65 5A D1 7C 19 86 24 9B D3 56 DF CC BE 73 6F 
52 62 4A 9D E3 CC 40 6D A9 48 DA 5C D0 68 8A 04 

K^2:
2F EA 8D 57 2B EF B8 89 42 54 1B 8C 1B 3F 8D B1 
84 F9 56 C7 FE 01 11 99 1D FB 98 15 FE 65 85 CF 

K*_3:
18 F0 B5 2A D2 45 E1 93 69 53 40 55 43 70 95 8D 
70 F0 20 8C DF B0 5D 67 CD 1B BF 96 37 D3 E3 EB 

K^3:
53 C7 4E 79 AE BC D1 C8 24 04 BF F6 D7 B1 AC BF 
F9 C0 0E FB A8 B9 48 29 87 37 E1 BA E7 8F F7 92  

...

K*_126:
A3 6D BF 02 AA 0B 42 4A F2 C0 46 52 68 8B C7 E6 
5E F1 62 C3 B3 2F DD EF E4 92 79 5D BB 45 0B CA 

K^126:
6C 4B D6 22 DC 40 48 0F 29 C3 90 B8 E5 D7 A7 34 
23 4D 34 65 2C CE 4A 76 2C FE 2A 42 C8 5B FE 9A 

K*_127:
84 5F 49 3D B8 13 1D 39 36 2B BE D3 74 8F 80 A1 
05 A7 07 37 BA 15 72 E0 73 49 C2 67 5D 0A 28 A1 

K^127:
57 F0 BD 5A B8 2A F3 6B 87 33 CF F7 22 62 B4 D0 
F0 EE EF E1 50 74 E5 BA 13 C1 23 68 87 36 29 A2 

K*_128:
52 F2 0F 56 5C 9C 56 84 AF 69 AD 45 EE B8 DA 4E 
7A A6 04 86 35 16 BA 98 E4 CB 46 D2 E8 9A C1 09 

K^128:
9B DD 24 7D F3 25 4A 75 E0 22 68 25 68 DA 9D D5 
C1 6D 2D 2B 4F 3F 1F 2B 5E 99 82 7F 15 A1 4F A4                

                    

                        
CTR-ACPKM mode with AES-256
***************************
k = 256
n = 128
c = 64
N = 256

Initial key K:
88 99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77 
FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF 

Plain text P:
11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88 
00 11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 
11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 
22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 
33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 
44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33 
55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33 44 

ICN:
12 34 56 78 90 AB CE F0 A1 B2 C3 D4 E5 F0 01 12 
23 34 45 56 67 78 89 90 12 13 14 15 16 17 18 19 

D_1:
80 81 82 83 84 85 86 87 88 89 8A 8B 8C 8D 8E 8F 

D_2:
90 91 92 93 94 95 96 97 98 99 9A 9B 9C 9D 9E 9F 

ACPKM's iteration 1
Process block 1
Input block (CTR):
12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 00 

Output block (G):
FD 7E F8 9A D9 7E A4 B8 8D B8 B5 1C 1C 9D 6D D0 

Plain text block:
11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88 

Cipher text block:
EC 5C CB DE 8C 18 D3 B8 72 56 68 D0 A7 37 F4 58 

Process block 2
Input block (CTR):
12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 01 

Output block (G):
19 98 C5 71 76 37 FB 17 11 E4 48 F0 0C 0D 60 B2 

Plain text block:
00 11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 

Cipher text block:
19 89 E7 42 32 62 9D 60 99 7D E2 4B C0 E3 9F B8 

Updated key:
F6 80 D1 21 2F A4 3D F4 EC 3A 91 DE 2A B1 6F 1B 
36 B0 48 8A 4F C1 2E 09 98 D2 E4 A8 88 E8 4F 3D 

ACPKM's iteration 2
Process block 1
Input block (CTR):
12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 02 

Output block (G):
E4 88 89 4F B6 02 87 DB 77 5A 07 D9 2C 89 46 EA 

Plain text block:
11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 

Cipher text block:
F5 AA BA 0B E3 64 F0 53 EE F0 BC 15 C2 76 4C EA 

Process block 2
Input block (CTR):
12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 03 

Output block (G):
BC 4F 87 23 DB F0 91 50 DD B4 06 C3 1D A9 7C A4 

Plain text block:
22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 

Cipher text block:
9E 7C C3 76 BD 87 19 C9 77 0F CA 2D E2 A3 7C B5 

Updated key:
8E B9 7E 43 27 1A 42 F1 CA 8E E2 5F 5C C7 C8 3B 
1A CE 9E 5E D0 6A A5 3B 57 B9 6A CF 36 5D 24 B8 

ACPKM's iteration 3
Process block 1
Input block (CTR):
12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 04 

Output block (G):
68 6F 22 7D 8F B2 9C BD 05 C8 C3 7D 22 FE 3B B7 

Plain text block:
33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 

Cipher text block:
5B 2B 77 1B F8 3A 05 17 BE 04 2D 82 28 FE 2A 95 

Process block 2
Input block (CTR):
12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 05 

Output block (G):
C0 1B F9 7F 75 6E 12 2F 80 59 55 BD DE 2D 45 87 

Plain text block:
44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33 

Cipher text block:
84 4E 9F 08 FD F7 B8 94 4C B7 AA B7 DE 3C 67 B4 

Updated key:
C5 71 6C C9 67 98 BC 2D 4A 17 87 B7 8A DF 94 AC 
E8 16 F8 0B DB BC AD 7D 60 78 12 9C 0C B4 02 F5 

ACPKM's iteration 4
Process block 1
Input block (CTR):
12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 06 

Output block (G):
03 DE 34 74 AB 9B 65 8A 3B 54 1E F8 BD 2B F4 7D 

Plain text block:
55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33 44 

Cipher text block:
56 B8 43 FC 32 31 DE 46 D5 AB 14 F8 AC 09 C7 39 

Cipher text C:
EC 5C CB DE 8C 18 D3 B8 72 56 68 D0 A7 37 F4 58 
19 89 E7 42 32 62 9D 60 99 7D E2 4B C0 E3 9F B8 
F5 AA BA 0B E3 64 F0 53 EE F0 BC 15 C2 76 4C EA 
9E 7C C3 76 BD 87 19 C9 77 0F CA 2D E2 A3 7C B5 
5B 2B 77 1B F8 3A 05 17 BE 04 2D 82 28 FE 2A 95 
84 4E 9F 08 FD F7 B8 94 4C B7 AA B7 DE 3C 67 B4 
56 B8 43 FC 32 31 DE 46 D5 AB 14 F8 AC 09 C7 39 

                    

                        
OMAC-ACPKM-Master mode with AES-256
***********************************
k = 256
n = 128
c = 64
N = 256

Initial key K:
88 99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77 
FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF 

Plain text P:
11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88 
00 11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 
11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 
22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 
33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 
44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33 
55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33 44 

ICN:
12 34 56 78 90 AB CE F0 A1 B2 C3 D4 E5 F0 01 12 
23 34 45 56 67 78 89 90 12 13 14 15 16 17 18 19 

D_1:
80 81 82 83 84 85 86 87 88 89 8A 8B 8C 8D 8E 8F 

D_2:
90 91 92 93 94 95 96 97 98 99 9A 9B 9C 9D 9E 9F 

ACPKM's iteration 1
Process block 1
Input block (CTR):
12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 00 

Output block (G):
FD 7E F8 9A D9 7E A4 B8 8D B8 B5 1C 1C 9D 6D D0 

Plain text block:
11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88 

Cipher text block:
EC 5C CB DE 8C 18 D3 B8 72 56 68 D0 A7 37 F4 58 

Process block 2
Input block (CTR):
12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 01 

Output block (G):
19 98 C5 71 76 37 FB 17 11 E4 48 F0 0C 0D 60 B2 

Plain text block:
00 11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 

Cipher text block:
19 89 E7 42 32 62 9D 60 99 7D E2 4B C0 E3 9F B8 

Updated key:
F6 80 D1 21 2F A4 3D F4 EC 3A 91 DE 2A B1 6F 1B 
36 B0 48 8A 4F C1 2E 09 98 D2 E4 A8 88 E8 4F 3D 

ACPKM's iteration 2
Process block 1
Input block (CTR):
12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 02 

Output block (G):
E4 88 89 4F B6 02 87 DB 77 5A 07 D9 2C 89 46 EA 

Plain text block:
11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 

Cipher text block:
F5 AA BA 0B E3 64 F0 53 EE F0 BC 15 C2 76 4C EA 

Process block 2
Input block (CTR):
12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 03 

Output block (G):
BC 4F 87 23 DB F0 91 50 DD B4 06 C3 1D A9 7C A4 

Plain text block:
22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 

Cipher text block:
9E 7C C3 76 BD 87 19 C9 77 0F CA 2D E2 A3 7C B5 

Updated key:
8E B9 7E 43 27 1A 42 F1 CA 8E E2 5F 5C C7 C8 3B 
1A CE 9E 5E D0 6A A5 3B 57 B9 6A CF 36 5D 24 B8 

ACPKM's iteration 3
Process block 1
Input block (CTR):
12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 04 

Output block (G):
68 6F 22 7D 8F B2 9C BD 05 C8 C3 7D 22 FE 3B B7 

Plain text block:
33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 

Cipher text block:
5B 2B 77 1B F8 3A 05 17 BE 04 2D 82 28 FE 2A 95 

Process block 2
Input block (CTR):
12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 05 

Output block (G):
C0 1B F9 7F 75 6E 12 2F 80 59 55 BD DE 2D 45 87 

Plain text block:
44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33 

Cipher text block:
84 4E 9F 08 FD F7 B8 94 4C B7 AA B7 DE 3C 67 B4 

Updated key:
C5 71 6C C9 67 98 BC 2D 4A 17 87 B7 8A DF 94 AC 
E8 16 F8 0B DB BC AD 7D 60 78 12 9C 0C B4 02 F5 

ACPKM's iteration 4
Process block 1
Input block (CTR):
12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 06 

Output block (G):
03 DE 34 74 AB 9B 65 8A 3B 54 1E F8 BD 2B F4 7D 

Plain text block:
55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33 44 

Cipher text block:
56 B8 43 FC 32 31 DE 46 D5 AB 14 F8 AC 09 C7 39 

Cipher text C:
EC 5C CB DE 8C 18 D3 B8 72 56 68 D0 A7 37 F4 58 
19 89 E7 42 32 62 9D 60 99 7D E2 4B C0 E3 9F B8 
F5 AA BA 0B E3 64 F0 53 EE F0 BC 15 C2 76 4C EA 
9E 7C C3 76 BD 87 19 C9 77 0F CA 2D E2 A3 7C B5 
5B 2B 77 1B F8 3A 05 17 BE 04 2D 82 28 FE 2A 95 
84 4E 9F 08 FD F7 B8 94 4C B7 AA B7 DE 3C 67 B4 
56 B8 43 FC 32 31 DE 46 D5 AB 14 F8 AC 09 C7 39 

                    

Appendix B. Contributors

Appendix C. Acknowledgments

We thank Mihir Bellare, Scott Fluhrer, Dorothy Cooley, Yoav Nir, Jim Schaad, Paul Hoffman and Dmitry Belyavsky for their useful comments.

Author's Address

Stanislav Smyshlyaev (editor) CryptoPro 18, Suschevsky val Moscow, 127018 Russian Federation Phone: +7 (495) 995-48-20 EMail: svs@cryptopro.ru