# Deterministic Authenticated Encryption Scheme for Memory Constrained Devices

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Preliminaries

**Definition**

**1**

**(DAE).**

**Definition**

**2**

**(Deterministic Privacy (detPriv)).**

**Definition**

**3**

**(Deterministic Authenticity (detAuth)).**

**Definition**

**4**

**(PRF**

**Advantage).**

**Definition**

**5**

**(rka-ind-advantage).**

## 4. Specifications of dAELM

#### 4.1. An Operational Scenario for the Proposed Scheme

#### 4.2. Description

Algorithm 1: Encryption ${\mathcal{E}}_{K}(A,M)$ |

Algorithm 2: Decryption ${\mathcal{D}}_{K}$$(A,C,T)$ |

## 5. Security Results for the New Construction dAELM

#### 5.1. Privacy

**Theorem**

**1.**

**Proof.**

**Adversary ${\mathcal{B}}_{A}$**: In this game, the challenger chooses the key K, and the adversary ${\mathcal{B}}_{A}$ is provided with the oracles $\mathcal{O}\left(RK\right(*,K),\xb7)$, which are simply a related key oracle and $\pi $ oracle. For each $(A,M)$, query adversary $\mathcal{B}$ internally calls oracle $\mathcal{O}$ and replies to adversary A with answer $(C,T)$.

**Lemma**

**1.**

**Adversary ${D}_{A}$**: In this game, the challenger chooses the key K, and the adversary ${D}_{A}$ is provided with the oracles $\mathcal{O}(\xb7)$, which is either a $MA{C}^{g(K,\xb7)}(\xb7)$ or $\$(\xb7)$ oracle. For each $(A,M)$, query adversary D internally calls oracle $\mathcal{O}$ and replies to adversary A with answer $(C,T)$.

**Lemma**

**2.**

#### 5.2. Authenticity

- Adversary A can query encryption oracle $\mathcal{E}$ and decryption oracle $\mathcal{D}$ at most ${q}_{enc}and\phantom{\rule{4pt}{0ex}}{q}_{dec}$ times, respectively.
- All the query responses of encryption oracle $\mathcal{E}$ are stored in a set, say, R. This set contains an $(A,C,T)$ tuple, which has been returned by an encryption query.
- If adversary A is able to generate a new $(A,C,T)$ pair ∉R that produces valid message M, then he wins and output is 1; otherwise, output is 0, after trying ${q}_{dec}$ number of queries.

**Theorem**

**2.**

**Proof.**

- An adversary generates a forgery pair $({A}^{\prime},{C}^{\prime},{T}^{\prime})$ which produces a valid ${M}^{\prime}$ such that $(\ast ,\ast ,{T}^{\prime})\in R$; i.e., he chooses a new ($A,C$) pair, which corresponds to the same tag ${T}_{i}\phantom{\rule{3.33333pt}{0ex}}(={T}^{\prime})$ that has been previously used for some other $({A}_{i},{C}_{i})$, where $1\le i\le {q}_{enc}$. T and ${T}^{\prime}$ will correspond to the same session key ${K}^{*}$ for a given key K. This same ${K}^{*}$ will result in the same output for further block cipher calls. This can be divided into cases.
- (a)
- ${C}^{\prime}={C}_{i}$: In this case, necessarily ${A}^{\prime}\ne {A}_{i}$. As all block cipher outputs are the same, XORing them with $C(={C}_{i})$ will produce the same $M(={M}_{i})$. For a successful forgery, we need at least one parameter to be different in the $({A}^{\prime},{C}^{\prime},{T}^{\prime})$ tuple. Since ${C}^{\prime}={C}_{i}$ and ${T}^{\prime}={T}_{i}$, ${A}^{\prime}$ should be different from ${A}_{i}$. Therefore, the probability of happening this case will be same as the birthday bound on the MAC output.
- (b)
- ${C}^{\prime}\ne {C}_{i}$: In this case, for ${T}_{i}\phantom{\rule{3.33333pt}{0ex}}(={T}^{\prime})$, ${C}^{\prime}\ne {C}_{i}$ will result in ${M}^{\prime}\ne {M}_{i}$. Therefore, the probability of this case will again become the same as the birthday bound on the MAC output.

Therefore, the overall probability of this case will be given by $\frac{{q}_{dec}^{2}}{{2}^{n}}$. - An adversary generates a forgery pair $({A}^{\prime},{C}^{\prime},{T}^{\prime})$, where ${T}^{\prime}$ is new, i.e. $(\ast ,\ast ,{T}^{\prime})\notin R$. This case is further possible in two scenarios:
- (a)
- An adversary is able to guess the key K correctly; i.e. he has access to the system and can generate any number of valid queries. The probability of this case will be equivalent to guessing the key, which is $\frac{1}{{2}^{n}}$.
- (b)
- Suppose adversary guesses the tag ${T}^{\prime}$ correctly for a given key K and $({A}^{\prime},{M}^{\prime})$ pair. Then, for further calculations of ciphertext, he also needs to guess the session key ${K}^{\prime}$. Then only he can generate the forgery. Therefore, the probability of happening this case will be the product of the probability of guessing the tag ${T}^{\prime}$ and session key ${K}^{\prime}$ that is equal to $\frac{1}{{2}^{2n}}$.

Therefore, the overall probability of this case will be $\frac{1}{{2}^{n}}$.

## 6. Comparison

## 7. Software Implementation

Algorithm 3: Decryption ${\mathcal{D}}_{K}(A,C,T)$ |

Algorithm 4:$AES\_CTR\_DEC\_4BLOCKS(K,C,CTR)$ |

$C={C}_{1}\left|\right|{C}_{2}\left|\right|{C}_{3}\left|\right|{C}_{4}$ where $|{C}_{i}|=128$ for $i=1,2,3,4$ ${M}_{1}={C}_{1}\oplus AE{S}_{K}\left(CTR\right)$ ${M}_{2}={C}_{2}\oplus AE{S}_{K}(CTR+1)$ ${M}_{3}={C}_{3}\oplus AE{S}_{K}(CTR+2)$ ${M}_{4}={C}_{4}\oplus AE{S}_{K}(CTR+3)$ $M={M}_{1}\left|\right|{M}_{2}\left|\right|{M}_{3}\left|\right|{M}_{4}$ Return M |

## 8. Discussion

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Games for Security Proof

#### Appendix A.1. Games for Privacy and Authenticity Proofs for dAELM

## References

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**Figure 1.**sp-AELM construction [10].

**Figure 6.**A graph showing memory usage by the cryptomodule during the decryption process, where the x-axis represents the length of message and the y-axis represents the memory usage in bytes. This is the case when we used HMAC-SHA1 as a MAC function, which processes the message in 512 bit blocks.

**Table 1.**Results of a comparison of dEALM with sp-AELM, OAE2, and SIV, where $\tau $ represents the tag length.

Parameters | sp-AELM [10] | OAE2 [18] | SIV [8] | dAELM [This Paper] |
---|---|---|---|---|

Type of AE | Randomized | Randomized | Deterministic | Deterministic |

Online Encryption | Yes | Yes | No | No |

Online Decryption | No | Yes | No | No |

Low memory support | Yes | Yes | No | Yes |

Misuse-resistant | No | No | Yes | yes |

Segmentation | Fixed size block | Variable size segment | Fixed size block | Fixed size block |

Ciphertext expansion | $\tau $ bits | $\tau $ bits per segment | $\tau $ bits | $\tau $ bits |

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**MDPI and ACS Style**

Agrawal, M.; Chang, D.; Kang, J. Deterministic Authenticated Encryption Scheme for Memory Constrained Devices. *Cryptography* **2018**, *2*, 37.
https://doi.org/10.3390/cryptography2040037

**AMA Style**

Agrawal M, Chang D, Kang J. Deterministic Authenticated Encryption Scheme for Memory Constrained Devices. *Cryptography*. 2018; 2(4):37.
https://doi.org/10.3390/cryptography2040037

**Chicago/Turabian Style**

Agrawal, Megha, Donghoon Chang, and Jinkeon Kang. 2018. "Deterministic Authenticated Encryption Scheme for Memory Constrained Devices" *Cryptography* 2, no. 4: 37.
https://doi.org/10.3390/cryptography2040037