SCIENCE CHINA Information Sciences, Volume 63, Issue 1: 112106(2020) https://doi.org/10.1007/s11432-019-1507-1

## Cryptanalysis of PRIMATEs

Wei WANG1,2,*
• AcceptedAug 2, 2019
• PublishedDec 24, 2019
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### Abstract

PRIMATEs is a family of authenticated encryption design submitted to competition for authenticated encryption: security, applicability, and robustness. The three modes of operation in PRIMATEs family are: APE, HANUMAN, GIBBON with security levels: 80, 120 bits. APE is robust despite the nonce misusing. In this study, we revise the algebraic model and find new integral distinguishers for both PRIMATE permutation and its inverse permutation. Moreover, we construct a zero-sum distinguisher for full 12-round PRIMATE-80/120 permutation with the $2^{100}/2^{105}$ complexity, improving over previous work. We also perform an integral attack on 8-round finalization of APE-80/120 with $2^{30}$ chosen messages. The key recovery process is optimized using the FFT technique presented by Todo and Aoki. Our work is the best attack against APE, demonstrating the practical attack on 8-round finalization of APE-80. The new integral distinguishers apply to create forgeries on 5/6-round finalization of APE and HANUMAN that require $2^{15}/2^{30}$ chosen messages, which is the first forgery attack against APE and HANUMAN.

### Acknowledgment

This work was supported by National Cryptography Development Fund (Grant No. MMJJ20170102), National Natural Science Foundation of China (Grant Nos. 61572293, 61502276, 61692276), Major Scientific and Technological Innovation Projects of Shandong Province (Grant No. 2017CXGC0704), National Natural Science Foundation of Shandong Province (Grant No. ZR2016FM22), and Open Research Fund from Shandong Provincial Key Laboratory of Computer Network (Grant No. SDKLCN-2017-04).

### References

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• Figure 1

The encryption of APE.

• Figure 2

The encryption of HANUMAN.

• Figure 3

(Color online) Zero-sum distinguisher for PRIMATE-80.

• Figure 4

(Color online) The 8-round integral attack on APE-80.

• Table 1   Results for PRIMATEs
 Attack type Variants Rounds Data complexity Time complexity Method Scenario Source Distinguisher PRIMATE-80 12/12 – $2^{130}$ Zero-sum – Ref. [11] 12/12 – $2^{100}$ Section sect. 4 PRIMATE-120 12/12 – $2^{130}$ Ref. [11] 12/12 – $2^{105}$ Section sect. 4 Key recovery APE-80 8/12 $2^{33}$ $2^{61}$ Cube Nonce-misuse Ref. [11]* 8/12 $2^{35}$ $2^{61}$ Cube Ref. [11] 8/12 $2^{35}$ $2^{39.29}$ Integral Nonce-misuse Section sect. 5 8/12 $2^{30}$ $2^{39.29}$ Integral Section sect. 5 Key recovery APE-120 8/12 $2^{33}$ $2^{71}$ Cube Nonce-misuse Ref. [11]* 8/12 $2^{35}$ $2^{71}$ Cube Ref. [11] 8/12 $2^{35}$ $2^{50.26}$ Integral Nonce-misuse Section sect. 5 8/12 $2^{30}$ $2^{50.26}$ Integral Section sect. 5 Forgery APE-80 5/12 $2^{15}$ $2^{15}$ Integral Nonce-misuse Section sect. 6 APE-80 6/12 $2^{30}$ $2^{30}$ Section sect. 6 APE-120 5/12 $2^{15}$ $2^{15}$ Section sect. 6 APE-120 6/12 $2^{30}$ $2^{30}$ Section sect. 6 Key recovery HANUMAN-120 6/12 $2^{65}$ $2^{65}$ Cube-like Nonce-respecting Ref. [12] Forgery HANUMAN-80 5/12 $2^{15}$ $2^{15}$ Integral Nonce-misuse Section sect. 6 HANUMAN-80 6/12 $2^{30}$ $2^{30}$ Section sect. 6 HANUMAN-120 5/12 $2^{15}$ $2^{15}$ Section sect. 6 HANUMAN-120 6/12 $2^{30}$ $2^{30}$ Section sect. 6

*

•

Algorithm 1 Distinguisher searching algorithm for 5-round PRIMATE permutation

for $0<s<\lceil~1.0294(n+1)\rceil$

Pick $x$ randomly;

Compute ${\rm~tmp}=\sum_{i=0}^{2^{15}-1}f(x\oplus~i)$;

if ${\rm~tmp}~\neq~0$ then

Output “Proposition 3.3 does not hold";

end if

end for

• Table 2   Notations
 Symbol Definition $x\in~\{0,1\}^k$ Bitstring $x$ of length $k$ (variable if $k=*$) $x\oplus~y$ XOR of bitstrings $x$ and $y$ $x||y$ Concatenation of bitstrings $x$ and $y$ $x_i$ $i$-th bit of the state used in PRIMATE permutation $X_i$ 5-bit state word of the state used in PRIMATE permutation $K$, $N$, $T$ Secret key $K$, nonce $N$, tag $T$ $P$, $C$, $A$ Plaintext $P$, ciphertext $C$, associated data $A$ (in blocks $P_i$, $C_i$, $A_i$) $p_1$, $p_2$, $p_3$, $p_4$ Four different permutations used in PRIMATEs
•

Algorithm 2 Distinguisher searching algorithm for 6-round PRIMATE permutation

for $0<s<\lceil~1.0294(n+1)\rceil$

Pick $x$ randomly;

Compute ${\rm~tmp}=\sum_{i=0}^{2^{30}-1}f(x\oplus~i)$;

if ${\rm~tmp}~\neq~0$ then

Output “Proposition 3.4 does not hold";

end if

end for

• Table 3   The $S$-box of PRIMATEs
 $x$ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 $S(x)$ 1 0 25 26 17 29 21 27 20 5 4 23 14 18 2 28 $x$ 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 $S(x)$ 15 8 6 3 13 7 24 16 30 9 31 10 22 12 11 19
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