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Precisely, there is an implicit independence assumption in Muller’s attack that each collection of eight consecutive keystream words is independent to each other. Unfortunately, this assumption does not always hold. We ﬁnd that under some conditions, a state collision can propagate for more than 8 rounds, thus the independent assumption is violated. This results in many invalid pairs in Muller’ attack. Based on this ﬁnding, we propose our distinguishing attack and key recovery attack on Helix without nonce reused.
Compared to 2 modular exponentiations with (n/2)-bit moduli in the RSA, this leads to a theoretic speed-up of a factor by up to r2 /4. 73 for 3-MPRSA (the modulus has 3 prime factors) with a 1024-bit modulus. ) become less eﬀective as r increases. (However, r can not be unlimitedly large because for the elliptic curve factorization method the diﬃculty of factoring an MPRSA modulus decreases with increasing r. In most cases, r is set 3,4 or 5. ) Therefore, MPRSA might be a practical alternative to RSA when decryption costs need to be lowered.
The authors like to thank the anonymous reviewers for their helpful suggestions. The ﬁrst three authors like to acknowledge the Centre of Excellence in Cryptology, Indian Statistical Institute for supporting this research. References 1. The ECRYPT Stream Cipher Project. eSTREAM Portfolio of Stream Ciphers (Revised on September 8, 2008) 2. : A New Version of Grain-128 with Authentication. In: Symmetric Key Encryption Workshop 2011, DTU, Denmark (February 2011) A Chosen IV Related Key Attack on Grain-128a 25 3.