Paper 2021/972

Partial Key Exposure Attack on Short Secret Exponent CRT-RSA

Alexander May, Julian Nowakowski, and Santanu Sarkar


Let $(N,e)$ be an RSA public key, where $N=pq$ is the product of equal bitsize primes $p,q$. Let $d_p, d_q$ be the corresponding secret CRT-RSA exponents. Using a Coppersmith-type attack, Takayasu, Lu and Peng (TLP) recently showed that one obtains the factorization of $N$ in polynomial time, provided that $d_p, d_q \leq N^{0.122}$. Building on the TLP attack, we show the first Partial Key Exposure attack on short secret exponent CRT-RSA. Namely, let $N^{0.122} \leq d_p, d_q \leq N^{0.5}$. Then we show that a constant known fraction of the least significant bits (LSBs) of both $d_p, d_q$ suffices to factor $N$ in polynomial time. Naturally, the larger $d_p,d_q$, the more LSBs are required. E.g. if $d_p, d_q$ are of size $N^{0.13}$, then we have to know roughly a $\frac 1 5$-fraction of their LSBs, whereas for $d_p, d_q$ of size $N^{0.2}$ we require already knowledge of a $\frac 2 3$-LSB fraction. Eventually, if $d_p, d_q$ are of full size $N^{0.5}$, we have to know all of their bits. Notice that as a side-product of our result we obtain a heuristic deterministic polynomial time factorization algorithm on input $(N,e,d_p,d_q)$.

Available format(s)
Public-key cryptography
Publication info
A minor revision of an IACR publication in ASIACRYPT 2021
CRT-RSACoppersmith’s methodPartial Key Exposure
Contact author(s)
julian nowakowski @ rub de
2021-11-15: revised
2021-07-22: received
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Creative Commons Attribution


      author = {Alexander May and Julian Nowakowski and Santanu Sarkar},
      title = {Partial Key Exposure Attack on Short Secret Exponent CRT-RSA},
      howpublished = {Cryptology ePrint Archive, Paper 2021/972},
      year = {2021},
      note = {\url{}},
      url = {}
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