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I am reading a paper by Shacham and Waters on Compact Proofs of Retrievability. In this paper on page number 3 under section 1.1 in the very first line the author states that an encoded file is broken into n number of blocks $m_1,m_2...m_n \in Z_p$ for some large prime p.

I am not able to understand that in terms of programmimg what does it exactly mean to have a chunk of file data in a group/field. On the same page they mention an equation:

$σ_i = f_k(i) + αm_i \in Z_p$

What is the value of this $m_i$ in this equation. Is it the number of bytes/bits in each block? What value of $m_i$ would I take if I have to calculate this $σ_i$ in the equation ? The calculated value of sigma lies in the group of prime order $Z_p$.

I am not sure if I have posted the question in the right forum but I could not get any comments in stack overflow hence I have posted it to a forum I find closely related to. Any help will be valuable.

Thanks and Regards

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You have to split your file in such way that each block will be smaller then prime $p$ you're using. After that you have $n$ such blocks.

You calculate $\sigma_i$ for every block so $i \in \{1,2,...,n\}$ and $m_i$ means block number $i$

Example:

Assume you're working in field $Z_p$ with $p=2^{255}-19$ (used in Curve25519 for example) and your file is 254 bytes or $254 \times 8$ bits long. In practice you can split it to blocks of (at most) 254 bits as $2^{254} \ll 2^{255}-19$. You can split your file to 8 parts, each 254bits long. So you have $n$=8, and each $m_1,m2,m3,m4,m5,m6,m7,m8$ are respective blocks from file.

Example2:

Assume you're working in field $Z_p$ with $p=2^8+1$ and your file is 5 bytes long. You can split it to blocks of 8 bits as $2^8 \lt 2^8+1$. So you have 5 such parts and $n=5$ and $m1,m2,m3,m4,m5$ are respective bytes from file.

Please note that second example is in no way secure as target field is way too small.

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  • $\begingroup$ what is the significance of -19 here ? I am actually working in a field Fe where e = 257. $\endgroup$ – skii Jan 12 '18 at 9:43
  • $\begingroup$ $2^{255}-19$ is just example prime I've chosen, something with actual cryptographic use, in your case $p=257=2^8+1$ so you can split your file in 8bit/1byte blocks BUT this is nowhere near secure field as it is really really small $\endgroup$ – Przemko Robakowski Jan 12 '18 at 11:28
  • $\begingroup$ Ok I have a doubt in the answer you say that p = (2^255) - 19. But in the comment for field you are pointing that p = 257? I am confused here. so in the last case was p =255? or p = (2^255) - 19 $\endgroup$ – skii Jan 12 '18 at 11:35
  • $\begingroup$ I've updated my answer, please check if it's clearer now $\endgroup$ – Przemko Robakowski Jan 12 '18 at 11:51

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