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Quantifying the success probability of brute force attack against (search) LPN

I’ve been trying to learn about attacks on LPN ($n$-bit secret, noise rate $eta$), and have found several allusions to a brute force algorithm that runs in time exponential in $n$ and requires a linear number of samples in $n$. For example, the third slide of this deck or the original BKW paper. That said, I have not been able to find the success probability of this algorithm, nor derive it myself.

LPN challenger:

  • upon initialization, chooses random $s in mathbb{Z}_2^n$
  • when queried, chooses random $a in mathbb{Z}_2^n$, computes $c = a cdot s mod{2}$ and returns the true $(a, c)$ with probability $1 – eta$, or the corrupted $(a, 1 oplus c)$ with probability $eta$).

Bruteforce algorithm:

  • query the LPN challenger for $m$ labelled samples $(x_1, l_1),…,(x_m, l_m) in mathbb{Z}_2^n times mathbb{Z}_2$
  • for each potential secret $s_j in mathbb{Z}_2^n$, compute the empirical error rate $r_j = (1/m) (sum_{i=1}^m (x_i cdot s_j) + l_i mod{2})$ (note each term of the sum is computed $mod{2}$ but the sum itself is computed over the integers)
  • output potential secret $s_t$ with empirical error rate $r_t$ closest to the noise rate $eta$ (i.e. $t = arg,min_t lvert r_t – eta rvert$)

Intuitively, it seems this algorithm should succeed with probability significantly better than the trivial $frac{1}{2^n}$ probability of guessing at random. But I’m not really sure how to quantify the success probability here. Any pointers to the literature or direct answers appreciated.

ENJ/USD Price Prediction for 14TH August: Enjin Made a 1.67% Rise in Value

ENJ/USD’s Daily Chart Shows Price Action on a Steady Bullish Move. The rise in the value of ENJ/USD is steady as the price action rises with the 9-day moving average. But on the 4-hour chart, the price activities are quite erratic. It is quite a struggle between the bulls and the bears, but the bulls::Listen

ENJ/USD’s Daily Chart Shows Price Action on a Steady Bullish Move.

The rise in the value of ENJ/USD is steady as the price action rises with the 9-day moving average. But on the 4-hour chart, the price activities are quite erratic. It is quite a struggle between the bulls and the bears, but the bulls are driving the price to the upside steadily. For now, they have managed to fix the price at $0.71472620. The 4-hour chart also confirms the slight bullish move which is pushing up. However, if today closes in favor of the bears, the trend may reverse briefly.

 

ENJ/USD Price Statistic Data

  • ENJUSD price for now: – $0.7159
  • ENJUSD market cap:- $641 million
  • ENJUSD circulating supply: – 896million ENJ
  • ENJUSD total supply:- 1 billion ENJ
  • ENJUSD total coin market cap ranking:- #74

Key Levels:

Support levels:- $0.55564848, $0.62671578, $0.70644982
Resistance levels:- $0.66831615, $0.52964875, $0.46551435

ENJ/USD Price Prediction for 14TH August: Enjin Made a 1.67% Rise in Value

ENJ/USD Price Prediction: What the Indicators Are Saying

The signal line of the Relative Strength Index is almost moving horizontally, while the RSI curve moves in a very erratic manner. The signal line, being a slow indicator, could not catch up in response to the movement of the RSI curve. The RSI line at the time of writing this analysis Holds continuously at 57% on the 4-hour chart. But the RSI indicator on the daily chart is fairly on the rise. And it is holding at 62.9%.

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ENJ/USD Price Prediction for 14TH August: Enjin Made a 1.67% Rise in Value

Price Prediction For ENJIN:

Going by the steady bullish trend of the market, the price may not reach $0.800000 as quickly as we may expect it to rise. I think ENJUSD’s next move may reach $0.71604474.

Related:

Quantifying the success probability of brute force attack against (search) LPN

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