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I knew what NAF was in theory but writing the actual algorithm under pressure was a different story.
Explain the NAF algorithm: repeatedly divide n by 2, but if n is odd, choose the remainder (1 or -1) that makes the quotient even, ensuring no adjacent nonzeros. For fromNAF, compute the sum of digit * 2^i. Emphasize handling negatives and returning least-significant-first.
Pro tip: Mention that NAF minimizes the number of nonzero digits, which is useful in cryptography (e.g., elliptic curve scalar multiplication) for efficiency. Also, note that the algorithm works for negative n by using floor division and adjusting the remainder.
Confirm that digits are least-significant-first and that n can be negative. Discuss edge cases like n=0, n=1, and negative numbers.
Use a loop: while n != 0, if n is odd, compute r = 2 - (n mod 4) to get 1 or -1, then n = n - r; else r = 0. Append r to digits, then n = n // 2. This ensures no adjacent nonzeros.
Iterate over digits with index i, compute sum += digit * (2**i). Return the sum.
Test with positive, negative, and zero values. Verify that fromNAF(toNAF(n)) == n and that no two adjacent digits are both nonzero.
Mention time complexity O(log n) and space O(log n). Compare with binary representation and note NAF's minimal weight property.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by defining NAF and the signed-binary representation, then prove correctness by showing the algorithm produces a valid representation with no two adjacent nonzeros. To prove minimality, use a weight argument: assign each nonzero digit a weight based on its position and show any representation with fewer nonzeros would violate the no-adjacent-nonzeros property or the value constraint.
Pro tip: Emphasize that NAF is unique and that the no-adjacent-nonzeros property is key to minimality; mention that this is why NAF is used in elliptic curve cryptography for efficient scalar multiplication.
Clearly state what a signed-binary representation is (digits in {-1,0,1}) and define NAF as the unique representation with no two adjacent nonzeros. Mention that NAF is computed by repeatedly dividing by 2 and adjusting remainders to -1, 0, or 1.
Show that the algorithm terminates and produces a valid representation equal to the original integer. Argue that the no-adjacent-nonzeros property holds by construction: when a remainder of 1 is chosen, the next quotient becomes even, ensuring the next digit is zero.
Assign a weight to each nonzero digit: for a digit at position i, weight = 2^i. Show that any signed-binary representation with fewer nonzeros would require a nonzero at a position where NAF has a zero, leading to a contradiction with the no-adjacent-nonzeros property or the value constraint.
Summarize that NAF is the unique representation with no adjacent nonzeros, and that this property directly implies it has the minimal number of nonzeros among all signed-binary representations.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, define what NAF (Non-Adjacent Form) is and how your toNAF and fromNAF functions work, then analyze their time and space complexity in terms of the number of bits in the input. Finally, discuss edge cases such as n = 0, negative numbers, and very large magnitudes, explaining how your implementation handles them and any potential pitfalls.
Pro tip: Mention that NAF is unique and has minimal weight, which is why it's used in cryptography; also note that the average density of non-zero digits is 1/3, which can affect performance in practice.
Briefly explain that NAF is a signed binary representation with no two adjacent non-zero digits, and describe your toNAF and fromNAF algorithms at a high level.
State that both toNAF and fromNAF run in O(k) time, where k is the number of bits in the input, because they process each bit once.
Explain that the space complexity is O(k) for the output array (or O(1) extra space if modifying in place), and note that the output length is at most k+1 digits.
Cover n = 0 (should return an empty array or a single zero digit), negative numbers (NAF can represent them, but clarify if your implementation handles sign separately), and very large magnitudes (e.g., arbitrary precision integers, where k can be huge, but complexity remains linear).
Summarize that the algorithms are efficient and suitable for cryptographic applications, and mention any optimizations or alternative representations if relevant.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Explain that NAF (Non-Adjacent Form) represents k with digits in {-1, 0, 1} and no two consecutive non-zero digits, minimizing the number of non-zero digits. Then describe how to compute n*k by iterating through the NAF digits from most significant to least, doubling the accumulator and adding or subtracting n based on the digit, which reduces the number of additions/subtractions compared to binary.
Pro tip: Mention that NAF is especially beneficial in elliptic curve cryptography for scalar multiplication, and that the average density of non-zero digits is 1/3, leading to about 1/3 fewer additions than binary. Also note that the trade-off is the need to precompute the NAF representation, which is cheap.
State that NAF is a signed binary representation with digits in {-1, 0, 1} and no two consecutive non-zero digits, and that it has minimal weight among all signed binary representations.
Explain the algorithm to compute NAF: repeatedly divide k by 2, and if k is odd, choose the remainder as 1 or -1 to make the quotient even, ensuring no adjacent non-zero digits.
Iterate through the NAF digits from most significant to least: for each digit, double the accumulator; if digit is 1, add n; if digit is -1, subtract n; if 0, do nothing.
Compare the number of additions/subtractions: binary uses about (bit length)/2 additions on average, while NAF uses about (bit length)/3, reducing operations by roughly 1/6 of the bit length.
Mention that NAF requires precomputing the representation and handling negative digits, but the reduction in additions is beneficial for performance-critical applications like cryptography.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.