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PROPRIETARY AND CONFIDENTIAL Choosing NTRUEncrypt Parameters William Whyte NTRU Cryptosystems March 2004

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Agenda Parameter Generation –How to pick parameters to obtain a given security level? We present a recipe for parameter generation Will use this recipe, or simply the constraints that come out of it? –Multiple parameter forms Standard form, product form –Possible bandwidth savings – NTRU-KEM Key validation But first…

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Review: NTRU parameters N, dimension of polynomial ring –NTRU works on polynomials of degree N-1 –Polynomial multiplication is convolution multiplication: terms of degree > N are reduced mod N. –For 80-bit security, N = 251. Increases roughly linearly with k for k-bit security q, “big” modulus –All coefficients in polynomial are reduced mod q –For 80-bit security, q = 239. Increases roughly linearly with k for k-bit security p, “small” modulus –Reduce mod p during decryption –p = 2, 2+X or 3 for all security levels. Sizes: –Public key, ciphertext size = N log 2 q = 2004 bits for 80-bit security –message size (bits) = N log 2 ||p|| = 251 bits for 80-bit security

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Review: NTRUEncrypt Operations Key Generation –Generate f, g, “small” polynomials in Z q [X]/(X N -1). –Public key h = p*f -1 *g mod q; private key = (f, f p = f -1 mod p). Encrypt (Raw operation) –Encode message as “small” polynomial m. –Generate “small” random polynomial r –Ciphertext e = r*h + m mod q. Decrypt (Raw operation) –Set a = f*e mod q. “mod q” = in range [A, A+q-1]. –Set m = f p * a mod p.

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Review: Why Decryption Works a= f * e(mod q) = f * (r*h + m)(mod q) = f * (r*p*g*F q + m)(mod q) = p*r*g + f*m(mod q) since f*F q = 1 (mod q) All of the polynomials r, g, f, m are small, so coefficients of p*r*g + f*m will (usually) all lie within q of each other. If its coefficients are reduced into the right range, the polynomial a(x) is exactly equal to p*r*g + f*m. Then f p * a = p*r*g*f p (mod p) + f p *f*m (mod p) = m (mod p). Current parameter sets for 2 80 security include means for choosing this range. Choice of range fails on validly encrypted message one time in –“Decryption failures” –Attatcker gains information from decryption failures: wants to choose funky r, m.

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © HashXOR r*h + m’ e Hash r Review: SVES-3 encryption mb m’ r*h mLen00… ID

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Parameter Generation Input: k, the desired level of security Process: –Choose N Set N to give necessary bandwidth –Choose form of f, g, r Ensure combinatorial security –Choose q, p Set q to prevent decryption failures –Ensure that these parameters give appropriate lattice security There are many different ways of making these choices. –These are the proposed ones for X9.98 Note: extremely provisional and may change as the analysis proceeds –Currently writing up a paper to formalize them

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Choose N –With binary messages, N is the number of bits that can be transported –For k bits of security for key transport, want to transport 2k bits of material Prevent birthday-like attacks based on future use of material –For SVES-3, want to use at least k bits of random padding Gives security against enumeration attacks if encryption scheme is used to transport low-entropy messages –Set N to be the first prime greater than 3k.

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Choose form of f, g, r Our choice: –Take f = 1+pF Speeds up decryption: f -1 mod p= 1, so we eliminate a convolution –Take F, g, r to be binary with df, dg, dr 1s respectively. Number of additions necessary for convolution is df*N. Alternatives: –Take f not to be of form 1+pF Slows down decryption but reduces q (see next choice) –Take f (or F), g, r to be of the form (e.g.) f = f 1 * f 2 + f 3. “Product form”: Number of additions necessary for convolution is (f 1 + f 2 + f 3 )*N. –Performance benefit

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © F, g, r have df, dg, dr 1s respectively Brute force-like search on F, g, r can be speeded up by meet- in-the-middle techniques. Using these techniques, number of binary convolution multiplications needed to break f is –Each multiplication requires df.N additions … perhaps divided by 2-8 if we use wordsize cleverly In general, use number of multiplications as security measure Attacker will go for easiest of (f, g), (r, m); pick df = dr = dg. Binary F, g, r: Combinatorial Security

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © df, dg, dr for different security levels N and df, using the above criteria: df ~= N. kNdf

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Pick q, p Our choice: –Pick p = 2, q to be the first prime greater than p.min(dr, dg) p.min(df, N/2) with large order mod N. This gives zero chance of decryption failures Minimum q to do so consistent with choice of p, df. –Best lattice security Alternatives: –Take p = 2+X or 3, q = first power of 2 greater than p(1).min(dr, dg) p(1).min(df, N/2) Taking q to be power of 2 speeds up reductions Larger value of p leads to larger q and worse lattice security –Take p = 2, q = largest prime less than first power of 2 greater than p(1).min(dr, dg) p(1).min(df, N/2) Speeds up reductions at expense of lattice security –Allow a non-zero chance of decryption failures, if it can be determined to be less than 2 -k. Reduces q, improves bandwidth and lattice security

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © df, q for different security levels N, df, q, using the above criteria: kNdfq

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Check Lattice Strength We characterize the lattice by two variables: –c = (2N). (2)||f||/ . = 2||f|| ( e / q) Length of shortest vector [ (2)||f|| ]… Divided by expected length of shortest vector for lattice of the same determinant [ = (N q/ e) ]… Scaled by (2N). –a = N/q. Experimentally, breaking time is very sensitive to c, somewhat sensitive to a. Experimentally, for fixed c, a, breaking time is exponential in N. For all the parameter sets given in the previous slide, we have a >= 1.25, c >= 2.58.

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Lattice Strength Based on the above experiments: Neglecting zero-forcing; also neglecting fact that the lattices under consideration are stronger than the ones experimented on. kNdfqcLattice bitstrength

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © More Notes on Parameter Generation Note how df affects lattice strength: –For these parameters, q ~= 2p df, ||f|| ~= df, c ~= ||f||/ q c is ~independent of df! More precisely: ||f|| >= df/2). Run expts for c = ( e / p) = 2.066? Rounding q up to next prime reduces min(c) slightly, not much. If we use the number of additions, not multiplications, as measure of combinatorial security, we can reduce df by typically 10-25% –Gain decreases as N increases –Reducing df reduces q potentially improves bandwidth Using product form (f = f 1 * f 2 + f 3 ) improves efficiency but increases q –Increased bandwidth, but typically only by one bit Using trinary (f, g) gives greater combinatorial security So many appealing choices…

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Product Form Interested in meet-in-the-middle attacks on product form –df1 = df2 = df3 Standard search, described in Tech Note 4, takes on (f1*f2) and (f3) –Could also remove a 1 from f2, add df 1s to f3 df =~ N for N = 251, N for N = 769

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Product Form Speedups, Keygen, Lattice Consideration Speed up typically factor of 2 over standard form Keygen will get slow for larger N -- haven’t done calculations For even larger N, guessing zeroes becomes better than guessing f Increased df doesn’t affect lattice strength (much) –c = 2.05 experiments would still be fine Bandwidth increases only for k=160 and 192, by 1 bit per coeff kNstandard df product df speedup factor effective df qc = =

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Hash2XOR r*h + m’ e Hash1 r NTRU-KEM? m’ r*h ID b Hash3 K

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Parameters for NTRU-KEM Still taking p = 2. Now, only have to transmit about 2k bits, so can save bandwidth For k-bit security: –Pick N = 2k –Pick df = N/2 –Increase N until combinatorial security is > 2 k. –Take df, dr, dg, to be the same –Take f=1+pF –Set q as before

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Parameter Sets This gives the following (note: c >= 2.05, so we omit it). In some cases, slightly increasing N decreases log 2 (q); we’ve done this where it helps. Note: q needs rounding up. kNdfqNumber of adds SVES-3 adds bwdthSVES-3 bwdth RSA bwdth ~

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Speeding up? The parameters above successfully reduce bandwidth Can we improve speeds? Taking small polynomials to be {-1, 0, 1} improves combinatorial security –Taking them to be {-2, -1, 0, 1, 2} would do even better, but… –The wider the polynomials are, the wider 2 their products are

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Trinary polynomials Take p to be 3. f, g, r, m could be trinary Two different forms: –Balanced: Equal +1s and –1s –Biased: Minimum possible number of –1s Set N/2 1s, N/2 0s If this doesn’t give enough combinatorial security, set some of the 0s to –1s. Once there is adequate combinatorial security, see if we can reduce the number of 1s End with dg+ 1s, dg- -1s Combinatorial security estimated as sqrt ((N pick dg+)(N pick dg-)) / N –This needs to be made more precise

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Polynomial width Consider a*b: –a has da+ +1s, da- -1s –b has db+ +1s, db- -1s –Maximum value if all +1s line up, all –1s line up. –Minimum value if all +1s line up with –1s. –Maximum width is Min(da+, db+) + Min(da-, db-) + Min(da+, db-) + Min(da-, db+) Advantage of having one balanced, one biased is we reduce this width compared to two balanced or two biased. Take f, r to be balanced trinary –Gives lowest Hamming weight Take g to be biased trinary Consider m on next slide

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Choosing N and encoding m Say we choose N to be ~2k –Then biased polynomials have very few –1s. Want to transmit k bits of entropy –Attacker can meet-in-the-middle on m’ Could draw m’ from a space of 3 N polynomials –But this might be tiresome –Open question: exactly how tiresome? Certainly tiresome in that output of Hash2 needs to be encoded as random trinary vector, involving repeated mod 3 divides of big integer Suggestion: Take b, output of Hash2, m’ to be binary –Once m’ is generated, flip some terms (only 1s or only 0s) to –1s to obtain combinatorial security –If more than (say) 4 need to be flipped, generate another b.

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Recipe Choose N to be first prime > 2k Choose F, r to be balanced trinary –(Actually, choose dr+ = dr- +1 for invertibility) Choose g to be biased trinary Choose f = 1+pF –f(1) should not be 0 mod 2 Say m’ will have no more than 4 –1s Maximum width is df + dr + dg- + 4 Set q = the first power of 2 greater than this width

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Parameter Sets This gives the following (note: c >= 2.11, so we omit it) df ~= N, compared to for SVES-3; time = N 2. kNdfdg+dg-qaddsSVES3 adds bwdthSVES3 bwdth RSA bwdth ~

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Notes: We gain a speedup of a factor of about 2 over standard SVES-3 –Comparable to (though slightly worse than) speedup from move to product form –Could consider product form here too, but there seem to be few advantages Bandwidth is about 0.65 of SVES-3 bandwidth –Bandwidth is between 16 k and 18.3 k for security k –Goes slightly worse than k, slightly better than k ln k. Lattice strength is a BIG question here –Not clear that you can get 80 bits of lattice strength at N=163 –Equally, not 100% clear that you can’t…. –Can increase lattice strength by beefing up dg+, but this only goes so far Requires an additional SHA at the end –But on fewer SHA compression blocks –End up with about 2*0.65 = 1.3 as many SHA calls

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Parameter Generation: Summary Outlined a possible parameter generation routine for NTRU –Put in k, turn the handle, out come the parameters –Parameters can be validated by third parties Specific parameter generation routine may change, but basic method remains the same: –Choose N –Choose form of f, g, r, m –Choose q –Check lattice strength; if too low, increase N to next prime and try again.

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Open Questions How many parameter sets do we want? –Optimize speed –Optimize bandwidth –Optimize for 8-bit processors? Can be done by increasing N decreasing q < 256 (or 128) Do we ever want to allow decryption failures for k > 80?

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Open Research Questions Is it okay to use SHA-160 as the core hash function for k > 80? –I think yes, but this needs discussion

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Key Validation What can go wrong with an NTRUEncrypt public key h? Should be random mod q. –Might be all zeroes Reveals message immediately –If q is composite and gcd(h i, q) != 1 for all h i, might be possible to recover message from ciphertext by simple modular reduction. –If h is too thin, such that r*h will have very few mod q reductions (< 2 k effort to guess reduction locations), can recover message from ciphertext by linear algebra. Possible simple key validation procedure: –Check that keys are not all the same value –Check that sufficient number of h i have gcd(h i, q) = 1 –Check that width of h > c. q/df, c > 1 some parameter set dependent constant.

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Key Validation (2) More sophisticated: –Measure how “random” h looks. For example: Chance that a given mod q value does not occur anywhere in h = (1-1/q) N Find value l such that for random h the probability that l distinct values do not occur anywhere in h is less than 2 -k. –A different bound may be appropriate Count the number of distinct values that do not occur in h and reject if greater than l. Next draft will contain suggested text.

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Issue: Forward Secrecy (1) NTRU key generation is efficient –Can generate ephemeral keypairs easily This + next slide propose three ways of getting perfect forward secrecy using this fact –Do these actually give forward secrecy? –Do they give mutual authentication? –Should they be included in the standard? (1) Say Alice has static keypairs (a s, A s ). –Bob generates ephemeral keypair (b e, B e ), sends Alice E As (B e ). This may have to be signed –Alice uses B e as the public key for key transport or key agreement –Afterwards, Bob disposes of (b e, B e ).

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © Issue: Forward Secrecy (2) (2) Say Alice and Bob have static, certified keypairs (a s, A s ), (b s, B s ). –Bob generates ephemeral keypair (b e, B e ), sends Alice E As (B e ). –Alice uses both B s and B e in two runs of a key transport or key agreement mechanism, combines the two transported keys to get a single shared key. –Afterwards, Bob disposes of (b e, B e ). (3) Say Alice and Bob have static, certified keypairs (a s, A s ), (b s, B s ). –Bob generates ephemeral keypair (b e, B e ), sends Alice E As (B e ). –Alice generates ephemeral keypair (a e, A e ), sends Bob E Bs (A e ). –Alice uses B e to transport secret k 1 to Bob –Bob uses A e to transport secret k 2 to Alice –Bob and Alice combine k 1, k 2 to get shared secret k. Note: need to define encryption carefully above: will probably be symmetric+public-key operation

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PROPRIETARY AND CONFIDENTIALNTRU CRYPTOSYSTEMS, INC. COPYRIGHT © That’s it! Questions?

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