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Let n,m, p,q,d,h be positive integers with n € {1,d} . The hard problem underlying the security of our schemes is decision-GLWR with sparse-trinary secrets (see Section 2.3). For the above parameters, we define the GLWR oracle Om,xs,s for a secret distribution yg that returns m GLWR samples as follows:

Table 12 Let n,m, p,q,d,h be positive integers with n € {1,d} . The hard problem underlying the security of our schemes is decision-GLWR with sparse-trinary secrets (see Section 2.3). For the above parameters, we define the GLWR oracle Om,xs,s for a secret distribution yg that returns m GLWR samples as follows:

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Figure 1: Submission overview
Figure 1: Submission overview
2.4.3 Submission proposal: Round2.KEM This section describes Round2.KEM, an IND-CPA-secure key encapsulation method. It builds on CPA-PKE (Section 2.4.2). In addition to the parameters and functions from CPA-PKE, it uses a system parameter \, and an injection bin, : Rilnxm x Zi — {0,1}, and a hash function H : {0,1}* > {0,1}#8. The system parameter A satisfies
2.4.3 Submission proposal: Round2.KEM This section describes Round2.KEM, an IND-CPA-secure key encapsulation method. It builds on CPA-PKE (Section 2.4.2). In addition to the parameters and functions from CPA-PKE, it uses a system parameter \, and an injection bin, : Rilnxm x Zi — {0,1}, and a hash function H : {0,1}* > {0,1}#8. The system parameter A satisfies
The mapping bin, concatenates a vector consisting of the binary representation of the coefficients of the polynomial entries d/n x ™ matrix, and the binary representations of the entries of a vector in Z/'. Round2.KEM also uses the bijection ¢: ZS, — {0, 1}+® that replaces each of the pu entries of its input by the B-bits binary representation of that entry.
The mapping bin, concatenates a vector consisting of the binary representation of the coefficients of the polynomial entries d/n x ™ matrix, and the binary representations of the entries of a vector in Z/'. Round2.KEM also uses the bijection ¢: ZS, — {0, 1}+® that replaces each of the pu entries of its input by the B-bits binary representation of that entry.
2.4.5 Submission proposal: Round2.PKE Round2.PKE is an IND-CCA public-key encryption scheme constructed by com- bining CCA-KEM (Section 2.4.4) and a data encapsulation mechanism (DEM).
2.4.5 Submission proposal: Round2.PKE Round2.PKE is an IND-CCA public-key encryption scheme constructed by com- bining CCA-KEM (Section 2.4.4) and a data encapsulation mechanism (DEM).
Algorithm 8: CCA-KEM.Keygen()
Algorithm 8: CCA-KEM.Keygen()
The DEM encapsulates a message M of length mlen bytes with the key K en- capsulated in the KEM, resulting in an encapsulated message c2 of length clen bytes. In the inverse operation, c2 is decapsulated to the original message M using K.
The DEM encapsulates a message M of length mlen bytes with the key K en- capsulated in the KEM, resulting in an encapsulated message c2 of length clen bytes. In the inverse operation, c2 is decapsulated to the original message M using K.
first main result of this section — a proof of IND-CPA security for Round2.KEM (Theorem 2.5.4.1), and a proof of IND-CCA security for Round2.PKE (Theo- rems 2.5.5.1 and 2.5.5.2), assuming the hardness of the above problems. Finally, Section 2.5.6 and Theorem 2.5.6.1 contain the second main result of this reduc- tion: a proof of the hardness for LWR,pt, the underlying problem of our schemes for the non-ring case (i.e., for m = 1) in the form of a polynomial-time reduc- tion to it from the Learning with Errors (LWE) problem with secrets uniformly chosen from Ze and errors drawn according to a Gaussian distribution. Figure 2: Summary of reductions involved in the security proofs for Round2.KEM and Round2.PKE. “SPT” refers to a variant of the problem in question where the secret is sparse-trinary, instead of uniform in Te.
first main result of this section — a proof of IND-CPA security for Round2.KEM (Theorem 2.5.4.1), and a proof of IND-CCA security for Round2.PKE (Theo- rems 2.5.5.1 and 2.5.5.2), assuming the hardness of the above problems. Finally, Section 2.5.6 and Theorem 2.5.6.1 contain the second main result of this reduc- tion: a proof of the hardness for LWR,pt, the underlying problem of our schemes for the non-ring case (i.e., for m = 1) in the form of a polynomial-time reduc- tion to it from the Learning with Errors (LWE) problem with secrets uniformly chosen from Ze and errors drawn according to a Gaussian distribution. Figure 2: Summary of reductions involved in the security proofs for Round2.KEM and Round2.PKE. “SPT” refers to a variant of the problem in question where the secret is sparse-trinary, instead of uniform in Te.
Table 2: IND-CPA game for KEM Definition 2.5.1.3 (IND-CPA Secure KEM). Let KEM=(Keygen, Encaps, Decaps) be a key encapsulation mechanism with key space K. Let X be a se- curity parameter. The IND-CPA game is defined in Table 2, and the IND-CPA advantage of an adversary A against KEM is defined as
Table 2: IND-CPA game for KEM Definition 2.5.1.3 (IND-CPA Secure KEM). Let KEM=(Keygen, Encaps, Decaps) be a key encapsulation mechanism with key space K. Let X be a se- curity parameter. The IND-CPA game is defined in Table 2, and the IND-CPA advantage of an adversary A against KEM is defined as
Table 3: IND-CPA games for CPA-PKE: Games Go and G In this equation, Adv! (B) is the advantage of B in distinguishing the output of the mapping fn from uniform, and Adu ni qa (2) is the advantage of adver- sary Z in distinguishing m GLWR samples (with sparse-trinary secrets) from uniform, with the GLWR problem defined for the parameters d,n, qi, q2. Finally, the adversary D distinguishes between
Table 3: IND-CPA games for CPA-PKE: Games Go and G In this equation, Adv! (B) is the advantage of B in distinguishing the output of the mapping fn from uniform, and Adu ni qa (2) is the advantage of adver- sary Z in distinguishing m GLWR samples (with sparse-trinary secrets) from uniform, with the GLWR problem defined for the parameters d,n, qi, q2. Finally, the adversary D distinguishes between
Table 4: IND-CPA games for CPA-PKE: Games G2 and G3 Table 5: IND-CPA games for CPA-PKE: Games G4 and G5
Table 4: IND-CPA games for CPA-PKE: Games G2 and G3 Table 5: IND-CPA games for CPA-PKE: Games G4 and G5
Game G2 is different from game G, only in the fact that (A, B) is a sample from the uniform distribution on Rey pode xRY. ,*” instead of from Og in sues TS Under the decision GLWR(ys) assumption, these two distributions are indis- tinguishable with a factor 7. Indeed, assume A can distinguish between G, and G2. We describe an algorithm C that distinguishes between samples from Od/n.ve.a.g and those from U(RYP*/) x U(RA/2*") using A.
Game G2 is different from game G, only in the fact that (A, B) is a sample from the uniform distribution on Rey pode xRY. ,*” instead of from Og in sues TS Under the decision GLWR(ys) assumption, these two distributions are indis- tinguishable with a factor 7. Indeed, assume A can distinguish between G, and G2. We describe an algorithm C that distinguishes between samples from Od/n.ve.a.g and those from U(RYP*/) x U(RA/2*") using A.
Table 6: IND-CPA games for Round2.KEM: Games Gp and Gy Similarly to the case for CPA-PKE, there is a distinguisher B between f,, and the uniform distribution
Table 6: IND-CPA games for Round2.KEM: Games Gp and Gy Similarly to the case for CPA-PKE, there is a distinguisher B between f,, and the uniform distribution
Table 7: IND-CPA games for Round2.KEM: Games G2 and G3 from game Gj; it is distributed uniformly if the input is from game G2. Algo- rithm C is thus a distinguisher between Og/n,y¢,7, and the uniform distribution with advantage equal to |Pr(S2) — Pr($1)|. Similar to the proof for IND-CPA security for CPA-PKE, it follows that there is an algorithm C’ such that
Table 7: IND-CPA games for Round2.KEM: Games G2 and G3 from game Gj; it is distributed uniformly if the input is from game G2. Algo- rithm C is thus a distinguisher between Og/n,y¢,7, and the uniform distribution with advantage equal to |Pr(S2) — Pr($1)|. Similar to the proof for IND-CPA security for CPA-PKE, it follows that there is an algorithm C’ such that
Table 8: IND-CPA games for Round2.KEM: Games Gy, Gs and Gg Games G', and Gs only differ in the generation of X. Consider the following algorithm F.
Table 8: IND-CPA games for Round2.KEM: Games Gy, Gs and Gg Games G', and Gs only differ in the generation of X. Consider the following algorithm F.
Figure 3: Summary of reductions used in Theorem 2.5.6.1. “SPT” refers to a variant of the problem in question where the secret is sparse-trinary, instead of uniform in Zi.
Figure 3: Summary of reductions used in Theorem 2.5.6.1. “SPT” refers to a variant of the problem in question where the secret is sparse-trinary, instead of uniform in Zi.
The first term in the cost above, i.e., (b- 2°) is the cost of running BKZ lattice reduction with block-size b, where c is the BKZ sieving exponent. Finally, note that the cost in Eq. 36 also accounts for the fact that the dual attack can be adapted against the sparse-ternary LWR problem by rescaling the dual lattice using an appropriate scaling factor w = o\/m/h [1] (so as to equalize the contributions of both parts w?s and v7 e in z), resulting in the following rescaled dual lattice: factor 6, using m samples is:
The first term in the cost above, i.e., (b- 2°) is the cost of running BKZ lattice reduction with block-size b, where c is the BKZ sieving exponent. Finally, note that the cost in Eq. 36 also accounts for the fact that the dual attack can be adapted against the sparse-ternary LWR problem by rescaling the dual lattice using an appropriate scaling factor w = o\/m/h [1] (so as to equalize the contributions of both parts w?s and v7 e in z), resulting in the following rescaled dual lattice: factor 6, using m samples is:
The term b-2° represents the cost of running BKZ lattice reduction with block- size b. The attack runs on a LWE problem of dimension d — k with m < d samples. Condition 40, which is essentially Condition 32, ensures that such an attack has chances of succeeding. Note that although the above applies to the primal attack, a similar analysis is possible for the dual attack, in which case Costrattice Reduction(d, k,h) is calculated as in Eq. 36, with the parameter d replaced by d—k.
The term b-2° represents the cost of running BKZ lattice reduction with block- size b. The attack runs on a LWE problem of dimension d — k with m < d samples. Condition 40, which is essentially Condition 32, ensures that such an attack has chances of succeeding. Note that although the above applies to the primal attack, a similar analysis is possible for the dual attack, in which case Costrattice Reduction(d, k,h) is calculated as in Eq. 36, with the parameter d replaced by d—k.
Table 10: uRound2.KEM,,—_ parameters
Table 10: uRound2.KEM,,—_ parameters
Table 11: nRound2.KEM,,—_ parameters
Table 11: nRound2.KEM,,—_ parameters
Table 15: uRound2.KEM,,—1, NIST level 5 — A generation variants Memory usage indication
Table 15: uRound2.KEM,,—1, NIST level 5 — A generation variants Memory usage indication
Table 16: uRound2.PKE,,—1 parameters
Table 16: uRound2.PKE,,—1 parameters
Table 17: uRound2.PKE,,—qa parameters
Table 17: uRound2.PKE,,—qa parameters
Table 18: nRound2.PKE,,—qa parameters
Table 18: nRound2.PKE,,—qa parameters
Table 19: uRound2.PKE,,—;,-=2 performance and memory usage
Table 19: uRound2.PKE,,—;,-=2 performance and memory usage
Table 20: uRound2.PKE,,—a performance and memory usage
Table 20: uRound2.PKE,,—a performance and memory usage
Table 22: uRound2.PKE,,—1, NIST level 5 — A generation variants Memory usage indication
Table 22: uRound2.PKE,,—1, NIST level 5 — A generation variants Memory usage indication
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